Transfer number

As transference number or Hittorf transference number or even (according to Johann Wilhelm Hittorf ) the fraction of the total is the electric current refers to the particular of an ion species in a solution with at least one other type of ion (cation and anion of a binary electrolyte) is transported: ${\displaystyle t_{i}}$ ${\displaystyle n_{i}}$ ${\displaystyle I}$ ${\displaystyle i}$

${\displaystyle t_{i}=n_{i}={\frac {I_{i}}{I_{\text{ges}}}}}$

The transfer number depends on the ion concentration (alternatively: on the molality ) and the ion mobility - which is not a speed - or on the migration speeds or the equivalent conductivities: ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle \lambda }$

• If the concentration of the ion species is high, a large proportion of the electrical current can be transported by these ions. ${\displaystyle i}$
• Ion mobility and migration speed: Faster ions (larger and larger ) are able to transport a larger part of the current than slow ones. ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle v}$
• Hydronium ions and hydroxide ions can transport much more current than other ions because they use a special charge exchange mechanism ("extra conductivity"). In real terms, they migrate much more slowly than theoretically calculated, and largely only pass their charges on to neighboring ions of the same solvent water. The maximum value of this extra conductivity is reached at around 150 ° C in water.

The English term for the transfer number is "transference number" or "ion transport number".

The product of the conversion number of an ion - at the respective concentration and temperature - (for a salt) with the equivalent conductivity of this salt is the equivalent conductivity of the corresponding ion (at the respective concentration and temperature). The limit conductivity of a salt and the transfer numbers of cation and anion result in the limit conductivities of the cations and anions of this salt:

${\displaystyle \lambda _{\text{Ion des Salzes}}^{\text{(c,T)}}=\Lambda _{\text{Salz}}^{\text{(c,T)}}\cdot n_{\text{Ion des Salzes}}^{\text{(c,T)}}=(\lambda _{\text{Kation des Salzes}}^{\text{(c,T)}}+\lambda _{\text{Anion des Salzes}}^{\text{(c,T)}})\cdot n_{\text{Ion des Salzes}}^{\text{(c,T)}}}$

This equation also applies to the molar conductivity instead of the equivalent conductivity .

Info:

• In the past, the term “transfer” or today “migration” refers to the migration of charged particles (ions or colloid particles) under the influence of an electric field (field gradient).
• " Cells / elements with transfer" are galvanic elements (or concentration elements) whose different electrolyte solutions (or the same electrolyte solutions of different concentrations for concentration elements ) are separated from one another by a diaphragm . Here, the cation and anion migrate from the more concentrated solution through the diaphragm into the more dilute solution due to the diffusion caused by the concentration . Diffusion potentials occur when the cation and anion have different ion mobilities and transfer numbers. The directly measurable cell voltage (potential difference) contains the diffusion voltages that occur (up to 30 mV are practically possible).
• "Cells / elements without overpass" are elements in which the two half-cells are coupled to one another by means of a power key. By using the electric key, there are almost no diffusion voltages. The directly measurable cell voltage (potential difference) contains practically no diffusion voltages and therefore corresponds to the difference between the two redox potentials according to the Nernst equation for redox reactions.

Application reference

In the case of salt bridges , care is taken to ensure that the transfer rates of cations and anions are approximately the same:

${\displaystyle t_{\text{kat}}\approx t_{\text{an}}}$

As a result, a salt bridge made of KCl consists of two different ions that have approximately the same ion mobility . ${\displaystyle v}$

If the specific electrolyte conductivity of a binary electrolyte has been measured, the equivalent conductivities of anion and cation can only be determined from this if, in addition to the molar concentrations of both ions and their valencies, at least one conversion number is known or has been measured.

If there are different ions in a multi- ion mixture , transfer numbers must be known or measured in order to determine the equivalent conductivities of all ions from the measured specific electrolytic conductivity of the solution. ${\displaystyle x}$ ${\displaystyle (x-1)}$

The equivalent conductivity of an ion type i of a salt (at the respective concentration and temperature) is the product of the conversion number of this ion i (at the respective concentration and temperature) and the equivalent conductivity of the salt (at the respective concentration and temperature): ${\displaystyle \lambda _{i}}$ ${\displaystyle n_{i}}$ ${\displaystyle \Lambda _{\text{Salz}}}$

${\displaystyle \lambda _{i}{\text{(c,T)}}=\Lambda _{\text{Salz}}{\text{(c,T)}}\cdot n_{i}{\text{(c,T)}}}$

This applies equally to cations and anions.

Example for application reference

• Potassium chloride solution of c = 1 [mol / liter] should have an equivalent conductivity of 98.3 [Scm ² / mol] at 19 ° C.
• This equivalent concentration ([1 mol / liter]) also corresponds to 0.001 [mol / cm ³ ].
• At 20 ° C, 1 molar KCL solution should have a specific conductivity of [S / cm]. ${\displaystyle \kappa =0.10202}$

The quotient of specific conductivity and equivalent concentration in [mol / cubic centimeter] is the equivalent conductivity of the salt: ${\displaystyle \kappa }$ ${\displaystyle c_{ev,*}}$

${\displaystyle \Lambda _{Kaliumchlorid}=\kappa _{Kaliumchlorid}/c_{ev,*}=0.10202/0.001=102.02}$

[Scm ² / mol] for a KCl solution of 1 [mol / liter] at 20 ° C. is the molar equivalent concentration per cubic centimeter. ${\displaystyle c_{ev,*}}$

• Since the conversion number for 1 molar potassium chloride solution could not be found in tables, it is assumed that a 0.8 molar potassium iodide solution has practically identical conversion numbers. Namely at 20 ° C for the cation (potassium) . And accordingly (iodide or assumed as chloride). ${\displaystyle n_{+}=0.4896}$ ${\displaystyle n_{-}=0.5104}$

Thus, 48.96 percent of the equivalent conductivity of potassium chloride is attributable to the potassium ion and 51.04% to the chloride ion. This only applies at 20 ° C and a concentration of approx. 1 [mol / liter] (or 0.8 mol / liter).

• The equivalent conductivities for the ions under these conditions are therefore:

${\displaystyle \lambda _{Kaliumion}=\Lambda _{Kaliumchlorid}\cdot n_{\text{Kaliumion}}=102.02\cdot 0.4896=49.95}$[Scm ² / mol]

and

${\displaystyle \lambda _{Chloridion}=\Lambda _{Kaliumchlorid}\cdot n_{\text{Chloridion}}=102.02\cdot 0.5104=52.07}$[Scm ² / mol]

• For ideal dilution (c = 0) and 18 ° C, Huebschmann gives the following values ​​of the limit conductivity : potassium: 65 and chloride: 66 [Scm ² / mol].
• If you extrapolate these values ​​with the known temperature coefficients (1.87% / K for potassium and 2.25% / K for chloride) to 20 ° C, you get 67.4 for potassium ions and 69.0 for chloride ions [Scm ² / mol] equivalent limiting conductivity (for c = 0).
• In another table from Huebschmann, the limit equivalent conductivity for potassium chloride solution is given as 130.0 for 19 ° C.

As you can see, the equivalent conductivity is significantly lower at higher concentrations than at ideal dilution. This is expressed in the so-called conductivity coefficient (quotient of actually measured conductivity and theoretical conductivity at ideal dilution). The following values ​​can now be calculated: ${\displaystyle f_{\lambda }}$

${\displaystyle f_{\lambda _{\text{KCl}}}=98{,}3/130{,}0=0{,}756}$ (130.0 at 19 ° C instead of 20 ° C!)

${\displaystyle f_{\lambda _{\text{Kaliumion}}}=49{,}95/67{,}4=0{,}741}$

${\displaystyle f_{\lambda _{\text{Chloridion}}}=52{,}07/69{,}0=0{,}755}$

These values ​​are only valid for a concentration of c = 1 [mol / l] at 20 ° C in potassium chloride solution!

For the entire salt, but also for the individual types of ions, the conductivity coefficient is around 0.75. About 75 percent of the ions in the solution effectively only contribute to conductivity. Because of the high concentration, many ions interfere with each other (friction and interactions!). If you multiply the conductivity coefficient by the limit conductivity, you get the lower equivalent conductivity for the higher concentration: ${\displaystyle f_{\lambda }}$

${\displaystyle \Lambda _{\text{Salz}}{\text{(c,T)}}=f_{\Lambda _{\text{Salz}}}{\text{(c,T)}}\cdot \Lambda _{\text{Salz}}^{0}{\text{(c=0,T)}}}$

${\displaystyle \lambda _{Ionx}{\text{(c,T)}}=f_{\Lambda _{\text{Ion x in diesem Salz}}}{\text{(c,T)}}\cdot n_{\text{Ion x in diesem Salz}}{\text{(c,T)}}\cdot \Lambda _{\text{Salz}}^{0}{\text{(c=0,T)}}\approx f_{\Lambda _{\text{Salz}}}{\text{(c,T)}}\cdot n_{\text{Ion x in diesem Salz}}{\text{(c,T)}}\cdot \Lambda _{Salz}^{0}{\text{(c=0,T)}}=\Lambda _{\text{Salz}}{\text{(c,T)}}/\Lambda _{\text{Salz}}^{0}{\text{(c=0,T)}}\cdot n_{\text{Ion x in diesem Salz}}{\text{(c,T)}}\cdot \Lambda _{\text{Salz}}^{0}{\text{(c=0,T)}}}$

Conversion numbers and conductivity coefficients are functions of concentration and temperature. For more concentrated solutions, they can only be determined by measurements (Hittorf test and measurement of the specific conductivity).

The ideal electrolyte

An ideal electrolyte has the conversion numbers 0.5 for cation and anion. Both ions contribute 50% to the flowing electrolysis current. Therefore, there is no compensatory migration (diffusion caused by concentration differences in the cathode compartment and anode compartment) of the slower migrating ion species from one electrode compartment to the other, since in the case of n (K +) = n (A -) = 0.5, both ion species (cation and anion ) hike quickly. In this case, with the Hittorf method and also in a power key , no concentration gradients would arise in the apparatus that would result in “concentration diffusions” of the ions. According to Milazzo (Elektrochemie, p. 30), the salts RbCl, RbBr, RbJ, CsCl, CsBr, CsJ have anion conversion numbers between 0.496 and 0.503 at a concentration of 0.02 mol / liter and a temperature of 18 ° C. Potassium chloride has a cation transfer number of 0.4847 at ideal dilution ( ) at 25 ° C. In addition to potassium chloride , the salts potassium nitrate , ammonium chloride and ammonium nitrate are also used . used as saturated solutions in an electricity key. In addition, in salts with conversion numbers , according to Henderson's equation (see below), there is no diffusion voltage for 1-1-valent electrolytes when the ions of this salt diffuse through a phase boundary from higher to lower concentration. Therefore, electricity keys are filled with appropriate saturated salt solutions. In the ideal case, just as many positive charges should pass the phase boundary per second as negative ones (ensured electrical neutrality!). ${\displaystyle c=0}$ ${\displaystyle n_{+}=n_{-}=0,5}$

Measurement of the number of deliveries

The Hittorf method

An electrolysis apparatus, which is composed of the cathode compartment, the central compartment and the anode compartment, is suitable for determining the transfer rate . Platinum electrodes are immersed in the cathode and anode compartment . The electrolysis rooms are filled with the electrolyte and connected with a bridge. If you look at the two electrode spaces, you will see that the concentrations of the cations and the anions change differently.

The transfer figures are calculated:

for the cations: ${\displaystyle t_{+}={\frac {q_{A+}-q_{E+}}{q_{\text{ges}}}}}$

and

for the anions: ${\displaystyle t_{-}={\frac {q_{A-}-q_{E-}}{q_{\text{ges}}}}}$

With

${\displaystyle q_{A}}$the amount of charge initially present in the corresponding space

${\displaystyle q_{E}}$ the amount of charge present at the end in the corresponding space

${\displaystyle q_{\text{ges}}}$ the amount of charge that has flowed through the electrolyte.

With the definition of the amount of charge:

${\displaystyle q=z\cdot F\cdot c\cdot V}$

With

${\displaystyle z}$= Number of charges

${\displaystyle F}$= Faraday constant

${\displaystyle c}$ = Concentration of the considered ion

${\displaystyle V}$= Volume of the electrolysis room

it follows from this:

${\displaystyle \Rightarrow t_{+}={\frac {z\cdot F\cdot V\left(c_{E+}-c_{A+}\right)}{q_{\text{ges}}}}+1}$

and

${\displaystyle \Rightarrow t_{-}={\frac {z\cdot F\cdot V\left(c_{A+}-c_{E+}\right)}{q_{\text{ges}}}}}$

According to this equation, the transfer number can be determined by determining the concentration, current and volume of an electrode chamber.

It is: ${\displaystyle t_{+}+t_{-}=1}$

During electrolysis, some ions migrate very quickly (e.g. H ⁺ , OH ^- ), while others migrate very slowly (Li ⁺ , CH ₃ COO ^- ). From knowledge of the molar limit conductivities of ions, these migration speeds and thus the transfer numbers of the ions in electrolysis can be determined: ${\displaystyle \Lambda _{0}}$

${\displaystyle t_{+}=\nu ^{+}\cdot {\frac {\Lambda _{0}^{+}}{\Lambda _{0}}}}$

and

${\displaystyle t_{-}=\nu ^{-}\cdot {\frac {\Lambda _{0}^{-}}{\Lambda _{0}}}}$

each with the stoichiometric number ${\displaystyle \nu .}$

A detailed description of the Hittorf method can be found in “Elektrochemie” (Giulio Milazzo, 1952) on pages 21-26.

The Hittorf apparatus and the processes involved in the experiment

The apparatus consists of three volumes - ideally the same size - the cathode compartment (KR), the central compartment (MR) and the anode compartment (AR). The connected "tubes" can be blocked from each other by taps. Ideally there are drain taps on each of the rooms. Ideally, the middle space also has an opening for immersing a conductivity probe. During the experiment the taps are open and at the beginning the equivalent concentrations of cations and anions are the same in all rooms . Electrolysis is now carried out with the lowest possible current and constant temperature. The following picture always emerges (with all possible electrolytes):

• In the cathode compartment, more cations are discharged at the cathode per second (cation removal from the cathode compartment) than in the same time from the anode compartment - due to the electrical field (voltage per electrode spacing) - migrate through the middle compartment into the cathode compartment at the same speed (cation supply to the cathode compartment ) ). The concentration of the cations in the cathode compartment therefore falls steadily. ${\displaystyle E}$ ${\displaystyle u(\mathrm {Kat} )}$

• In the anode compartment, more anions are discharged per second at the anode (anion extraction from the anode compartment) than they migrate from the cathode compartment through the central compartment into the anode compartment at the same time - due to the electric field (voltage per electrode spacing) - through the middle compartment into the anode compartment (anion supply to the anode compartment) ). The concentration of the anions in the anode compartment therefore decreases steadily. ${\displaystyle E}$ ${\displaystyle u(An)}$

If both ions had the same ion mobility (migration speeds at the respective field strength), this would already describe the entire effect. In this case, the equivalent concentrations of cations in the cathode compartment and anions in the anode compartment drop at the same rate. In the case of different ion mobilities (real case) it follows:

• The equivalent concentrations of cations in the cathode compartment and anions in the anode compartment decrease at different speeds or even increase due to the displacement of the fast ion in the associated electrode compartment. see also Walther Nernst # Elektrochemie
• As a result, differences arise in the equivalent concentrations of positive and negative ions in the volumes of the cathode compartment and the anode compartment . Since every salt solution must be electroneutral, there is now a compensating diffusion of the slow ion against the field direction through the central space to the "wrong" electrode space.

The concentration-related diffusion of the slow ions against the field direction immediately compensates for the charge difference (difference in the equivalent concentrations of cations and anions).

The experiment should be carried out long enough to achieve differences in the equivalent concentration of cations (or anions) that can be easily analyzed. It must be terminated at the latest when the concentrations in the central area begin to decrease. This can occasionally be checked using a conductivity probe (isoterm) in the middle room (to do this, switch off the electrolysis current for a short time). At the end of the experiment, the taps between the cathode compartment / middle compartment / anode compartment are closed, the samples taken from the cathode compartment and anode compartment and either the concentration of the cations or that of the anions is determined. It is the equivalent - molarity , or the equivalent molality used for calculation.

By finally combining the sample volumes taken from KR and AR, the mean final concentration (for the entire apparatus) can also be determined directly. The mean final concentration must always be lower than the initial concentration, since precious metal has been deposited on the cathode. The middle area is not balanced because its concentration should have remained constant.

True and apparent number of transfers to Hittorf

The transfer number of the cation is defined in a binary electrolyte as the quotient of the migration speed (or ion mobility) of the cation to the sum of the migration speeds of the cation and anion. This is also the true transfer number of the cation. In a multi-ion mixture, the true transfer number of any ion is defined as the quotient of the migration speed (or ion mobility) of the ion concerned to the sum of all migration speeds (or ion mobilities) of all other cations and anions. In the case of non-binary salts (more than two types of ions are formed during dissociation), the number of charge carriers "z" ("valence", taking into account the stoichiometric coefficients of the dissociation equation) exchanged by the respective ion must be multiplied for each migration speed (or ion mobility) . An ion with twice the charge transports twice as many charge carriers per unit of time at the same migration speed. In the case of binary electrolytes, z is the same for the anion and the cation (condition of the electrical neutrality of a solution) and is therefore canceled.

Above all with the method according to Hittorf, and to a small extent also with the method according to Mac Innes, measurement errors occur, since the migrating ions “pull” the water molecules of their hydration shell (hydrated ion radius!) With them. This is due to electrostatic forces of attraction as well as the amount of water (solvent) carried along in the solvation shell of the ion. The Hittorf method in particular can lead to measurement errors because the dipole water molecules that migrate with them due to electrostatic attraction can change the concentrations in the measurement apparatus. This can be taken into account by adding uncharged organic substances such as sugar to the solution. From the change in the sugar concentration in the apparatus parts, conclusions can be drawn that the water molecules were being transported (unwanted) and the resulting errors in the transfer figures according to Hittorf can be corrected mathematically. The correction to be made to the apparent conviction number in order to obtain the true one depends on concentration. For aqueous solutions with concentrations of up to 0.1 mol / liter, the difference between the true and apparent transfer number will hardly go beyond the units of the third decimal place (according to Milazzo, Elektrochemie, p. 28). At higher concentrations, apparent Hittorf's and true transfer numbers differ by up to 10%.

A comparison table for Hittorf and true transfer numbers (corrected Hittorf numbers) are provided by Näser / Lempe / Regen in "Physical chemistry for technicians and engineers" on page 340.

The transfer figures determined according to Hittorf are apparent transfer figures without correction and are marked with the superscript "H" (Hittorf). True conversion numbers are marked with the superscript "w" (true):

• (apparent) Hittorf conversion number n ^H
• true (corrected Hittorfian) transfer number n ^w

The conversion takes place according to the following model:

${\displaystyle n_{\text{Kat}}^{w}=n_{\text{Kat}}^{H}+Z\cdot y}$

such as

${\displaystyle n_{\text{An}}^{w}=n_{\text{Kat}}^{H}-Z\cdot y}$

This is supposed to be the change in the number of moles (concentration) of the water in the cathode compartment, which occurs when a charge of 1 F (1 Faraday = 96485.3 [As / mol]) is passed through, which obviously means the equivalent amount of substance). should be the number of equivalents (number of moles times ionic valency) of the electrolyte per number of moles of water. ${\displaystyle Z}$ ${\displaystyle y}$

The different degrees of hydration of ions (hydrated ion radius) can be seen from the difference between true and apparent transfer numbers.

Table with test data:

	HCl	LiCl	NaCl	KCl	RbCl	CsCl
Period of the cation	1	2	3	4th	5	6th
Test temperature [° C]	20th	20th	20th	20th	18th	20th
Trial concentration c [mol / l]	without specification	without specification	without specification	without specification	0.02	without specification
${\displaystyle n_{\text{Kat}}^{H}}$ (measured)	0.820	0.278	0.366	0.482	?	0.485
${\displaystyle n_{\text{Kat}}^{w}}$ (measured)	0.844	0.304	0.383	0.495	0.497 (uncertain)	0.491
( - ) = + (Z * y) ${\displaystyle n_{\text{Kat}}^{w}}$ ${\displaystyle n_{\text{Kat}}^{H}}$	0.024	0.026	0.017	0.013	?	0.006
${\displaystyle \Delta n_{\text{Wasser}}^{KR}}$ Change in the amount of substance n of the water in the cathode compartment KR	0.240	1,500	0.760	0.600	?	0.530
${\displaystyle n_{\text{An}}^{H}}$ (calculated from the values ​​of the cations)	0.180	0.722	0.634	0.518	?	0.515
${\displaystyle n_{\text{An}}^{w}}$ (calculated from the values ​​of the cations)	0.156	0.696	0.617	0.505	0.503 (uncertain)	0.509
( - ) = - (Z * y) ${\displaystyle n_{\text{An}}^{w}}$ ${\displaystyle n_{\text{An}}^{H}}$	−0.024	−0.026	−0.017	−0.013	?	−0.006

The data of the Hittorf and true transfer numbers come from the book "Physical Chemistry for Technicians and Engineers" (Näser / Lempe / Regen, p. 340). The test temperature was 20 ° C. The concentration was not mentioned. Likewise, the amount of charge that has flown was not mentioned, so that the values ​​for and cannot be calculated. The values ​​for rubidium chloride come from the book Elektrochemie (Milazzo, 1951, p. 30, tab. 3). ${\displaystyle Z}$ ${\displaystyle y}$

As can be seen, the true conversion numbers of the alkali metal cations are always greater than the Hittorfs, since too low changes in concentration occur in both electrode spaces due to the dragged-along hydration shell water. The larger the atom (period and atomic number), the lower the hydration shell and hydration number , the lower the “dilution effect” in the electrode spaces. The difference between Hittorf's and true transfer numbers becomes smaller as the period of the cation increases.

Walther Nernst was the first person to determine the true transfer figures by means of an experiment according to Hittorf with the addition of urea or sugar (non-electrolyte) from the changes in the concentration of the non-electrolyte in the cathode and anode compartment.

Evaluation of the Hittorf method

The apparent Hittorf conversion numbers result from the relative changes in concentration ( molarity , molality is used for this) in the anode compartment "AR" and cathode compartment "KR" after the end of the electrolysis / experiment ${\displaystyle \Delta (c)}$

${\displaystyle n_{Kat,H}={\frac {\Delta (c_{AR})}{\Delta (c_{AR})+\Delta (c_{KR})}}}$

and:

${\displaystyle n_{An,H}={\frac {\Delta (c_{KR})}{\Delta (c_{AR})+\Delta (c_{KR})}}}$

The symbol "H" as an index stands for the Hittorf method. Cation Cat. Anion An.

The volumes of the anode compartment and cathode compartment must be the same here. Otherwise, the changes in concentration must be replaced by changes in the amount of substance (per volume or per mass of solvent).

The concentrations only consider the cation / or the anion. Usually the cation when noble metals are deposited on the cathode and this deposited mass is determined by weighing. The corresponding amount of substance of the metal is calculated.

The following also applies to binary electrolytes:

${\displaystyle {\frac {n_{\mathrm {Kat} }}{n_{\mathrm {An} }}}={\frac {\lambda _{\mathrm {Kat} }}{\lambda _{\mathrm {An} }}}={\frac {\Delta (c_{\mathrm {AR} })}{\Delta (c_{\mathrm {KR} })}}={\frac {n_{\mathrm {Kat} }}{1-n_{\mathrm {Kat} }}}}$

As you can see from the last equation, the quotients of the transfer numbers behave like the quotients of the equivalent conductivities (or migration speeds or ion mobilities), but in the opposite direction to the changes in equivalent concentration. If the equivalent conductivities are the same, the transfer numbers are both 0.5, their quotient 1 and the changes in concentration in the cathode compartment and anode compartment are the same, their quotient is one.

Interpretation of the values ​​of an experiment according to Hittorf

Näser gives a practical example, which, however, raises some questions because it is not completely plausible.

The number of transfers of copper sulphate solution should be determined. The test temperature is not mentioned. Before the start of the experiment, the Hittorf apparatus had a copper concentration of b = 1.214 [g / kg water] ( molality ). At the end of the test, the copper concentration in the anode compartment has risen to a final value of 1.430 [g / kg water]. The concentration of the cathode space at the end of the experiment is not mentioned. A total of 0.300 g of copper is said to have deposited. However, this must mean 0.300 [g copper / kg water], since apples cannot be compared with pears and the volumes of the cathode compartment and anode compartment are not mentioned. From the separated "copper mass" 0.300 [g / kg water], the molar mass of copper ( M = 63.55 [g / mol]), the Faraday constant F = 96485 [As / mol] and the valence of copper (II) -Ions ( z = 2) the total amount of charge Q that has flowed is calculated: 910.95 [As]. In the anode compartment, 0.216 [g copper ions per kg water] should be "left behind" (difference between the final and initial concentration). This should correspond to the amount of charge transported by the sulfate ions, i.e. q (-) = 655.9 [As]. This means that n (-) = 655.9 / 910.95 = 0.72. n (+) = 1-n (-) = 0.28. The copper ion should therefore have a practical conversion number of 0.28 and the sulfate ion 0.72 in copper sulfate solution.

However, it is neither explained why the concentration difference is called “left behind”, nor why it should correspond to the amount of charge transported by the sulfate ions. Furthermore, the conversion numbers for copper sulfate in ideally diluted solution at 25 ° C are n (+) = 0.414 and n (-) = 0.586 (limit conductivities copper 56.6 and sulfate 80 [Scm ² / mol]). The cation transfer number from the Hittorf experiment therefore deviates by −32.8% from the theoretical value at ideal dilution. An explanation for this is not given.

Nevertheless, the test result is self-explanatory. As is known, the sulfate ion has a higher ion mobility / boundary conductivity / migration speed than the copper ion. The electrolyte concentration (copper concentration and copper sulfate concentration) therefore increases in the "electrode compartment of the sulfate ion" (anode compartment AR), while it decreases in the "electrode compartment of the slower copper ion" (cathode compartment KR). It turns out that the ratio of the ion mobilities or limit conductivities or migration speeds corresponds to the ratio of the final concentrations achieved in the electrode spaces at the end of the experiment (if the ion mobilities are the same the concentration ratio would be 1):

In this way, the transfer figures can be determined directly from the concentration ratio at the end of the experiment (which is not clearly stated in the specialist books). To this end, we consider the experiment mentioned again:

	Molality at the start of the experiment in [g copper / kg water]	Amount of copper deposited during the experiment in [g copper / kg water]	Molality at the end of the experiment in [g copper / kg water]
Volume cathode space KR	1,214	(−0.300) deposited in the KR, but based on the total volume of KR + AR	0.398
Volume of anode space AR	1,214	0	1.430
arithmetic mean of KR and AR (molality of the combined volumes of KR + AR)	1,214	−0.300	0.914

Since 0.3 g of copper (per kg of water) was deposited during the test, the mean arithmetic copper concentration of the combined volumes of the cathode compartment and anode compartment at the end of the test must be 0.300 less than the initial value (1.214):

${\displaystyle 1{,}214-0{,}300=0{,}914}$

From the definition of the arithmetic mean , the final concentration that must prevail in the cathode compartment now follows:

${\displaystyle 0{,}914=(1{,}430+b_{\text{KR, Versuchsende}})/2}$and . ${\displaystyle b_{\text{KR, Versuchsende}}=0{,}398}$

The transfer number n (+) = 0.28 calculated by Näser is also obtained by calculating the quotient of the final concentrations of copper:

${\displaystyle n_{\text{langsames Kupfer-Kation}}=n_{+}={\frac {b_{\text{Elektrodenraum des langsamen Kupfer-Ions, Versuchsende}}}{b_{\text{Elektrodenraum des schnellen Sulfat-Ions, Versuchsende}}}}={\frac {0{,}398}{1{,}430}}=0{,}2783}$

and consequently:

${\displaystyle n_{\text{schnelles Sulfat-Anion}}=n_{-}=1-n_{+}=1-0{,}2783=0{,}7217}$

Thus the following generally applies to every experiment:

${\displaystyle n_{\text{langsames Ion}}={\frac {{\text{Endkonzentration }}b_{\text{im Elektrodenraum des langsamen Ions}}}{{\text{Endkonzentration }}b_{\text{im Elektrodenraum des schnellen Ions}}}}}$

The final concentration in the "electrode compartment of the faster ion" is always greater than that in the "electrode compartment of the slow ion". The “electrode spaces of the ion” generally refer to the electrode space to which the respective ion migrates, due to the polarity of its charge.

In the example, the molality relates to the cation (copper), but in other examples can also be related to the anions to be deposited (e.g. chlorine as chloride and as chlorine gas). ${\displaystyle b}$

The Mac Innes and Smith migratory interface method

The interface between two electrolytes in contact shifts under the influence of an electric field . If you use a colored ion and if you succeed in keeping the interface reasonably sharp during the experiment, you can determine the ion mobility and the transfer numbers from the speed (migration speed of the ion at the prevailing field strength ) of this shift . A detailed description of the Mc method. Innes can be found in “Elektrochemie” (Giulio Milazzo, 1952) on pages 26-28. ${\displaystyle u}$ ${\displaystyle E}$ ${\displaystyle v}$ ${\displaystyle n}$

In principle, it is sufficient to measure the transfer number / ion mobility of any ion for a salt (solution) once. The ion mobilities of all other ions can then be determined from conductivity measurements by combining them with this ion .

Milazzo names the following as the basic conditions for applying the Mac Innes method:

• the colored indicator ion must always follow the (mostly colorless) ion to be measured
• the indicator ion must have a smaller ion mobility than the ion to be measured ${\displaystyle v}$
• the concentration of the solution with the indicator ion should be set (lower) so that a higher field strength (voltage drop) acts on it and both ions ultimately migrate at exactly the same speed
• the one of the two solutions with the higher density is to be filled in the measuring tube of the apparatus under the solution of the lower density (layering of both solutions of different concentrations and densities)

Furthermore, it makes sense to keep the current constant during the experiment using a constant current source. The displacement of the interface is proportional to the amount of charge flowing . With a constant current, it is directly proportional to the time (constant displacement speed of the interface). ${\displaystyle I}$

The trick of the Mac Innes and Smith method

Two different ions will normally always have two different values ​​of ion mobility v , and therefore migrate at different speeds with an electric field strength . Since the colored indicator ion "Ind" should move just as quickly in the test in the measuring tube of the apparatus according to Mac Innes as the ion "x" to be measured, a trick must be used: Two different field strengths must prevail in the apparatus, each only have an accelerating effect on one ion (and its counterion). ${\displaystyle E}$ ${\displaystyle E}$

The ion migration speed is the product of the field strength and the ion mobility : ${\displaystyle u}$ ${\displaystyle E}$ ${\displaystyle v}$

${\displaystyle u_{i}=E\cdot v_{i}}$

in the apparatus must therefore apply:

${\displaystyle u_{x}=E_{x}\cdot v_{x}=u_{\text{Ind}}=E_{\text{Ind}}\cdot v_{\text{Ind}}}$

and thus:

${\displaystyle {\frac {E_{x}}{E_{\text{Ind}}}}={\frac {v_{\text{Ind}}}{v_{x}}}}$

Instead of the ion mobility , the equivalent conductivities / limit conductivities can also be used. ${\displaystyle v}$ ${\displaystyle \lambda }$

In the Mac Innes apparatus, two "columns of liquid" of different concentrations and densities have been filled with layers in the measuring tube so that they do not mix. At the bottom of the vertical measuring tube is the solution with the higher density and above that with the lower density. Let us now assume that the two "columns of liquid" are the same height. Both columns have the same cross-section (cross-section of the measuring tube inside) and the same test current flows through them. The test current is ideally kept constant during the test by means of a constant current source . ${\displaystyle I}$ ${\displaystyle I}$

The field strength is the voltage that falls on the liquid column with the ion x to be measured, based on the height of this liquid column in [cm]. The same applies to the field strength . ${\displaystyle E_{x}}$ ${\displaystyle U_{x}}$ ${\displaystyle E_{\text{Ind}}}$

According to Milazzo, the selected colored indicator ion must always have a lower ion mobility than the (mostly colorless) ion to be measured. So that it can migrate at the same speed, "its" field strength must be set higher than that of the ion to be measured. To do this, the concentration of the solution with the indicator ion must be set lower than the concentration of the solution with the ion to be measured. (Note: The voltage drops across the liquid column are the product of the resulting resistance of the liquid column and the flowing -constant- current . With decreasing ionic concentration of anions and cations in the liquid column , the resistance and thus the voltage drop ( voltage ) increases constant current flow and thus the respective electric field strength ). It is therefore approximately true under the condition : ${\displaystyle v}$ ${\displaystyle I}$ ${\displaystyle E}$ ${\displaystyle v_{\text{Ind}}<v_{x}}$

${\displaystyle 1<{\frac {E_{\text{Ind}}}{E_{x}}}\approx {\frac {c_{x}}{c_{\text{Ind}}}}}$

For an exact calculation, the ion mobilities of the cation and anion in the respective solution must of course be taken into account, which both additively to the conductivity. In the two solutions, either the cation or the anion will generally be identical in both solutions. Depending on whether an anion or a cation is to be "measured".

Once the required concentrations of both solutions have been calculated, their densities must be determined at the desired measuring temperature. The solution with the higher density is poured into the bottom of the measuring tube and carefully covered with the lighter solution. The densities alone determine whether the colored indicator ion is ultimately at the top and migrates downwards in the experiment, or is at the bottom and then moves upwards. The colored / colorless interface of both liquid layers migrates with the migration speed during the experiment . Since charges are shifted when migrating, the distance covered by the shifted interface is proportional to the amount of charge that has flowed, i.e. the product of current times time. ${\displaystyle u_{x}=u_{\text{Ind}}}$

However, the method used to set the concentration ratio when the ion mobility of the ion x to be measured is still unknown (that would actually be the application) is not answered by Milazzo either. It can be assumed that the ion mobility has to be roughly known (and the concentrations are adjusted accordingly), for example from the Hittorf experiment, and that the Mac Innes method then only serves to precisely determine the true transfer number. It should be pointed out here that, according to Milazzo, the true transfer figures according to Hittorf and those according to Mac Innes show a comparable accuracy when working precisely (Tab. 2, p. 28). ${\displaystyle v}$

The method of the moving interface according to Walther Nernst for colored ions

Walther Nernst determined the speed of migration of colored ions by measuring the displacement of a visible interface. In his apparatus, practically only an identical electrical field strength E acted on all ions. This simple method was further developed by Smith and MacInnes for the measurement of colorless ions using two different field strengths.

By measuring the cell voltage of a concentration element with transfer / diaphragm

According to Milazzo there is the possibility of measuring the electromotive force (cell voltage) of a concentration element to determine the conversion numbers of a binary electrolyte with preferably monovalent / monovalent ions (cation and anion are monovalent). The concentration element is the interconnection of two half-elements with identical electrode material for cathode and anode . The electrodes are immersed in a salt solution of the same salt, whereby different concentrations ( and ) of this salt must be present in both half-elements . Both half-cells are not connected by a power key , but by a diaphragm . Therefore this is a cell with an overpass . According to the Nernst equation, the concentration difference results in a cell voltage (potential difference) to which any diffusion voltages that may occur are added. The transfer numbers should be according to the equations ${\displaystyle c_{1}}$ ${\displaystyle c_{2}}$

${\displaystyle n_{\text{Kation}}={\frac {E_{\text{Anode}}}{(E_{\text{Anode}}-E_{\text{Kathode}})}}}$

and

${\displaystyle n_{\text{Anion}}={\frac {E_{\text{Kathode}}}{(E_{\text{Kathode}}-E_{\text{Anode}})}}}$

be calculated. is the electromotive force (potential difference / voltage between the respective electrode and the reference electrode ). ${\displaystyle E}$

The prerequisite is that the value of the transfer numbers between the various concentrations and cannot be changed (concentration-dependent) and that both concentrations do not change during the measurement. ${\displaystyle c_{1}}$ ${\displaystyle c_{2}}$

Näser gives the following formula:

${\displaystyle n_{\text{Anion}}={\frac {E_{n\pm }}{E}}={\frac {E_{\text{mit Überführung/Diaphragma}}}{E_{\text{ohne Überführung/mit Stromschlüssel}}}}}$

The method of determining the transfer numbers from the diffusion stresses is less precise than the methods according to Hittorf and Mc Innes.

Diffusion stress as a function of the transfer numbers (Henderson's equation)

The basis of the measurement method mentioned is the diffusion voltage , which for 1-1-valent electrolytes is derived from Henderson's equation :

${\displaystyle E_{\text{Diff}}=(n_{\text{An}}-n_{\text{Kat}})\cdot {\frac {R\cdot T}{z\cdot F}}\cdot \ln \left({\frac {a_{1}}{a_{2}}}\right)=(2\cdot n_{\text{An}}-1)\cdot {\frac {R\cdot T}{z\cdot F}}\cdot \ln \left({\frac {a_{1}}{a_{2}}}\right)}$

with ( activities or alternatively molar concentrations), charge exchange number (eff. valence) , general gas constant , absolute temperature in [K], Faraday constant . ${\displaystyle a_{1}>a_{2}}$ ${\displaystyle z}$ ${\displaystyle R}$ ${\displaystyle T}$ ${\displaystyle F}$

and always occurs when monovalent cations and monovalent anions diffuse through a diaphragm at different speeds (different ion mobilities!). It is a function of the ion mobilities / transfer numbers. The equation mentioned should not be confused with the Henderson-Hasselbalch buffer equation.

There is also an equation for diffusion voltages of binary electrolytes with polyvalent ions:

${\displaystyle E_{\text{Diff}}={\frac {{\dfrac {u_{\text{Kat}}}{z_{\text{Kat}}}}-{\frac {u_{\text{An}}}{z_{\text{An}}}}}{u_{\text{Kat}}+u_{\text{An}}}}\cdot {\frac {R\cdot T}{F}}\cdot \ln \left({\frac {a_{1}}{a_{2}}}\right)}$

In these equations, the migration speeds can be replaced directly by ion mobility or equivalent conductivities (lambda), since the field strength and / or Faraday constant in the numerator and denominator of the quotient are reduced. If the migration speeds are replaced in the latter equation by the transfer numbers of the binary electrolyte, the equation for the dependence of the diffusion voltage in binary electrolytes of polyvalent ions on the transfer number (s) is obtained: ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle u}$

${\displaystyle E_{\text{Diff}}=\left({\frac {n_{\text{Kat}}}{z_{\text{Kat}}}}-{\frac {n_{\text{An}}}{z_{\text{An}}}}\right)\cdot {\frac {R\cdot T}{F}}\cdot \ln \left({\frac {a_{1}}{a_{2}}}\right)=\left(n_{\text{Kat}}\cdot \left({\frac {1}{z_{\text{Kat}}}}+{\frac {1}{z_{\text{An}}}}\right)-{\frac {1}{z_{\text{An}}}}\right)\cdot {\frac {R\cdot T}{F}}\cdot \ln \left({\frac {a_{1}}{a_{2}}}\right)}$

The “Inorganikum” (pp. 328–331) and “ABC Chemistry” may be mentioned as further literature with various equations on the relationship between diffusion stresses and transfer numbers (or ion mobilities).

Comparison of the three methods

• The method by measuring the diffusion voltage provides values ​​immediately as soon as the equilibrium voltages have been established after a few seconds. Unfortunately this method is the least accurate. It must also be measured with a high-resistance voltmeter (e.g. differential voltmeter , tube voltmeter or FET voltmeter), since any current draw from the cell must be avoided (the original voltage should be measured). It makes sense to set the concentrations of the two half-cells as different as possible (very high and very low), as this increases the measurement accuracy (the diffusion voltage). ${\displaystyle E_{\mathrm {d} }}$

• The Hittorf method is easy to use. However, if possible, a low current must be used and the temperature of the apparatus must be kept constant (isothermal measurement). The cathode compartment and the anode compartment must ideally have exactly the same volume, since only then can the changes in concentration (easily converted) be transferred. Because of the low currents / current density, the experiment takes longer. The ampere hours or milliampere hours that have flowed (total current of cations and anions) must be measured. Therefore a constant current source makes sense (stop trial time). Ultimately, the deposited precious metal mass must be determined by weighing the cathode. Any gases separated must be determined volumetrically with a gas burette . Cations of base metals such as potassium and sodium can also be deposited if a mercury cathode is used and the apparatus is operated with a higher current density . Ultimately, the mercury is then weighed out in order to determine the base metal mass bound in it as amalgam. This is only possible with metals that form alloys ( amalgams ) with mercury . Milazzo depicts such a Hittorf apparatus with a mercury cathode (Fig. 3, “Jahnscher Apparat”, p. 25).

• The method according to Mac Innes / Smith is the safest for determining the true transfer number, since the migration speed of the ion is determined directly. However, it is obviously not easy to implement. A colored indicator ion has to be found, the speed of which migrates as quickly as possible (i.e. not faster or slower) as the ion to be determined through a suitable choice of the concentrations of two layered solutions (the one to be determined and the colored indicator ion). Two solutions with different densities are overlaid. Only one contains the colored indicator ion. Ultimately, there is a migration of the interface between the two solutions, the migration speed of which is determined by visual observation (with high-precision magnification devices and apparently a vernier or some other scale). The migration speed of the interface corresponds to that of the ion to be determined and that of the indicator ion.

The accuracy of the Hittorf and Mac Innes methods is about the same.

Calculation methods

Basics

The physical basis of the transfer number is the current (partial current) produced by each type of ion in the electric field , which is only produced by the ions actually discharging at the electrodes (see decomposition potential ). The current (partial current) is made up of the product of Faraday constant , field strength (or electrical voltage per electrode spacing ), electrode surface (in square centimeters), the molar ionic concentration (amount of the ion based on cubic centimeters), the charge number (charge exchange number) (or ) and the ion mobility (which is not a speed!) together: ${\displaystyle n_{i}}$ ${\displaystyle i}$ ${\displaystyle J_{i}}$ ${\displaystyle J_{i}}$ ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle U}$ ${\displaystyle l}$ ${\displaystyle A}$ ${\displaystyle i}$ ${\displaystyle z}$ ${\displaystyle n}$ ${\displaystyle v}$

${\displaystyle J_{i}=F\cdot E\cdot A\cdot c_{i,mol/cm3!}\cdot z_{i}\cdot v_{i}}$

With

${\displaystyle E={\frac {U}{l}}}$

The total current I _tot is made up of the partial currents J _i . The transfer number is the respective partial current J _{i of} the ion i normalized to the total current / total current (or the partial charge quantity q _{i transported} by the ion i normalized to the total charge quantity Q transported of all ions).

The relationships between ion mobility in [Scm ² / As], migration speed in [cm / s] in the electric field in [V / cm] and the isothermal equivalent conductivity in [Scm ² / mol], which is constant for every ion concentration, are: ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle E}$ ${\displaystyle \lambda }$

ionic migration speed ${\displaystyle u_{\pm }=v_{+/-}*E={\frac {E\cdot \lambda _{\pm }}{F}}}$

and

Ion mobility ${\displaystyle v_{\pm }={\frac {u_{\pm }}{E}}={\frac {\lambda _{\pm }}{F}}}$

For E = 1 [V / cm] (or 1V / m, depending on the units of ion mobility v used), u and v are numerical values ​​of the same size (but still have different units)! The limit equivalent conductivity (lambda-infinite) only applies to c = 0 mol / liter, approximately also below c = 0.01 mol / liter for monovalent ions (with existing polyvalent ions below an ionic strength of 0.01 mol / liter).

Notes on the units of ion mobility v

Warning: the ion mobility is not a speed although it has the symbol . (It's a normalized speed) ${\displaystyle v}$

The relationship applies to their units:

${\displaystyle 1\,\mathrm {\frac {m^{2}}{Vs}} =1\,\mathrm {\frac {Sm^{2}}{As}} =10000\,\mathrm {\frac {cm^{2}}{Vs}} =10000\,\mathrm {\frac {Scm^{2}}{As}} }$

At a field strength of 1 V / m or 1 V / cm, the ion mobility is in terms of value identical to the speed of movement of the ion in the correct unit (m / s or cm / s). Because it applies: ${\displaystyle v}$ ${\displaystyle E}$ ${\displaystyle u}$

${\displaystyle v={\frac {u}{E}}={\frac {u\cdot l}{U}}}$

1 Siemens: 1S (= 1 / 1Ohm = 1A / V ). Second s. Tension . Migration speed . Electrode gap . ${\displaystyle U}$ ${\displaystyle u}$ ${\displaystyle l}$

In the physical context, the SI units for in [m ² / Vs] = [Sm ² / As] are common. In technical applications, on the other hand, calculations are often made with cgs units [Scm ² / As] = [cm ² / Vs], since the electrode areas and electrode spacings are often given in [cm ² ] and [cm]. ${\displaystyle v}$

Notes on the importance of ion mobility v

The equivalent ion mobilities of most ions in aqueous solution are: ${\displaystyle \mu }$

${\displaystyle \mu \approx 3\dots 10\cdot 10^{-3}\mathrm {\frac {m^{2}}{val\cdot \Omega }} }$with the units Val and Ohm .

The unit val is out of date and must no longer be used. The ion mobility of hydronium ions at 25 ° C is 0.00362 [Scm ² / As] (also corresponds to [cm ² / Vs]) or in the units used in the physical context: [m ² / Vs] (also corresponds to [Sm ² / As]). ${\displaystyle 326,7\cdot 10^{-9}}$

see ion mobilities in current units under Ion mobility # numerical values .

Only the proton and the hydroxide ion are significantly more mobile, which is based on the special movement mechanism of these two ions: the normal ion migration is accompanied by the Grotthuss mechanism , a synchronous, abrupt change in position of many protons via intermolecular hydrogen bridges .

In principle, it is sufficient to measure the transfer number / ion mobility of any ion for a salt (solution) once. The ion mobilities of all other ions can then be determined from conductivity measurements by combining them with this ion .

Notes on outdated symbols of the linked quantities

today the following applies in the SI system:

• time ${\displaystyle t}$
• speed ${\displaystyle v}$
• Amount of substance ${\displaystyle n}$
• Equivalent conductivity , limiting conductivity of individual ions or (often with +/- indice) ${\displaystyle \lambda }$ ${\displaystyle \Lambda }$

Technicians still use w as a symbol of speed even today. In electrochemistry, u is the symbol of the speed of migration of the ion. Physicists and technicians also use the symbol v (physical speed of the SI system) as an alternative. In electrochemistry, however, v is the migration speed normalized to the field strength, the so-called ion mobility v. The ion mobility is an isothermal constant for each ion. The ion migration speed u increases with the electric field strength E (voltage U per electrode spacing l), while the ion mobility v is / remains constant. Today one should use n + and n- for the transfer number if possible, since t is the time in the SI system. But n is also the valency (alternatively: z) of an ion and in the SI system the amount of substance n (mole). The small ( ) or large lamba ( ) today is the equivalent conductivity for the respective concentration (with the index “c”) or for ideal dilution (limit conductivity, with the index “infinite”). The large lambda is sometimes used as the sum of the equivalent conductivity of all ions. So today, outside of the SI system in electrochemistry, etc. Physical chemistry: ${\displaystyle \lambda }$ ${\displaystyle \Lambda }$ ${\displaystyle \Lambda }$

• Migration speed u of the ion
• Ion mobility v (which is not a velocity!)
• Transfer numbers n +, n-
• Charge number, valence or charge exchange number , ${\displaystyle n}$ ${\displaystyle z}$
• often the total equivalent conductivity of all ions is named with the large lambda (without index +/–) ${\displaystyle \Lambda }$

earlier:

• time ${\displaystyle t}$
• Speed (physics) or (technology), rarely ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle u}$
• Transfer number or ${\displaystyle n}$ ${\displaystyle t}$
• Ion mobility (also known as "ion conductivity"): w, w +, w- or u (cation) and v (anion) ("u" and "v" in Milazzo in "Elektrochemie") and u (for cation and anion)
• Ion migration speed (also called "migration ability"): w +, w- (with Milazzo in “Elektrochemie” p. 42 and with Keune in “chimica” p. 139) and v (as physical speed; Keune “chimica”)
• Equivalent conductivity , limiting conductivity of individual ions or (often with +/- index), also called ion conductivity ${\displaystyle \lambda }$ ${\displaystyle \Lambda }$
• Often the total equivalent conductivity of all ions is named with the large lambda (without index +/-) ${\displaystyle \Lambda }$

Because of these many changes and ambiguities, such an overview is urgently needed. In Milazzo's book, too, translations from Italian into German were incorrectly translated. Ion mobility and migration speed are often confused. Always note the units. At a field strength of 1 V / cm (or 1 V / m) the migration speeds of the ions with the ion mobilities in [Scm ² / As] (or [Sm ² / As]) are numerically the same!

Quotient of two transfer numbers

The ratio of two transfer numbers of any ions i and k involved in the current transport of an electrolyte is the quotient of the products of molar concentration (of the respective ion) in [mol / cm3], charge exchange number of the respective ion and the migration speed of the ion in [cm / s] . Instead of the migration speed in [cm / s], the ion mobility in [Scm ² / As] or the equivalent conductivity in [Scm ² / mol] can also be used. Since this is a quotient, the normal molar concentration of the ions in [mol / liter] can also be used instead of the special molar concentrations (based on cubic centimeters!) (Factor 1000 or 1/1000 is shortened): ${\displaystyle z}$ ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle \lambda }$

${\displaystyle {\frac {n_{i}}{n_{k}}}={\frac {c_{i,spez}\cdot z_{i}\cdot v_{i}}{c_{k,spez}\cdot z_{k}\cdot v_{k}}}={\frac {c_{i,spez}\cdot z_{i}\cdot u_{i}}{c_{k,spez}\cdot z_{k}\cdot u_{k}}}={\frac {c_{i,spez}\cdot z_{i}\cdot \lambda _{i}}{c_{k,spez}\cdot z_{k}\cdot \lambda _{k}}}}$

The concentrations of the ions are not the molecular concentration of the dissolved salt ( otherwise the degree of dissociation and stoichiometric coefficients would have to be taken into account). Instead of the concentrations, the activities should actually be taken into account or the individual activity coefficient of each ion at each concentration must be multiplied.

For binary electrolytes (one cation and one anion) (charge exchange number) can be shortened, since it is always the same size for cation and anion. This no longer applies when dissociating into more than two ions. If the ion concentrations are the same, for example when a binary 1-1-valent electrolyte is dissociated, the ionic concentrations can also be reduced. ${\displaystyle z_{i}}$

Calculation of the theoretical transfer number for ideally diluted solutions

Since the ion mobility , ionic migration speeds (or out of date:) and limit equivalent conductivities are linearly proportional to each other (only coupled by natural constants), the following applies to ideally diluted solutions ( = 0 mol / liter): ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle w}$ ${\displaystyle \lambda }$ ${\displaystyle c}$

${\displaystyle n_{\text{Kat}}={\frac {J_{\text{Kat}}}{(J_{\text{Kat}}+J_{\text{An}})}}={\frac {u_{\text{Kat}}}{(u_{\text{Kat}}+u_{\text{An}})}}={\frac {v_{\text{Kat}}}{(v_{\text{Kat}}+v_{\text{An}})}}={\frac {\lambda _{\text{Kat}}}{(\lambda _{\text{Kat}}+\lambda _{\text{An}})}}={\frac {q_{\text{Kat}}}{(q_{\text{Kat}}+q_{\text{An}})}}={\frac {q_{\text{Kat}}}{Q}}}$

and analogously for the anion:

${\displaystyle n_{\text{An}}={\frac {J_{\text{An}}}{(J_{\text{Kat}}+J_{\text{An}})}}={\frac {u_{\text{An}}}{(u_{\text{Kat}}+u_{\text{An}})}}={\frac {v_{\text{An}}}{(v_{\text{Kat}}+v_{\text{An}})}}={\frac {\lambda _{\text{An}}}{(\lambda _{\text{Kat}}+\lambda _{\text{An}})}}={\frac {q_{\text{An}}}{(q_{\text{Kat}}+q_{\text{An}})}}={\frac {q_{An}}{Q}}}$

For non-binary electrolytes, the valence / charge exchange number of the respective ion has to be multiplied at each term in these equations . For binary electrolytes, the charge exchange number is the same for all ions, due to the electroneutrality of a solution, and can therefore be removed from the equations. ${\displaystyle z}$

It should be noted that the equivalent conductivities of ions listed in tables have already been standardized to the equivalent concentration and thus to the molar concentration and ionic valency . That is why the values ​​of ions are given as (1/2) · limit conductivity value etc. The limit conductivity values ​​of multivalent ions are therefore multiplied by the effective charge exchange number / ionic valency in the same way as the monovalent ions that are contained several times in the formula. How, for example, nitrate is contained twice in barium nitrate, i.e. the effective valence (charge exchange number) of the nitrate ion here is 2. Ultimately, as with all binary electrolytes, the charge exchange numbers are reduced here too, since they are the same (here 2 for nitrate and barium). ${\displaystyle z}$ ${\displaystyle z=2}$

In practice, one can assume the validity of these equations for 1-1-valent electrolytes up to molar concentrations of a maximum of = 0.01 mol / liter (without generating major errors). This is the scope of the Kohlrausch square root law . Experience has shown that Kohlrausch's law of square roots applies to total concentrations (diss. + Undiss.) Below 0.01 mol / liter for 1-1-valent electrolytes. If polyvalent ions are present, experience has shown that the calculated ionic strength is below 0.01 mol / liter ${\displaystyle c}$ ${\displaystyle I}$

From the (constant) limit conductivities of the ions for 25 ° C listed in tables, the transfer figures for 25 ° C and ideal dilution ( ) can now be calculated. However, other values ​​are to be expected for other concentrations and temperatures. Only the ions that can actually be discharged actually contribute to the transfer figures. Only these are to be taken into account. See chapter Redox potential. ${\displaystyle c=0}$

Calculate from the limit conductivities for 25 ° C of hydronium ion (349.8 Scm ² / mol), picration (31 Scm ² / mol) and acetate ion (40.9 Scm ² / mol) the transfer numbers of cation and anion for ideally diluted ( = 0 mol / l) acetic acid and picric acid, we get: ${\displaystyle c}$

• Acetic acid n (+) = 0.895 / n (-) = 0.105
• Picric acid n (+) = 0.919 / n (-) = 0.081

Milazzo gives the values for 0.1–1.0 molar acetic acid and picric acid: 0.892 / 0.108 (acetic acid) and 0.910 / 0.090 (picric acid). ${\displaystyle c}$

The calculated value only applies to the respective temperature (reference temperature of the limit conductivity, etc.).

Example barium nitrate

Ba (NO ₃ ) ₂ barium is divalent. The nitration is monovalent. The charge exchange number is that two nitrate ions have to be discharged for each barium ion. The charge exchange number can be reduced, since the salt is binary (consists of a cation and an anion). n (Ba ⁺⁺ ) = 2 x 63.6 / (2 x 63.6 + 2 x 71.5) = 63.6 / (63.6 + 71.5) = 0.471 ${\displaystyle z=2}$

Example iron (III) sulfate

Fe ₂ (SO ₄ ) ₃ Iron is trivalent here. The sulfate ion is bivalent. The charge exchange number is 2 · 3 = 3 · 2 = 6 and can be abbreviated (binary salt). n (Fe ⁺⁺⁺ ) = 2 x 3 x 68 / (2 x 3 x 68 + 3 x 2 x 80) = 68 / (68 + 80) = 0.460

Example mercury (I) nitrate

[Hg ₂ ] (NO ₃ ) ₂ The mercury (I) ion has the valence +2 (not +1). The nittration is monovalent. The charge exchange number is 2 and can be reduced. n ([Hg ₂ ] ⁺⁺ ) = 2 x 68.6 / (2 x 68.6 + 2 x 71.5) = 68.6 / (68.6 + 71.5) = 0.490 ${\displaystyle z}$

Example of the trinary salt sodium potassium sulfate

NaKSO ₄ In the case of non-binary salts, the charge exchange number cannot be reduced in every case. ${\displaystyle z}$

For the example, at least three (borderline) cases must be distinguished.

Case 1) At a mercury cathode, sodium and potassium are simultaneously discharged in a molar ratio of 1: 1 at a high current density. The charge exchange number is 2 for the sulfate ion, but 1. It is not possible to shorten it for sodium and potassium. n (Na +) = 1 x 50.1 / (1 x 50.1 + 1 x 73.5 + 2 x 80) = 0.177 and n (K +) = 1 x 73.5 / (1 x 50.1 + 1 73.5 + 2 80) = 0.259 and n (SO4--) = 2 80 / (1 50.1 + 1 73.5 + 2 80) = 0.564

Case 2) Only the sodium ion is discharged at a mercury cathode. The charge exchange number is because two monovalent sodium ions are discharged for each sulfate ion. The charge exchange number can be shortened. Potassium ions are not discharged. n (Na +) = 2 50.1 / (2 50.1 + 2 80) = 50.1 / (50.1 + 80) = 0.385 n (SO4--) = 2 80 / (2 50.1 + 2 * 80) = 80 / (50.1 + 80) = 0.615 ${\displaystyle z=2}$

Case 3) Only the potassium ion is discharged from a mercury cathode. The charge exchange number is that for each sulfate ion, two monovalent potassium ions are discharged. The charge exchange number can be shortened. Sodium ions are not discharged. n (K +) = 2 73.5 / (2 73.5 + 2 80) = 73.5 / (73.5 + 80) = 0.479 n (SO4--) = 2 80 / (2 73.5 + 2 * 80) = 80 / (73.5 + 80) = 0.521 ${\displaystyle z=2}$

Case 4) The ratio of discharged amounts of sodium ions and potassium ions varies. The conversion numbers of cations and anions range between cases 2 and 3.

Calculation of the theoretical transfer number for real dilute solutions

Under real diluted solutions are to be understood here electrolyte solutions with concentrations ( would be ideally diluted). How much current (per electrode surface) an ion can transport at one temperature in a solvent depends not only on its ion mobility / migration speed / boundary conductivity and the voltage (more precisely the field strength ), but also on its ionic molar concentration (in [mol / cm ³ ]) in which it is present in the solution. ${\displaystyle c>0}$ ${\displaystyle c=0}$

From the equation for the ion current (partial current) of the ion : ${\displaystyle J_{i}}$ ${\displaystyle i}$

${\displaystyle J_{i}=F\cdot E\cdot A\cdot c_{i,mol/cm^{3}}\cdot z_{i}\cdot v_{i}}$

With

${\displaystyle E={\frac {U}{l}}}$

it now follows for the transfer number for cation: ${\displaystyle n_{i}}$

${\displaystyle n_{Kat}={\frac {c_{Kat,aequi}\cdot z_{Kat}\cdot v_{Kat}}{(c_{Kat}\cdot z_{Kat}\cdot v_{Kat}+c_{An}\cdot z_{An}\cdot v_{An})}}={\frac {c_{Kat}\cdot z_{Kat}\cdot u_{Kat}}{(c_{Kat}\cdot z_{Kat}\cdot u_{Kat}+c_{An}\cdot z_{An}\cdot u_{An})}}={\frac {c_{Kat}\cdot z_{Kat}\cdot \lambda _{Kat,c}}{(c_{Kat}\cdot z_{Kat}\cdot \lambda _{Kat,c}+c_{An}\cdot z_{An}\cdot \lambda _{An,c})}}={\frac {q_{Kat}}{(q_{Kat}+q_{An})}}={\frac {q_{Kat}}{Q}}}$

and for the anion:

${\displaystyle n_{An}={\frac {c_{An,aequi}\cdot z_{An}\cdot v_{An}}{(c_{Kat}\cdot z_{Kat}\cdot v_{Kat}+c_{An}\cdot z_{An}\cdot v_{An})}}={\frac {c_{An}\cdot z_{An}\cdot u_{An}}{(c_{Kat}\cdot z_{Kat}\cdot u_{Kat}+c_{An}\cdot z_{An}\cdot u_{An})}}={\frac {c_{An}\cdot z_{An}\cdot \lambda _{An,c}}{(c_{Kat}\cdot z_{Kat}\cdot \lambda _{Kat,c}+c_{An}\cdot z_{An}\cdot \lambda _{An,c})}}={\frac {q_{An}}{(q_{Kat}+q_{An})}}={\frac {q_{An}}{Q}}}$

Since there is a relationship here, ion mobility , migration speed or equivalent conductivity can alternatively be used in the formula. ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle \lambda }$

Strictly speaking, the individual activity coefficient of the ion i must be multiplied for each molar ion concentration in order to get from the concentration to the activity . For low and medium concentrations (below 0.1 [mol / liter] ionic strength ) these values ​​can be calculated from known formulas. ${\displaystyle c_{i,aequi}}$ ${\displaystyle f_{i}}$

In order to obtain correct values ​​for the conversion numbers to the respective ionic concentrations , the values ​​for the respective ionic concentration (s) would have to be known. Otherwise, the respective limit conductivities can only be roughly estimated. The calculated value only applies to the respective temperature (reference temperature of the limit conductivity, etc.). ${\displaystyle c_{i,{\text{aequi}}}}$ ${\displaystyle \lambda _{c}}$ ${\displaystyle c_{i}}$ ${\displaystyle \lambda _{i,{\text{unendlich}}}}$

Dependencies of the number of deliveries

The transfer number of an ion in a solution of different ions cannot be regarded as a characteristic quantity of an ion insofar as its value depends on all other types of ions. It is largely independent of the current strength / current density with which the electrolysis is carried out (unless concentration-related electrochemical polarization occurs due to one of the ion types on only one electrode).

Dependence on the dissociation equation

An example is given here for clarification. The complex substance H ₂ [PtCl ₆ ] (hexachloroplatinic acid) could theoretically dissociate in water according to two different equations:

1. ${\displaystyle \mathrm {H_{2}[PtCl_{6}]\rightleftharpoons \ 2H^{+}+Pt^{4+}+6Cl^{-}} }$
2. ${\displaystyle \mathrm {H_{2}[PtCl_{6}]\rightleftharpoons \ 2H^{+}+[PtCl_{6}]^{-2}} }$

It is obvious that cases 1 and 2 would lead to different types of ions and probably also different concentrations of hydronium ions . This would lead to different total conversion numbers of the cations and anions. Equation 2) was determined to be applicable by measurements.

On the other hand, the complex salt ammonium tetrachlorozincate (NH ₄ ) ₂ [ZnCl ₄ ] is known to be composed of the two ammonium ions NH ₄ ⁺ and the complex tetrachlorozincate (II) anion [ZnCl ₄ ] ⁻² only as a crystalline salt . In water, however, it is said to be present almost completely when it dissociates into the ions ammonium NH ₄ ⁺ , zinc Zn ⁺² and chloride Cl ^- . So this salt dissociates into two cations and one anion.

Further examples with mixed cations are sodium potassium sulfate NaKSO ₄ and sodium potassium tartrate . There are also salts with mixed anions.

Dependence on the degree of dissociation (alpha)

The degree of dissociation refers to the proportion (molar concentration) of the dissociated particles (ions), based on the total molar concentration of all dissolved particles (undissociated and dissociated dissolved) of the dissolved salt. Since an improved or worsened dissociation always affects the concentration of the cations and the anions of a binary electrolyte at the same time, the ratio of the concentrations of anions to cations always remains the same, even if the degree of dissociation changes (depending on the concentration). Therefore, in the solution of a salt, there is no dependence of the conversion numbers on the degree of dissociation. It must be noted here, however, that the degree of dissociation often depends on the temperature. If the temperature coefficients of the cation and anion are clearly different, a change in temperature can cause a change in the degree of dissociation and a change in the conversion numbers.

In solutions of several salts there are several "independent" degrees of dissociation (via the ion concentrations all colligative properties influence one another somewhat). Therefore, the transfer numbers of the individual ions in such multi-ion mixtures can change depending on the degree of dissociation (change in the concentration ratios).

Milazzo points out that the isothermal ion mobilities (migration speeds per field strength) are not 100% constants of an ion type, but also depend on the respective "counterion" and the concentrations. For the dependence of the migration speed of the potassium ion at a field strength of 1 V / cm (i.e. the ion mobility), Milazzo gives a table with different potassium salts at 18 ° C and 25 ° C and a (apparently deliberately chosen high) molar concentration of c = 0 , 1 mol / liter of the salts. At 25 ° C, for example, the potassium ion migrates in 0.1N potassium sulfate solution at 0.000540 cm / s and in 0.1N potassium chloride solution at 0.000654 cm / s. In potassium bromide has 0.000656 and 0.000631 potassium chlorate cm / s rate of migration . In more dilute solutions, the differences would have been much smaller. Because of this low dependency on “counterions” and concentrations (at low concentrations), there may also be a low dependency of the conversion numbers on the degree of dissociation (alpha). The lower the concentration of the ions, the less likely it is that the ion mobilities (migration speeds ) will influence each other in order to be completely eliminated in the case of ideal dilution ( ). Therefore, the limit equivalent conductivities of the ions at a concentration of c = 0 are isothermal constants. ${\displaystyle c=0}$

Concentration dependence

There is generally a concentration dependency of the transfer numbers, which can be well described in analogy to Kohlrausch's law (square root law). This law generally applies to molar concentrations (what is meant here is the total molecular concentration of the particles present in ionically and non-ionically dissolved form) below 0.01 mol / liter. The transfer number at a concentration (total concentration of ionic and undissociated dissolved particles) is linked to the transfer number at ideal dilution ( ): ${\displaystyle n(c)}$ ${\displaystyle c}$ ${\displaystyle n(c=0)}$ ${\displaystyle c=0}$

• ${\displaystyle n_{i,(c)}=n_{i,(c=0)}{-A\cdot {\sqrt {c}}}}$

Here, A is an empirical constant that can take negative or positive values. If the constant A is positive for the anion / cation, it is negative for the cation / anion.

• Example: According to Milazzo (Table 3, p. 31), aqueous calcium nitrate solution Ca (NO3) 2 at a total molar concentration of c = 0.005 mol / liter has a conversion number of cations / anions of: 0.450 / 0.550 (without temperature information). From the known limit equivalent conductivities for calcium ions / nitrate ions (59.5 / 71.5 Scm ² / mol), the conversion numbers for ideally diluted solutions (c = 0) are calculated as: 0.454 / 0.546 (at 25 ° C).

Now the values ​​of each ion are inserted into the Kohlrausch equation:

• ${\displaystyle n_{Ca,(c=0{,}005)}=n_{Ca,(c=0)}{-A\cdot {\sqrt {0{,}005}}}}$ so

${\displaystyle 0{,}450={0{,}454}{-A\cdot {\sqrt {0{,}005}}}}$

• ${\displaystyle n_{NO3,(c=0{,}005)}=n_{NO3,(c=0)}{-A\cdot {\sqrt {0{,}005}}}}$ so

${\displaystyle 0{,}550={0{,}546}{-A\cdot {\sqrt {0{,}005}}}}$

This gives the constant A for the calcium ion as A = + 0.0566 and for the nitrate ion as A = -0.0566 (in calcium nitrate solution). The conversion figures for calcium ions and nitrate ions in calcium nitrate solution (from 25 ° C) can now also be calculated for other molecular concentrations below c = 0.01 mol / liter. If you insert the ionic concentrations into the equation (that of the nitrate is twice as high as that of calcium because of the stoichiometric coefficient of the nitrate in calcium nitrate!), Then the constants of both equations also differ in amount. Nevertheless, the equations remain valid.

For the salt rubidium iodide, a constant A of +0.262 for the rubidium ion and −0.262 for the iodide ion is calculated (c = 0.02 mol / liter, T = 18 ° C, n (Rb) = 0.498). At ideal dilution, rubidium iodide has a rubidium ion conversion number of 0.535 at 18 ° C (calculated from the limit conductivities for rubidium 77 and iodide 66.8 at 18 ° C).

The unit of constant A depends on the unit of c. This law is only valid if its graphic representation (diagram with the conversion number of the respective ion plotted on the linear y-axis and the square root of the molecular concentration plotted on the linear x-axis) depicts falling or rising straight lines.

Experience has shown that Kohlrausch's law of square roots applies to total concentrations (diss. + Undiss.) Below 0.01 mol / liter for 1-1-valent electrolytes. If polyvalent ions are present, experience has shown that the calculated ionic strength I is below 0.01 [mol / liter]. Are multivalent ions in front of the square root of the molar concentration in the is Kohlrausch equation today often by the square root of the ionic strength I is replaced. The ionic strength is defined as half of the sum of the products of all molar ion concentrations (dissociated ion concentrations?) Of the ion types present with the squares of their valencies.

If, on the other hand, there is a significantly stronger dependence of the transfer numbers - at low concentrations - on the concentration than according to root (c) (invalidity of the square root law), then complex ions are definitely present in the solution. With increasing concentrations of complex ions (and the non-complex ions necessary for their formation), the transfer number of the non-complex ions can drop so far that it becomes negative.

If the ions of a salt are present in extremely low concentrations, for example in the case of very poorly soluble salts (e.g. heavy metal sulfides), then in the limit case they (almost) no longer contribute to the conductivity, so that the conductivity is separated from the dissociated hydronium and hydroxide -Ions of the solvent water is determined solely. The conductivity and the transfer numbers now correspond to those of the solvent water and can be calculated from the temperature-dependent ion product of the water and the limit conductivities of hydronium ion and hydroxide ion and their temperature coefficients (alpha).

Ionic strength

In 1921 GN Lewis and M. Randall defined the ionic strength because it had been shown that polyvalent ions (quadratically) have a stronger influence on the equivalent conductivity values ​​or molar conductivity values ​​(compared to monovalent ones). In the presence of multivalent ions, the square root law of electrolytic conductivity empirically found by Kohlrausch remained valid when the concentration is replaced by the calculated ionic strength I:

${\displaystyle \lambda _{i,(c,I)}=\lambda _{i,(c=0)}{-A\cdot {\sqrt {I}}}}$

In this respect, Kohlrausch's law of square roots could be confirmed to be valid for low values ​​of ionic strength I. In this form, it describes the dependence of the equivalent conductivities on the charge numbers (valencies) and concentrations of all ions present in the solution. According to the definition equations of the transfer number, this also affects the transfer numbers.

The ionic strength also determines the mean activity coefficient .

Dependence on the defined conductivity coefficient f _λ

The conductivity coefficient f _λ is defined as the quotient of the equivalent conductivity of one type of ion at the molar concentration and the limit equivalent conductivity of this type of ion in an ideally diluted solution (c = 0 mol / liter): ${\displaystyle c}$

${\displaystyle f_{\lambda }={\frac {\lambda _{\pm },(c)}{\lambda _{\pm },(c=0)}}}$

It is mainly used with strong electrolytes (degree of dissociation close to one). For strong electrolytes, this coefficient is therefore a function of the concentration of the ion. Its values ​​range from 0 (at very high concentrations) to 1 (ideal dilution, ). ${\displaystyle c=0}$

Empirically and theoretically according to Debye-Hückel-Onsager, the following relationship to the effective concentration / ionic strength I was found:

• ${\displaystyle f_{\lambda }=1-{K\cdot {\sqrt {I}}}}$

For 1-1-valent electrolytes, this equation could already be confirmed from Kohlrausch's law of the square root of equivalent conductivity (by dividing by the limit equivalent conductivity).

The values ​​of the degree of dissociation and the osmotic coefficient also move accordingly between 0 and 1. There are physical links to these quantities. The conductivity coefficient is a measure of the deviation of a solution from the ideal state (ideally diluted solution, c = 0). It flows as a multiplicative value together with the degree of dissociation of a salt and the total concentration of the salt in the solution, in the applied electrolysis current (since the equivalent conductivity with real dilution is lower than the limit equivalent conductivity with ideal dilution).

Furthermore, the conductivity coefficient depends on the charge numbers (valencies) of the ion (s). On page 342 of the textbook "Inorganikum" there is a table with the dependencies of the conductivity coefficient on the concentration (s) and charge numbers of both types of ions (cation and anion). Since the ions interact with one another, the individual conductivity coefficients are not independent of the respective "counterion" in the electrolyte. It can therefore make sense to define the conductivity coefficient as the quotient of the total conductivities of cation and anion (again for concentration c based on the value at c = 0). All ion concentrations and valences are then immediately incorporated into this value. Higher coefficients of conductivity of individual ions lead to higher induced currents of this ion. Only if the conductivity coefficients of all ions increase or decrease uniformly do the transfer numbers of all ions remain the same, although the electrolysis current increases or decreases. If a conductivity coefficient of the multi-ion electrolyte changes differently than that of the other types of ions, then all the transfer numbers change. From the change in the total conductivity coefficient, however, it cannot be concluded that there has been a change in the conversion numbers of cation and anion (disadvantage of this definition). The increase / decrease in the total conductivity coefficient only says something about the increase / decrease in the total current, but not how it is composed.

Since for each ion both the equivalent conductivity (at ) and the limit equivalent conductivity (c = 0) are functions of temperature that have practically the same temperature coefficient (alpha), the conductivity coefficients themselves are not functions of temperature. ${\displaystyle c>0}$

The definition of the conductivity coefficient is an alternative calculation method to the application of the Kohlrausch square root law for the determination of the concentration dependence of the transfer numbers (or equivalent conductivities ). In Kohlrausch's law, the valency of the ions is contained in the constant A (negative increase in the straight line in the diagram, with a higher valency the increase is steeper, so the constant has a higher value). ${\displaystyle f(\lambda )}$

See also chapter 1.1.

Concentration dependence in multi-ion mixtures with different concentrations of the different ions

In multi-ion mixtures, the transfer number is also dependent on the concentration of the respective ion (this transfer number), since the amount of charge that can be transported by this ion is the product of an empirical constant, the molar equivalent concentration of the ion and its migration speed (or ion mobility).

Temperature dependence

The temperature dependence of the transfer numbers results from the temperature dependency of the ion mobilities (or migration speeds or proportional limit equivalent conductivities) of the ions, which is explained by the temperature dependence of the solvation of the ions and thus their ion diameter in solution as well as the temperature dependence of the viscosity of the solvent. For increasing temperatures, the transfer numbers of cation and anion approach each other in order to reach the numerical value 0.5 in the theoretical limit case.

The temperature dependency of the transfer numbers can be calculated for ideally diluted solutions from the known temperature coefficients (alpha) of the limit equivalent conductivities of the individual ions. The alpha values ​​are constants that should apply to the solvent water at an ideal dilution between 18 ° C and approx. 90 ° C. The alpha value is the difference between two limit conductivities (usually from 25 ° C and 18 ° C) based on the limit conductivity at the reference temperature (here 18 ° C) and based on the temperature difference (here 7K). Multiplying the alpha value by 100 gives the percentage increase in the limit conductivity of the ion with a temperature increase of 1 Kelvin. Hübschmann names these temperature coefficients for many ions at ideal dilution and a reference temperature of 18 ° C (25 ° C) in "Tables on chemistry" on page 61. The temperature coefficients alpha (here in percent per Kelvin) are between 1.59 (hydronium ion) and 8.21 [% / K] (aluminum ion) based on the limit conductivity at reference temperature (here 18 ° C; recently 25 ° C is mostly used). Potassium / sodium / ammonium ions have 1.87 / 1.98 / 1.87 [% / K]. What is striking here is the same size of the alpha values ​​for potassium and ammonium, which also have the same hydrated ionic radii and limit conductivities. Hydroxide ion 2.06% / K. Alkali ions , alkaline earth ions and many anions have values ​​from 1.9 to 2.2% / K. Heavy metal cations such as iron (II) (2.89% / K), manganese (II), copper (II), nickel (II), cobalt (II) etc. are between 2.1 and 2.9 [% / K ]. Iron (III) has 1.64% / K, halide ions are between 1.85 (bromide) and 2.34 (iodide). Sulphate has 2.40% / K. Chloride 2.25% / K. The calculable equivalent conductivities (and conversion numbers) should apply to the temperature range 18 to 90 ° C.

The temperature coefficient (alpha value) of the equivalent conductivity of an ion i is defined, for example:

${\displaystyle \alpha _{i}={\frac {\lambda _{i,25\,^{\circ }\mathrm {C} }-\lambda _{i,18\,^{\circ }\mathrm {C} }}{\lambda _{i,18\,^{\circ }\mathrm {C} }\cdot 7[K]}}}$

Other definitions (reference temperatures / reference equivalent conductivities) are also possible. Today, 25 ° C "room temperature" is often used as the reference temperature.

For the equivalent conductivity of an ion i at a different temperature (T2) with a reference temperature of 18 ° C, the following applies:

${\displaystyle \lambda _{i,T2}=\lambda _{i,18\,^{\circ }\mathrm {C} }\cdot (1+(\alpha _{i,18\,^{\circ }\mathrm {C} })\cdot {(T_{2}-18\,^{\circ }\mathrm {C} ))}}$

The temperature coefficient of an ion should not be confused with the temperature coefficient k of the entire electrolyte (with at least two different ions contained). However, both values ​​can be converted into one another. ${\displaystyle \lambda }$

General formula of the temperature dependence of the change in the transfer number

If one formally subtracts the defining equations of the conversion numbers of an ion - for two different temperatures - in a binary electrolyte from one another and introduces the temperature coefficients of the two types of ions into these, one obtains after transformation and abbreviation for the cation: ${\displaystyle \alpha _{i}}$

${\displaystyle \Delta {(n_{i})}^{(\Delta T)}=n_{i}^{(T_{2})}-n_{i}^{(T_{0})}=\lambda _{\text{Kation}}^{(T_{0})}\cdot \lambda _{\text{Anion}}^{(T_{0})}\cdot (\alpha _{\text{Kation}}^{(T_{0})}-\alpha _{\text{Anion}}^{(T_{0})})\cdot (T_{2}-T_{0})/A}$

and for the anion:

${\displaystyle \Delta {(n_{i})}^{(\Delta T)}=n_{i}^{(T_{2})}-n_{i}^{(T_{0})}=\lambda _{\text{Kation}}^{(T_{0})}\cdot \lambda _{\text{Anion}}^{(T_{0})}\cdot (\alpha _{\text{Anion}}^{(T_{0})}-\alpha _{\text{Kation}}^{(T_{0})})\cdot (T_{2}-T_{0})/A}$

In both formulas, only the difference between the alpha values ​​in the sign differs. is the new temperature and the reference temperature with the associated (limit) equivalent conductivity values ​​and the alpha values ​​of the ions related to this temperature. ${\displaystyle T_{2}}$ ${\displaystyle T_{0}}$

The numerical value for A is calculated as follows (a common numerical value for both ions):

${\displaystyle A=(\lambda _{\text{Kation}}^{(T_{0})})^{2}+(\lambda _{\text{Anion}}^{(T_{0})})^{2}+(\lambda _{\text{Kation}}^{(T_{0})})^{2}\cdot \alpha _{\text{Kation}}^{(T_{0})}\cdot (T_{2}-T_{0})+(\lambda _{\text{Anion}}^{(T_{0})})^{2}\cdot \alpha _{\text{Anion}}^{(T_{0})}\cdot (T_{2}-T_{0})+2\cdot \lambda _{\text{Kation}}^{(T_{0})}\cdot \lambda _{Anion}^{(T_{0})}+\lambda _{\text{Kation}}^{(T_{0})}\cdot \lambda _{\text{Anion}}^{(T_{0})}\cdot \alpha _{\text{Kation}}^{(T_{0})}\cdot (T_{2}-T_{0})+\lambda _{\text{Kation}}^{(T_{0})}\cdot \lambda _{\text{Anion}}^{(T_{0})}\cdot \alpha _{\text{Anion}}^{(T_{0})}\cdot (T_{2}-T_{0})}$

The validity of the formula can be checked on ideally diluted electrolytes with their limit conductivity. The A value for ideal dilute potassium chloride solution with reference temperature 18 ° C for all conductivity values, alpha values and the reference transport number is . From 25 ° C to 18 ° C, the conversion number of the potassium ion in potassium chloride solution therefore changes by 0.490-0.496 = -0.006. That of the chloride therefore around 0.5097-0.5038 = +0.006. ${\displaystyle n_{i}^{(T_{0})}}$ ${\displaystyle A=19637{,}3585}$

Example potassium ion in potassium chloride, reference temperature is 18 ° C, is 25 ° C: ${\displaystyle T_{2}}$

${\displaystyle \Delta n_{\text{Kalium}}^{(\Delta T)}=n_{\text{Kalium}}^{(25^{\circ }\mathrm {C} )}-n_{\text{Kalium}}^{(18^{\circ }\mathrm {C} )}=\lambda _{\text{Kalium}}^{(18)}\cdot \lambda _{\text{Chlorid}}^{(18)}\cdot (\alpha _{\text{Kalium}}^{(18)}-\alpha _{\text{Chlorid}}^{(18)})\cdot (25-18)/A}$

and in numerical values:

${\displaystyle \Delta n_{\text{Kalium}}^{(\Delta T)}=0{,}490-0{,}496=65\cdot 66\cdot (0{,}0187-0{,}0225)\cdot (25-18)/(4225+4356+4225\cdot 0{,}0187\cdot 7+4356\cdot 0{,}0225\cdot 7+2\cdot 65\cdot 66+65\cdot 66\cdot 0{,}0187\cdot 7+65\cdot 66\cdot 0{,}0225\cdot 7)=-0{,}006}$

The validity of these formulas is only given if it is assumed that the alpha values ​​are constants. So they only apply to temperature differences that are not too great.

Example calculations for ideal dilution

The basis of the following values ​​is the booklet "Tables on Chemistry" (Hübschmann, 1991). The limit conductivities for 90 ° C were calculated from the values ​​for 18 ° C and 25 ° C and the temperature coefficient calculated from them (based on the value of 18 ° C). Attention: the temperature coefficients at Huebschmann refer to different (not mentioned) temperatures and are therefore unusable.

Conversion figures of pure neutral water at 18, 25 and 90 ° C

electrolyte	formula	cation	Anion	Temperature [° C]	lambda + [Scm 2 / mol]	lambda- [Scm 2 / mol]	n (+)	n (-)
water	${\displaystyle H_{2}O}$	Hydronium ion ${\displaystyle H^{+}}$	Hydroxide ion ${\displaystyle OH^{-}}$	18th	315	173	0.645	0.355
water	${\displaystyle H_{2}O}$	Hydronium ion ${\displaystyle H^{+}}$	Hydroxide ion ${\displaystyle OH^{-}}$	25th	350	198	0.639	0.361
water	${\displaystyle H_{2}O}$	Hydronium ion ${\displaystyle H^{+}}$	Hydroxide ion ${\displaystyle OH^{-}}$	90	approx. 675.6 (calculated)	approx. 429.6 (calculated)	approx. 0.611 (calculated)	approx. 0.389 (calculated)

Since the hydroxide ion has a higher temperature coefficient (2.06% / K based on the value of the limiting conductivity of 18 ° C, i.e. 173 Scm ² / mol) than the hydronium ion (1.59% / K), the transfer number of the increases with the temperature Hydroxide ion, while that of hydronium ion decreases.

Conversion figures of aluminum chloride at 18, 25 and 90 ° C

Aluminum chloride obviously has one of the greatest possible temperature dependencies of the conversion numbers of a salt, since the temperature coefficients of aluminum (III) ion (8.21% / K based on the limiting conductivity of 18 ° C) and chloride ion (2.25% / K) are mutually exclusive differ greatly.

electrolyte	formula	cation	Anion	Temperature [° C]	lambda + [Scm 2 / mol]	lambda- [Scm 2 / mol]	n (+)	n (-)
Aluminum chloride	${\displaystyle AlCl_{3}}$	Aluminum ion ${\displaystyle Al^{3+}}$	Chloride ion ${\displaystyle Cl^{-}}$	18th	40	66	0.377	0.623
Aluminum chloride	${\displaystyle AlCl_{3}}$	Aluminum ion ${\displaystyle Al^{3+}}$	Chloride ion ${\displaystyle Cl^{-}}$	25th	63	76.4	0.452	0.548
Aluminum chloride	${\displaystyle AlCl_{3}}$	Aluminum ion ${\displaystyle Al^{3+}}$	Chloride ion ${\displaystyle Cl^{-}}$	90	approx. 276.4 (calculated)	approx. 172.9 (calculated)	approx. 0.615 (calculated)	approx. 0.385 (calculated)

The proportion of current transported by the aluminum ion rises sharply with temperature, while that of the chloride ion falls, since the temperature coefficients are correspondingly different (Al: 8.21% / K; Cl: 2.25% / K).

Transfer figures of potassium nitrite at 18, 25 and 90 ° C

electrolyte	formula	cation	Anion	Temperature [° C]	lambda + [Scm 2 / mol]	lambda- [Scm 2 / mol]	n (+)	n (-)
Potassium nitrite	${\displaystyle KNO_{2}}$	Potassium ion ${\displaystyle K^{+}}$	Nitrition ${\displaystyle NO_{2}^{-}}$	18th	65	59	0.529	0.471
Potassium nitrite	${\displaystyle KNO_{2}}$	Potassium ion ${\displaystyle K^{+}}$	Nitrition ${\displaystyle NO_{2}^{-}}$	25th	73.5	72	0.505	0.495
Potassium nitrite	${\displaystyle KNO_{2}}$	Potassium ion ${\displaystyle K^{+}}$	Nitrition ${\displaystyle NO_{2}^{-}}$	90	approx. 152.5 (calculated)	approx. 192.8 (calculated)	approx. 0.442 (calculated)	approx. 0.558 (calculated)

This example shows what happens when the equivalent conductivity of one ion is first lower (here with the nitrite anion), but then because of a higher temperature coefficient (nitrite: 3.15% / K; potassium: 1.87% / K) when the temperature rises more strongly per Kelvin and finally the value of the counterion (here potassium) reaches and ultimately exceeds it. The equivalent conductivity and transfer number of the nitrite is only smaller at 18 ° C than the equivalent conductivity and transfer number of potassium, reaches the same value for potassium at a (calculable) temperature Tx (transfer number = 0.5 for cation and anion, same equivalent conductivities of both ions) in order to ultimately assume greater values ​​than those of potassium.

The temperature at which the transfer numbers become the same, i.e. 0.5, is calculated in this case:

${\displaystyle T_{n=0{,}5}={\frac {(\lambda _{Kat,18\,^{\circ }\mathrm {C} }-\lambda _{An,18\,^{\circ }\mathrm {C} }+18\cdot (\lambda _{An,18\,^{\circ }\mathrm {C} }\cdot \alpha _{An,18\,^{\circ }\mathrm {C} }-\lambda _{Kat,18\,^{\circ }\mathrm {C} }\cdot \alpha _{Kat,18\,^{\circ }\mathrm {C} }))}{(\lambda _{An,18\,^{\circ }\mathrm {C} }\cdot \alpha _{An,18\,^{\circ }\mathrm {C} }-\lambda _{Kat,18\,^{\circ }\mathrm {C} }\cdot \alpha _{Kat,18\,^{\circ }\mathrm {C} })}}}$

electrolyte	formula	cation	Anion	Temperature [° C]	lambda + [Scm 2 / mol]	lambda- [Scm 2 / mol]	n (+)	n (-)
Potassium nitrite	${\displaystyle KNO_{2}}$	Potassium ion K +	Nitrition ${\displaystyle NO_{2}-}$	approx. 27.33	approx. 76.30	approx. 76.28	approx. 0.500	approx. 0.500

Depending on the viscosity of the solvent

The dynamic viscosity of every solvent decreases with increasing temperature . Therefore the migration speed / ion mobility of all ions increases with increasing temperature. The transfer figures only change if the increases in the migration rates of cation and anion are different. This is mostly to be expected. See temperature dependence.

Influence of complexing agents

If complexing agents and the ions that can be incorporated into the complex are present in the necessary concentrations, at least two ions together form a complex ion with a new valence and a different hydrated ion diameter (usually a smaller one). Complex ions have better ion mobility / equivalent conductivity. Their formation or their decay change the transfer numbers of all other ions in the solution (as far as the other ions can also be discharged). The transfer numbers of all dischargeable ions (with the prevailing current densities and electrode potentials) are functions of the concentrations of all ions involved in the formation of the complex ions if complexing agents are present (complex-forming ions). In this case, Kohlrausch's law (square root law) is no longer valid for the dependence of the equivalent conductivity of an ion on its concentration in the solution. It can therefore no longer be used for the conversion of transfer numbers (ideally diluted and real diluted solutions).

Dependence on dissolved non-ionic substances

If non-ionic substances are present in the solution at the same time, this can change the individual equivalent conductivities or transfer numbers if the migration speeds of individual types of ions (with large ion diameters) are reduced by hindrance / friction (at high concentrations of the non-electrolyte), or if this non-electrolyte with the existing ions can form complex ions. According to Milazzo, the latter applies, for example, to sulfuric acid in the presence of acetone or sugar. If, on the other hand, the dissolved non-ionic substance only changes the viscosity of the solvent, it will simultaneously have a uniformly reducing effect on the migration speeds of all types of ions, which causes the electrolysis current (conductivity) to drop, but does not change the transfer figures.

Dependence on the ion radius and the charge number of the ion ${\displaystyle n}$

According to the definition of Stokes' law , ions are seen as spheres with a spherical radius (ionic radius) r, which diffuse through the solvent with a velocity (ion velocity ) proportional to the electric field strength E - here the ion mobility is not meant. Due to the dynamic viscosity of the solvent, they have to migrate against a frictional force F _r at temperature , which increases with the migration speed. Therefore, in the end - practically immediately - there is a constant migration speed (for any field strength or voltage ). The product of the speed of migration and the number of charges (valence) determines the charge transported per unit of time, i.e. the partial current brought by the ion to the total current (electrolysis current) . The migration speed increases with the number of charges (valence of the ion), with the field strength (voltage per electrode spacing), with increasing temperature (due to the decreasing viscosity of the solvent) and with decreasing (hydrated) ionic radius . It must be pointed out here that the ion radius is more of a virtual variable that is determined for crystals using X-ray diffraction . Solvent is deposited in solutions by these ions, which are now dissolved. This “hydrate shell” increases the “real” ion radius in the solvent. It is known that small atoms (elements with a small atomic number ) greatly increase their crystal ionic radius during hydration, while large atoms (elements with larger atomic numbers) only slightly increase their crystal ionic radius during hydration. In molten salts there cannot be a hydrate shell due to the lack of a solvent, but the addition of uncharged molecules with a dipole element would be conceivable. ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle T}$ ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle E}$ ${\displaystyle U}$ ${\displaystyle u}$ ${\displaystyle z}$ ${\displaystyle u}$ ${\displaystyle E}$ ${\displaystyle r}$

The force accelerating the ion in the electric field is: ${\displaystyle i}$ ${\displaystyle E}$ ${\displaystyle F_{i}}$

${\displaystyle F_{i}=z_{i}\cdot e\cdot E}$

The frictional force of the ion depends on its migration speed and the dynamic viscosity of the solvent : ${\displaystyle u}$ ${\displaystyle \eta }$

${\displaystyle F_{R,i}=6\cdot \pi \cdot \eta \cdot r_{i}\cdot u_{i}}$

${\displaystyle z}$is here the valence (charge) of the ion . is the elementary charge of an electron . Dynamic viscosity of the solvent: . Voltage , electrode spacing (in cm). ${\displaystyle i}$ ${\displaystyle e}$ ${\displaystyle \eta }$ ${\displaystyle U}$ ${\displaystyle l}$

By equating the forces and adjusting according to the migration speed of the ion i, one obtains: ${\displaystyle u_{i}}$

${\displaystyle u_{i}={\frac {z_{i}\cdot e\cdot E}{6\cdot \pi \cdot \eta \cdot r_{i}}}={\frac {z_{i}\cdot e\cdot U}{6\cdot \pi \cdot \eta \cdot r_{i}\cdot l}}}$

By inserting the transfer number into the definition equation (ratio of the migration speed of the ion under consideration to the sum of all migration speeds), this leads to:

${\displaystyle n({\text{Kat}})={\dfrac {z({\text{Kat}})}{r({\text{Kat}})\cdot \left({\dfrac {z({\text{Kat}})}{r({\text{Kat}})}}+{\dfrac {z({\text{An}})}{r({\text{An}})}}\right)}}}$

and

${\displaystyle n({\text{An}})={\dfrac {z({\text{An}})}{r({\text{An}})\cdot \left({\dfrac {z({\text{Kat}})}{r({\text{Kat}})}}+{\dfrac {z({\text{An}})}{r({\text{An}})}}\right)}}}$

The transfer numbers are thus composed of the quotients of ionic valencies and ionic radii. In the case of the ionic radii, of course, the hydrated ionic radius is meant.

The effective (hydrated) ion radius of an ion at the respective temperature can be calculated from the above equation of the migration speed by rearranging if the dynamic viscosity of the solvent is known and the migration speed or ion mobility or the equivalent conductivity / limit conductivity is known: ${\displaystyle r}$ ${\displaystyle i}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle \lambda _{i}}$

${\displaystyle r_{i}={\frac {z_{i}\cdot e\cdot E}{6\cdot \pi \cdot \eta \cdot u_{i}}}={\frac {z_{i}\cdot e}{6\cdot \pi \cdot \eta \cdot v_{i}}}={\frac {z_{i}\cdot e\cdot F}{6\cdot \pi \cdot \eta \cdot \lambda _{i}}}}$

${\displaystyle F}$Faraday constant, elementary charge of the electron, charge number of the ion, electric field strength (voltage per electrode gap), dynamic viscosity of the solvent. ${\displaystyle e}$ ${\displaystyle z}$ ${\displaystyle E}$ ${\displaystyle \eta }$

Solvation is discussed further in “chimica, Volume II”, p. 142. In the “Textbook of Inorganic Chemistry” (Hollemann-Wiberg, edition 90), a solvated lithium ion is shown schematically on p. 708. According to chimica, the solvation takes place in at least two shells, an inner solid hydrate shell and an outer “loose” hydrate shell. As soon as the ion migrates in the electric field, it largely loses its outer shell and thus only takes with it the water of hydration that is contained in the inner shell. When calculating the ion mobility / migration speed, one therefore obtains the hydrated radius of the inner hydration shell of the ion.

Example: alkali metals, hydronium ion and ammonium ion as chlorides

Crystal ion radii, hydrated ion radii and the conversion numbers of the chlorides for alkali metals, hydronium ions (hydrochloric acid) and ammonium chloride resulting in water in ideally diluted solution at 18 and 25 ° C:

Cation:	H +	Li +	Na +	K +	NH 4 +	Rb +	Cs +		Anion: Cl -
Atomic number Z of the ion	1	3	11	19th	without	37	55		17th
Period of the ion	1	2	3	4th	without	5	6th		3
Crystal ion radius in [Å] / and [pm] (picometer)	0.000013 / 0.0013 2)	0.60 / 60	0.95 / 95	1.33 / 133	1.43 / 143	1.48 / 148	1.69 / 169		1.81 / 181
calculated partially hydrated ionic radius in [Å] / and [pm] (picometer) for 25 ° C in water 1)	?	2.38 / 238	1.84 / 184	1.25 / 125	1.25 / 125	1.20 / 120	1.20 / 120		1.21 / 121
hydrated ionic radius in [Å] / and [pm] (picometers)	?	3.40 / 340	2.76 / 276	2.32 / 232	?	2.28 / 228	2.28 / 228		?
k-parameter of the ion according to Kielland in [Å] (mean hydrated ion diameter)	9	6th	4th	3	3	3	3		3
Hydration number, number of attached water molecules per dissolved ion	?	approx. 25	approx. 17	approx. 11	?	?	approx. 10		?
Limit conductivity at 18 ° C in [Scm 2 / mol]	315	32.55	42.6	63.65	63.6	66.3	66.8		66.3
Limit conductivity at 25 ° C in [Scm 2 / mol]	349.8	38.69	50.11	73.52	73.4	77	77		76.34
Temperature coefficient of the limit conductivity at 18 ° C (alpha, 18)	0.01578	0.02695	0.02518	0.02215	0.02201	0.02306	0.02181		0.02163
Conversion number n + of the cation in chloride at 18 ° C in water ( ) ${\displaystyle c=0}$	0.826	0.3293	0.391	0.4901	0.4896	0.500	0.502		= 1-n +
Conversion number n + of the cation in chloride at 25 ° C in water ( ) ${\displaystyle c=0}$	0.8209	0.3363	0.3963	0.4906	0.4902	0.5021	0.5022		= 1-n +

Sources of the table: Ionic radii and hydration number from the “Textbook of Inorganic Chemistry” by Hollemann / Wiberg on pages 117, 287, 731 and 736. The ionic radius for the hydrogen ion was given on page 117 as 1.3 · 10 ⁻¹³  cm and converted to Ångström. 1 angstrom = 0.1 nm = 100 pm. 2) According to Hübschmann (chemistry tables) the hydrogen ion does not have a fixed ionic radius. The values ​​of the limit conductivities for 18 and 25 ° C come from the book "Elektrochemie" by Giulio Milazzo, Springer-Verlag, 1951, p. 43. The temperature coefficients (alpha) were calculated from the latter values ​​for a reference temperature of 18 ° C. The transfer numbers were calculated from the limit conductivities of the respective temperature for infinite dilution. No reference temperature was given for the ionic radii. The k parameter of the ions comes from the book Udo Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. Thieme Verlag, Stuttgart 1996, ISBN 3-13-585804-9 , p. 321 and p. 47. The k parameter is used to calculate individual activity coefficients of ions according to the Debye-Hückel equation and is intended to be the mean hydrated ion diameter of the ion be (safely rounded). 1) Calculated hydrated ionic radius (partially hydrated when moving in an electric field), based on the derived formula of the ionic radius and the limit conductivities for 25 ° C in water (dynamic viscosity of water: 0.891 mPas at 25 ° C).

As the table shows, only a few water molecules can be attached to large crystalline ion radii (large atomic number, large period) (hydration number), consequently the hydrated ion diameter (ion in aqueous solution) with such large atoms is no longer very much larger than the crystalline ion diameter . Small ions therefore solvate strongly, large ones only slightly. The hydrated ion diameter determines equivalent conductivity and transfer numbers. It is not constant, but a function of temperature.

In molten salts there is logically only the crystalline ion radius, which determines the specific conductivity, equivalent conductivity and transfer numbers there.

Unedited subsection on ionic radius and charge number

• Example 1: According to Hollemann-Wiberg, the monovalent ions of lithium ( atomic number : 3) and cesium (atomic number: 55) have crystal ionic radii of 0.60 and 1.69 angstroms . Their limit equivalent conductivities are 39 and 77.2 [Scm ² / mol] at 25 ° C. This means that the lithium ion, which is significantly smaller in the crystal, has a much poorer equivalent conductivity in ideally diluted aqueous solution than the much larger cesium ion in the crystal. This is explained by a strong increase in the ion envelope (hydrated ion radius) when the lithium salt is dissolved, but only a slight increase in the ion envelope (hydrated ion radius) when the cesium salt is dissolved in water. The hydrated lithium ion thus has a correspondingly larger ion shell than the hydrated cesium ion, which results in its limit equivalent conductivities.
• Example 2: According to Hollemann-Wiberg, the ions of divalent and trivalent iron have the crystal ionic radii 0.76 and 0.64 Angstroms. The atomic number of iron is 26, which is a bit higher. The limit equivalent conductivities for 25 ° C in water are 53.5 and 68.0 [Scm ² / mol]. This shows that, because the atomic number (same element) is the same, no major differences are to be expected in the hydration of the various ions (shells). The larger ion in the hydrated and unhydrated state is iron (II) with the correspondingly lower limit equivalent conductivity, as would be expected according to Stokes' law .

The “ion radii in crystals” mentioned in the literature are therefore only comparatively useful for assessing the influence of the hydration of an ion in the solvent water and thus the influence on its equivalent conductivity and its conversion number in an electrolyte (ion mixture).

In relation to the influence of the number of charges, Milazzo values ​​for the melts of the salts InCl, InCl2, InCl3 are mentioned: their total equivalent conductivities (chloride ions and indium ions) are given as 130, 29 and 17 [Scm ² / mol]. Milazzo also names equivalent conductivities for molten salts of the chlorides of the metals mercury, thallium, tin and lead in various valence levels. These salts each show a dramatic decrease (over several powers of ten) in the total equivalent conductivity (cations and anions) of the molten salts with increasing valency of the metal cation. The melting temperature (s) are unfortunately not mentioned.

the following text still needs to be revised:

If a neutral atom with an atomic radius absorbs electrons, a negatively charged ion, an anion, is formed. The more electrons the atom absorbs (number of charges ), the larger the "electron shell" of the anion formed will be. If electrons are given off by the atom, a positively charged ion, a cation, is formed. The more electrons were given off (charge number “n” of the cation), the smaller the ionic radius of the cation in relation to the atomic radius of the uncharged element. The more electrons that an atom absorbs when an anion is formed, the greater the ionic radius of the anion in relation to the atomic diameter. Since larger ions migrate more slowly in a solvent (migration speed or ion mobility is lower) than small ions, larger ions contribute less to charge transport (current flow) than small ions. The transfer number of a larger ion is therefore smaller (with the same ion concentration and the same number of charges ) than that of a small ion. ${\displaystyle r}$ ${\displaystyle n}$ ${\displaystyle z}$

Conclusion:

• In the case of cations (atoms with released electrons), an increased number of charges leads to a smaller ion diameter, increased migration speed (ion mobility) and thus to a disproportionately higher current flow (taking into account the increased transport of charges due to the larger number of charges), which is why the transfer number of the cation increases disproportionately with increasing number of charges . ${\displaystyle n}$ ${\displaystyle n}$
• In the case of anions (atoms with absorbed electrons), a larger number of charges leads to a larger ion diameter, to a lower migration speed (ion mobility) and thus to an increased but relativistically too small current flow (also taking into account the increased charge transport due to the larger number of charges). The transfer number of an anion with a larger number of charges has also increased, but less than the increase in the number of charges is due, since the speed of migration has decreased somewhat with increasing . ${\displaystyle n}$ ${\displaystyle n}$

A graphic representation (diagram) of atomic and ion diameters of the most important (monatomic) cations and anions of all elements from atomic number 1 (hydrogen) to 96 (curium) is given in "Brockhaus ABC Chemie", GDR 1965, p. 591 under the heading "Ion radius “Pictured.

A doubling of the charge number of an ion would basically lead to a doubling of the current produced by this ion if the ionic concentration remained the same if the migration speed remained the same. However, this cannot be the case, since the change in the number of charges results in a change in the ion radius. The partial current contributed by an ion to the total current of the electrolysis is therefore directly proportional to the product of the migration speed (or ion mobility) and the number of charges n, but indirectly proportional to the ion diameter which depends on the number of charges (which has already been included in the migration speed). In addition, it must be taken into account that when the number of charges is doubled, the molar concentration of the "counter-ions" charged with a different polarity must also be doubled or their concentration has remained the same, but their number of charges must also have doubled, due to the condition of the electro-neutrality of a solution. Therefore, the influence of the charge number (s) on the transfer number (s) is limited to the change in the migration speeds due to an increase or decrease in the ionic radii.

In solvents (but not in melts), the ions are solvated, so they have a shell made of accumulated solvent. The diameter of this hydrate shell (solvent water) depends on the temperature. Changes in the number of charges (valencies) and changes in the hydration shell due to changes in the number of charges and the temperature are combined in the resulting change in the migration speeds / transfer numbers. In general, it can be stated that different types of ions with the same number of charges n with largely the same ion diameters also have largely identical migration speeds (ion mobilities, equivalent conductivities) and therefore have to have largely the same transfer numbers at the same concentrations (!) Of these different ions in the solution. This can be seen when comparing the limit conductivities of potassium (+1) (73.52 Scm ² / mol at 25 ° C) and ammonium (+1) (73.4 Scm ² / mol at 25 ° C) and when comparing rubidium (+1) (77 Scm ² / mol at 25 ° C) and cesium (+1) (77 Scm ² / mol at 25 ° C), which must each have almost identical ion diameters.

Exceptions: If one compares the limit equivalent conductivities of the monoatomic divalent ( ) mercury (II) cation Hg (+2) (63.6 at 25 ° C) with that of the “biatomic” likewise divalent ( , but statistically every mercury atom is univalent here ) Mercury (I) cation [Hg-Hg] (+ 2) (68.6 at 25 ° C), it turns out that the mercury (I) cation must migrate faster here. So biatomic ions do not behave like single-atom ions. This certainly also applies to the biatomic zinc (I) and cadmium (I) cations, which are only stable in melts. ${\displaystyle n=2}$ ${\displaystyle n=2}$

Dependence of the transfer numbers on electrolysis voltage, redox potential and current density

If a multi-salt electrolyte solution is exposed to an electric field , not all of the dissolved ion types usually take part in the current transport (simultaneously). If the terminal voltage (electrolysis voltage ) is increased slowly, initially only a very small current flows, the so-called residual current . It is transported by the hydronium ions and hydroxide ions ( autoprotolysis of the water !). Above a certain minimum voltage, the current flow then increases -näherungsweise linear- sharply, and finally at a higher voltage to a so-called limiting current to swing (the horizontal curve of the limit current, more specifically the limiting current density ), the I = f (U) diagram. In the diagram, however, the current density (instead of the current) is usually plotted on the y-axis, since all functions of electrolysis are also functions of current density.

Each type of ion has a characteristic redox potential (usually called redox voltage) for each ion concentration and temperature for its formation or discharge (reduction or oxidation) . For 25 ° C and c = 1 mol / liter, the so-called standard potentials (“redox voltages”, half-cell potentials, measured against the standard hydrogen electrode ) can be taken from specialist books on chemistry or “physical chemistry”. The difference between the more positive half-cell potential and the more negative half-cell potential is the discharge voltage / electrolysis voltage / decomposition voltage / redox voltage of the two interconnected half-cells (electrolysis apparatus ) under the prevailing conditions of concentration (s) and temperature. The standard potentials apply to 25 ° C and c = 1 mol / liter for each type of ion. For other concentrations and temperatures, the changes in the half-cell potentials (redox voltage of the ion type) can be calculated using the Nernst equation . The higher concentration is used in the numerator of the logarithmic ratio. For changed temperatures, all half-cell potentials must be converted using the Nernst equation, whereby the Nernst factor (for 25 ° C: 0.059) must be recalculated for the desired temperature in Kelvin. Attention: the Nernst factor also contains a correction constant, depending on whether the formula uses the natural or the decadic logarithm. However, the standard is the decadal (correction factor is ln10 here).

In practice, the measured decomposition voltage (electrolysis voltage) is often higher than the calculated difference (i.e. values ​​converted to temperature and concentration (s)) between the two redox potentials of cation and anion. The deviation is called overvoltage . Overvoltages can occur at the cathode, anode or both electrodes. The overvoltages increase the amounts of the redox potentials of the cations and / or anions. Positive potentials shift slo to even more positive values, negative to even more negative values. They therefore increase the decomposition voltage and counteract the flow of current. Overvoltages are usually proportional to the current density increasing. Overvoltages arise primarily when gases such as hydrogen, oxygen and chlorine are separated, but also when some metals such as chromium, nickel, iron and cobalt are separated. For the deposition of alkali metals on mercury electrodes, Milazzo points to the example of lowering the overvoltage of sodium and increasing the overvoltage of hydrogen, each on a mercury cathode. With regard to sodium, this should come about because the sodium is bound in the mercury ( amalgam formation ) and is thus largely withdrawn from the redox process. Only then is it possible to discharge the sodium before hydrogen in an aqueous solution.

Only the cations and anions that have the lowest redox potentials are the first to discharge (at low terminal voltage / low current density). Only later are those with higher redox potentials discharged. This fact is used in the diagrams of the polarography (graphic representation of the gradual discharge in the polarogram ).

So if there is a mixed solution of copper (II) chloride and copper (II) iodide, the iodides are first discharged, since their redox potentials (in terms of amount) are lower than those of the chlorides. The same also applies to the cations. Here, hydrogen is normally discharged from alkaline earth or alkali ions. Unless you use a mercury cathode . At very high current densities, redox systems that are somewhat further apart can also be partially discharged at the same time.

Only the ions that are being discharged contribute with their ion mobility / migration speeds / equivalent conductivities as well as with their molar concentration and their valence to the respective current flows and transfer numbers of the individual ion types. Ions that are currently not being discharged have the transfer number zero at this time (under these conditions).

Non-discharged cations / anions still migrate in the electric field and collect in front of the cathode / anode, which leads to a polarization voltage (formation of an overvoltage due to concentration polarization , lowering of the current as a result) at these electrodes.

The normal potentials for 25 ° C and activity a = 1 mol / liter already contain polarization voltages.

Example: The mixed salt potassium-sodium-sulfate KNaSO ₄ is in solution (or a mixture of sodium sulfate and potassium sulfate). The solution is electrolyzed. First of all, the sulfate is oxidized to peroxodisulfate at the anode. Initially, only the sulfate anion is present. Only hydrogen is discharged at the cathode, no sodium and no potassium. The current conversion numbers for cation (hydrogen ion) and anion (sulfate) are calculated as n (H) = (2 · 350 / (2 · 350 + 2 · 80)) = 0.814 and n (SO4) = 1-0.814 = 0.186. If a mercury electrode and high current densities are used, no more hydrogen is discharged, but instead all sodium (more positive redox potential than potassium!) Is discharged at the mercury cathode. The transfer numbers are now: n (Na) = (2 * 50.1 / (2 * 50.1 + 2 * 80)) = 0.385 and n (SO4) = 1-0.385 = 0.615 (discharge of sodium sulfate). As soon as all sodium has been discharged, the cathode potential drops a little further to more negative values ​​and the potassium (potassium sulfate) is discharged. The transfer numbers are now: n (K) = (2 * 73.5 / (2 * 73.5 + 2 + 80)) = 0.479 and n (SO4) = 1-0.479 = 0.521. It was assumed here that only sulphate ions are discharged at the anode and no hydroxide ions, which is certainly not entirely correct. The calculation can also be made with the hydroxide ions instead of the sulfate ions. It is crucial to know which ions are actually discharged. In addition, only c (OH) = 10 ⁻⁷ mol / liter hydroxide ions are present in a neutral salt solution , so that practically only sulfate can be discharged at the anode.

Current density-electrode potential curves

Current density-cathode potential curves (or anode potential curves) are recorded to graphically show which ions can be discharged under which conditions. The current density (the flowing electrolysis current divided by the constant electrode area in cm ² ) is shown on the y-axis and the electrode potential in volts is measured against a reference electrode (usually the normal hydrogen electrode ) on the x-axis of the diagram . Milazzo presents these diagrams in Figures 31–35 and explains their meaning. The electrode potential itself can be tapped with an electrolyte-filled glass capillary directly on the surface of the electrode, the so-called "Luggins capillary" (Milazzo, p. 130, Fig. 26) and against it via a power key - here a beaker with electrolyte - connected reference electrode can be measured as voltage in volts.

Variation of the electrode area (versus current density)

As already discussed, the current density has the main influence on the potential of an electrode because of the electrode polarization (overvoltages!). If only one electrode area (cathode or anode) is reduced while the electrolysis current is kept constant, the current density increases and the potential of the cathode / anode becomes more negative / positive. Accordingly, the transfer numbers can change if only one electrode is affected by this change in current density. The ratio of the electrode areas determines the ratio of the transfer numbers with the electrolysis current kept constant (in addition to the temperature and the concentration of the ions). This is used technically in many technical electrolyses . E.g. in the production of hydrogen peroxide H ₂ O ₂ via the intermediate stage of peroxodisulfuric acid H ₂ S ₂ O ₈ . While the current density at the platinum anode is 0.5 to 1 [A / cm ² ] ( anodic oxidation of sulfate ions to peroxodisulfate ions ), the lead cathode has only 0.1 [A / cm ² ] current density. The surfaces of the cathode and anode were therefore chosen to be of different sizes in order to optimize the electrolysis process in terms of energy efficiency. Undesired electrode reactions should not take place.

Influence of the pH value on redox potentials and conversion numbers

The redox potentials of most cations and anions are (possibly strongly) dependent on the pH value. The textbook of inorganic chemistry (Hollemann / Wiberg) gives on several pages tables for standard redox potentials in acidic and also in basic solutions. The redox potentials of the metals sometimes shift dramatically in the direction of more negative values ​​when a basic environment is present. Usually only the standard redox potentials for acidic environments are given in textbooks. The same potential reduction also occurs when complexing agents are present. Since the redox potentials determine which ions can be discharged, they determine the practically occurring transfer numbers depending on the pH value, the molar concentrations (activities) of the ions to be discharged - according to the Nernst equation - and the temperature (see Nernst factor the Nernst equation).

Depolarization, direct current and alternating current

With the same effective field strengths , different currents often flow if the electrolytic conductivity is measured once with direct voltage and once with alternating voltage of medium frequency (from 1 kHz). When using alternating current (from 1 kHz) the electrode polarization is largely or completely canceled. Each electrode is polarized umpteen times per second with a corresponding frequency , so it is alternating between cathode and anode. Accordingly, cations and anions change their directions of movement several times per second. Electrolysis no longer takes place insofar as the discharged ions immediately reunite with the "counter-ions" discharged shortly thereafter to form the original electrolyte. The concentration does not decrease with alternating current "electrolysis". If electrodes with different polarities were present when using direct voltage, a change to alternating current can theoretically change the transfer rates. In practice, however, transfer numbers cannot be determined for alternating current. The addition of chemical substances that lead to the depolarization of electrodes, the so-called depolarizers , often lead to a change in the transfer numbers of individual ions during electrolysis with direct current, as the discharge potentials of these ions change or the passivation of an electrode is canceled. The use of other electrode materials can also change the transfer rates (see chapter Overvoltages). If the (equilibrium) concentration / activity of the separated gas / substance / metal is reduced through chemical bonding or alloying with the metal electrode (e.g. formation of amalgam on a mercury cathode), the discharge potentials change and thus the transfer rates change with the current density kept constant .

pH value dependency for zwitterions

So-called zwitterions are mostly organic molecules that contain a basic ( proton-attaching ) and an acidic ( proton-releasing ) group of molecules. These molecules can therefore form cations and anions depending on the pH value. The concentrations of the cations and anions formed gradually change to larger or smaller values ​​when the pH value changes. Typical zwitterions are some (basic) amino acids , amino acid esters , amines , amine oxides , betaines and alkaloids . Those of the various amino acids, each with a basic amino group and an acidic carboxy group, are typical zwitterions. In the acidic medium, the amino group attaches a proton (several cannot be attached) and is thus simply positively charged. The negative counterion forms the acidic ion (acid residue, anion) of the added strong acid. Typical salts of this type are the so-called hydrochlorides . The amino acid thus forms a cation in the acidic medium . In a basic medium, however, it acts as an acid and splits off the proton of the carboxy group. Here it forms salts with the added alkali and thus functions here as an anion . The addition / cleavage of the proton on the amino group or carboxy group are therefore reversible and dependent on the pH value . In the acidic medium the zwitterion molecules are therefore primarily present as cations and in the basic medium as anions. The so-called isoelectric point here is the pH value at which the electrolytic conductivity of the hermaphrodite ions becomes zero, as they no longer migrate in the electric field (the migration speed has become zero because the same number of molecules are positively charged as there are negatively charged ones). At this pH value, the zwitterions in total therefore practically do not contribute to the current flow (if only zwitterions were present, the current flow would be zero, but this is practically impossible). At the isoelectric point the transfer number of the hermaphroditic ions becomes zero. Your transfer number is a function of pH. If the isoelectric point is undershot or exceeded, the conductivity / conversion number of the zwitterions increases in both cases.

Influence of larger and heavier atomic nuclei in isotopes on the conversion number and electrochemical properties

The three hydrogen isotopes protium (light hydrogen), deuterium (heavy hydrogen) and tritium (superheavy hydrogen) show most clearly of all isotopes the changed properties due to the change in the mass number with the same atomic number . The mass numbers of the mentioned isotopes behave like 1: 2: 3 despite the same atomic number. Protium contains only one proton in the atomic nucleus, deuterium one proton and one neutron, tritium one proton and two neutrons. The number of electrons in the atom and ion shells are each the same. However, the oxides of the isotopes (differ water , heavy water and over heavy water ) in many physical properties such as melting points, boiling points, density (H ₂ O: 18.0150 and D ₂ O: 20.0276; each at 25 ° C) , Dielectric constant, dissociation constant, ion product, but also in the electrolysis voltage ( redox voltage ) and the ion mobility (migration speed of the ions). Deuterium is the left of the hydrogen in the electrochemical voltage series (discharge potential -0.003 V based on the hydrogen). In addition, it should have a higher overvoltage than hydrogen. The same should apply to tritium. Therefore, deuterium oxide and tritium oxide are enriched during the electrolysis of water (especially at high current densities and low temperatures, since their overvoltage (that of deuterium and tritium) is then higher than that of hydrogen). Milazzo names hydroxide [OH] (-) and deuteroxide [OD] (-), a deuterated hydroxide ion, the limit conductivities for 25 ° C: 197.6 and 119 [Scm ² / mol]. If the limit conductivity of the sodium ion (50.11) is used to calculate the number of conversion of sodium into sodium hydroxide and sodium deuteroxide (deuterated sodium hydroxide) at 25 ° C, the result is: 50.11 / (50.11 + 197.6) = 0.2023 (Na in NaOH) and 50.11 / (50.11 + 119) = 0.2963 (Na in NaOD). In an ideally diluted solution of sodium hydroxide, the sodium ion thus contributes less to the transport of electricity than in sodium deuteroxide. Or to put it another way: The hydroxide ion migrates much faster than the heavier deuteroxide ion (as can be recognized immediately from the limit equivalent conductivities). This indicates a larger (hydrated) ionic radius of the deuteroxide ion compared to the hydroxide ion. According to Milazzo, the ions of isotopes of other elements (lithium, potassium, oxygen and chlorine are mentioned) with moderately higher atomic numbers also have electrochemical differences that can be used to separate the isotopes .

Additional discharge of H + and OH- ions to maintain the electrical neutrality of the solution

In the physics textbooks it is always stated that the total current results additively from the ion mobilities of the cation and anion (multiplied by the charge exchange number , Faraday constant , electrode surface , molar concentration and electric field strength ). But for all real cases in which the ion mobility of cation and anion differ, this cannot be true. This is because the same amount of charge must be absorbed by the discharged cations at the cathode every second as given off by the discharged anions at the anode. The electron current flowing to the cathode (physical current direction!) Is just as large as the electron current flowing away from the anode ( Kirchhoff's first rule ). Otherwise the principle of electroneutrality of the saline solution would be violated, as would Kirchhoff's first rule. ${\displaystyle z}$ ${\displaystyle F}$ ${\displaystyle A}$ ${\displaystyle c}$ ${\displaystyle E}$

Every salt solution always contains the same equivalents of positive charges (cations) as equivalents of negative charges (anions) and is therefore itself electronically neutral. This applies to every differentially small volume element.

There can only be one explanation for the practical solution to the problem: The lower partial current of the ion with the lower ion mobility v must be reduced to the level of the higher partial current (the ion with the higher ion mobility) due to the additional discharge of another ion - with the same polarity ) Are "increased". For this purpose, only the discharge of hydronium ions or the discharge of hydroxide ions are possible in an aqueous solution of a single salt . This would also correspond to the anodic oxidation of water molecules (to atomic oxygen and excess H + ions) or the cathodic reduction of water molecules (to atomic hydrogen and excess OH ions). In both cases the pH value changes constantly and the concentration of the water theoretically decreases.

In "Elektrochemie" (Milazzo, 1951, p. 151), Milazzo confirms that when precious metals are deposited from complex anions (in the example silver from dicyanoargentate anions), hydrogen is often deposited at the cathode at the same time, although silver Cations have a more positive (noble) deposition potential than hydrogen. In this example, the silver from the anions is to be deposited on the cathode together with hydrogen. This confirms the assumption that further ions would have to be discharged, in this case H ⁺ ions at the cathode. Complex anions have a much greater ion mobility than normal cations (here the silver ion). The silver cation to be discharged, which is only released from the complex anion in a two-stage process, therefore produces a significantly smaller partial flow (at the same concentration), which is separated by the additional discharge of hydronium ions (or the cathodic reduction of water) of hydrogen at the cathode is increased to the level of the higher partial flow of the anion discharge. Näser points out that ions can also be partially discharged during electrolysis (what is certainly meant is the partial discharge of multivalent ions).

Of course, this has to have an effect on the number of transports, which in practice will probably turn out differently when measured in the Hittorf test than theoretically calculated beforehand.

Field strength effect (Vienna effect)

At very high electrical field strengths (E = 100kV / cm and more) there is no longer any difference between ideal dilution ( ) and higher molar concentrations with regard to the practically ascertainable conductivities. At such high field strengths, the equivalent conductivity of the ions is the same at all concentrations, namely identical to the respective limit conductivity of the ion. (This does not apply to low field strengths. See chapter Conductivity coefficient .) ${\displaystyle c=0}$

With such high field strengths, the transfer numbers could therefore be calculated from the limit conductivities or ion mobilities for ideal dilution - even at high molar concentrations.

Dispersion effect (Debye-Falkenhagen effect)

At very high frequencies (above 1 MHz) of the alternating current used, the ion in question (central ion) no longer interacts with the surrounding ion cloud of "counter-ions". The central ion moves back and forth in the ion cloud at a high frequency. The counter-ions of the ion cloud, however, cannot carry out these rapid movements. The relaxation or asymmetry effect (see relaxation time ) does not apply. The central ion is no longer "slowed down" by its ion cloud at high frequencies.

Specific conductivity and equivalent conductivity and thus the current flow therefore assume a maximum value at such high frequencies. Theoretically, this also has an effect on the expected number of transfers, but in practice these cannot be determined with alternating current.

Values ​​are to be expected which at least correspond to those calculated from the limit conductivities , even at higher molar concentrations.

Negative conversion numbers of non-complex metal cations in solutions with complex anions of the same metal

Since the limit conductivity of complex ions can have much higher values ​​than that of non-complex “counter-ions”, the transfer number of non-complex ions in a solution with complex ions can even assume negative values. In this case too, the sum of all transfer numbers of the ions in the solution is equal to one.

Examples:

• During the formation of the tetraiodo-cadmat (II) anion when cadmium (II) iodide is dissolved in alkali iodide solution, the number of non-complex cadmium (II) cations transferred continues to decrease with increasing iodide concentration, so that they ultimately also become negative Values ​​(−2.5 at 5 mol / liter cadmium iodide concentration) were achieved.
• Even in concentrated solutions of zinc iodide (in alkali iodide solution) complex iodo-zincate anions are evidently present, because the conversion number of the non-complex zinc cation reaches negative values ​​over molalities of b = 3.5 mol / (kg solvent).

Conversion numbers in solid electrolytes, molten salts and colloidal solutions

Electrically conductive (non-melted) crystals - so-called solid electrolytes or crystal electrolytes - can be ionic and / or conductive through free electrons. Often only one type of ion contributes almost exclusively to conduction. Their transfer number is then one. In the case of ionic conduction and conduction by electrons, both effects must be taken into account when defining the transfer numbers. Milazzo gives a table of (unmelted) solid electrolytes with their specific conversion numbers for cations and anions. In some solid electrolytes, the value of the transfer numbers changes extremely as a function of temperature. For example, the cation transfer number for sodium chloride should drop from 1.00 to 0.12 between 557 ° C and 710 ° C. For the salts CuCL / CuBr / CuJ, below the following temperatures 300 ° C / 360 ° C / 390 ° C, in addition to ionic conductivity, there should also be electron conductivity, which becomes total electron conductivity below 18 ° C. Here is a quote from the literature mentioned (Chimica, GDR, 1972): “Real electrolytes (meaning salts) often show noticeable conductivity even in the solid state (as crystals), which increases with increasing temperature as a result of the loosening of the structure. If there are two types of ions with very different (crystalline) ion radii (crystal structure!), The smaller ions take care of practically all of the charge transport at lower temperatures (this means room temperature). The transfer number of this type of ion is then 1. This applies, for example, to the silver ion Ag + (n (Ag +) = 1) in silver iodide crystals AgI. "

In conductive aqueous colloids it is also possible to define transfer numbers for positively or negatively charged colloid particles. In colloids, the so-called isoelectric point denotes the concentration of a "foreign ion" (oppositely charged ion or colloid ion as the opposite pole to the actual colloid ion in the solution), when added - up to this value - the conductivity of the now changed colloid solution becomes zero. The colloid then usually disintegrates very quickly due to flocculation.

Determination of the transfer rates in molten salts

The determination of the conversion figures in molten salts is difficult because a molten salt (of a salt) has a certain constant density and thus constant molar concentration (or molar volume) at constant temperature. The transport of ions of this salt and their deposition on the electrodes do not change the constant molar concentrations. It is possible, but very expensive, to add the salt whose transfer numbers of its ions (under test conditions) are to be determined in a low concentration in a suitable "carrier salt" melt. The carrier salt must have a higher decomposition voltage or, ideally as a melt, be much less conductive than the salt to be determined. During electrolysis, there is then a decrease in the concentration of the salt to be determined in the molten carrier salt. The desired migration speeds of the salt (its ions) under test conditions can then be calculated from this, taking into account the conductivity / ion mobility of the carrier salt. For low-melting salts, aluminum chloride, aluminum bromide and aluminum iodide are suitable as carrier salts, since they are practically non-conductors until just above their melting point. These salts are molecularly present in the melt.

Determination of the conversion figures in solids according to the Tubandt method

Quotation: "Three crystals or pressed bodies (meaning cylindrical salt tablets pressed from salt powder with high pressure) are connected in series and placed between two electrodes (clamped). The change in mass of the two outer bodies (tablets) can result from the current flow, similar to the Hittorfschen Method by which the transfer figures are determined. " An ion exchange takes place here, which causes a very small change in mass and can be measured with a precision balance. The two outer salt tablets must consist of a different conductive salt, i.e. contain other ions, since only then can mass changes occur (?). A high-precision analytical balance with a microgram resolution is required. Since the conductivity of a solid salt is very low at room temperature, the experiment will take a long time.

Relationship between the transfer number and the temperature coefficient k of an ionic solution

For strong electrolytes (strongly dissociated substances) the following applies to the relationship between the temperature coefficient k of the conductivity of an ionic solution and the individual temperature coefficients of the ions i participating in the current transport: ${\displaystyle \alpha _{i}}$

${\displaystyle K^{(c,T0)}=n_{Kation}^{(c,T0)}*\alpha _{Kation}^{T0}+n_{Anion}^{(c,T0)}*\alpha _{Anion}^{T0}}$

The temperature coefficient k of the conductivity of an ionic solution (or a melt) is therefore composed proportionally according to the transfer numbers of the ions from their own (ionic) temperature coefficients . Since in melts sometimes only one ion determines the charge transport alone, the k value of the melt would then correspond in these cases to the alpha value of the ion (in the melt at the temperature of the melt!). ${\displaystyle \alpha _{i}}$

See: Ion mobility # The temperature dependence of the electrolytic conductivity . The temperature coefficients of the ions are listed in the table of ion mobility ( ion mobility # numerical values ). ${\displaystyle \alpha _{i}}$

Relationship between transfer number and diffusion coefficient of an ion

The diffusion coefficient of an ion i is defined according to Nernst as:

${\displaystyle D_{i}={\frac {R\cdot T\cdot v_{i}}{(z_{i}\cdot F)}}}$

${\displaystyle R}$ general gas constant , absolute temperature , ion mobility , valence of the ion, Faraday constant . Kat = cation, An = anion. (In the inorganic, the ion mobility is still mentioned , which would be wrong today!) ${\displaystyle T}$ ${\displaystyle v}$ ${\displaystyle z}$ ${\displaystyle F}$ ${\displaystyle v}$ ${\displaystyle u}$

With the equation of definition of the transfer number for any ion:

${\displaystyle n_{i}={\frac {u_{i}}{u_{\text{Kat}}+u_{\text{An}}}}={\frac {v_{i}}{v_{\text{Kat}}+v_{\text{An}}}}}$

(here are the migration speed and the ion mobility) ${\displaystyle u}$ ${\displaystyle v}$

the ion mobility of the ion in this equation can be replaced by the transfer number : ${\displaystyle v_{i}}$ ${\displaystyle i}$ ${\displaystyle n_{i}}$

${\displaystyle D_{i}={\frac {R\cdot T\cdot n_{i}\cdot (v_{\text{Kat}}+v_{\text{An}})}{(z_{i}\cdot F)}}}$

Thus there is a (formal) dependence of the ionic diffusion coefficient on the transfer number of the ion i. In the latter, by definition, the ion mobilities of all ions present have been incorporated, so that the diffusion coefficient becomes a function of the ion mobilities of all ions present in the electrolyte. ${\displaystyle D_{i}}$ ${\displaystyle n_{i}}$ ${\displaystyle D_{i}}$ ${\displaystyle v}$

Properties of ideal and non-ideal electrolyte

If a binary electrolyte has conversion numbers of almost 0.5 in aqueous solution, it can be regarded as an ideal electrolyte. Both types of ions now take about 50% of the current transport. This means that both types of ions in water must have largely the same hydrated ionic radii. If such an electrolyte is also soluble in non-aqueous solvents with a significantly different dielectric constant (polarization of the solvent than dielectric dipole ), then it is to be expected that the conversion numbers of the ions will not change or change only insignificantly.

However, if a binary electrolyte in aqueous solution has transfer numbers of both types of ions - cation and anion - that differ significantly from 0.5, i.e. if it is a non-ideal electrolyte, it is very likely that the transfer numbers will change when dissolving in another solvent (with a different dielectric constant) become.

While in the ideal electrolyte the hydration is similarly strong for both ions, in the non-ideal electrolyte it is clearly different, from which the constant or the change in the conversion numbers can be expected when the solvent is changed. However, tables usually only list conversion figures in aqueous solutions, which is why no example can be given here.

Table with measured transfer rates of aqueous solutions

The values ​​in the table below are measured values ​​from the book “Elektrochemie” (1952) by Milazzo (Tab. 3, pp. 30–31). The actually measured value is given here directly, the calculated value (n (+) + n (-) = 1) of the “counterion / counterions” is given here in brackets. Several "counterions" (opposite polarity) or "mitions" (same polarity) can be present in the case of acidic or basic dissociation of:

• Salting strong acids with weak bases or
• Salting of weak acids with strong bases.

This is certainly the case with the aforementioned acetates of potassium and sodium. Which is why hydroxide ions also determine the conductivity and total conversion numbers (which are not mentioned here). Silver acetate presumably dissociates acidic, so that hydronium ions are still present in the solution. is the effective valence (charge exchange number) of the ions in the table. ${\displaystyle z}$

electrolyte	formula	molecular equivalent concentration [mol / l]	Temperature [° C]	${\displaystyle z}$	n (+)	n (-)
Rubidium iodide	${\displaystyle {\ce {RbI}}}$	0.02	18th	1	(0.498)	0.502
Hydrobromic acid	${\displaystyle {\ce {HBr}}}$	0.1	25th	1	0.792	(0.208)
acetic acid	${\displaystyle {\ce {HAc}}}$ (Short name)	0.1-1	25th	1	(0.892)	0.108
Picric acid	${\displaystyle {\ce {HPi}}}$ (Short name)	0.1-1% by mass	without specification	1	0.910	(0.090)
sulfuric acid	${\displaystyle {\ce {H_2SO_4}}}$	0.02	without specification	2	(0.823)	0.177
Sodium bromide	${\displaystyle {\ce {NaBr}}}$	0.05	18th	1	(0.381)	0.619
Sodium hydroxide	${\displaystyle {\ce {NaOH}}}$	0.04	25th	1	(0,201)	0.799
Lithium hydroxide	${\displaystyle {\ce {LiOH}}}$	0.20	without specification	1	(0.152)	0.848
Potassium sulfate	${\displaystyle {\ce {K_2SO_4}}}$	0.008-0.018	25th	2	0.4829	(0.5171)
Copper sulfate	${\displaystyle {\ce {CuSO_4}}}$	0.053	16-19	2	0.375	(0.625)
Silver nitrate	${\displaystyle {\ce {AgNO_3}}}$	0.05	25th	1	0.4648	(0.5352)
Nickel sulfate	${\displaystyle {\ce {NiSO_4}}}$	0.1	40	2	0.366	(0.634)
Mercury (I) nitrate	${\displaystyle {\ce {[Hg]_2(NO_3)_2}}}$ ${\displaystyle {\ce {= [Hg-Hg](NO_3)_2}}}$	0.05	20th	2! (one electron per Hg)	0.480	(0.520)
Ammonium nitrate	${\displaystyle {\ce {NH_4NO_3}}}$	0.1	25th	1	(0.513)	0.487
Ammonium picrate	${\displaystyle {\ce {NH_4Pi}}}$ (Short name)	0.03-0.05	without specification	1	(0.708)	0.292
Potassium iodide	${\displaystyle {\ce {KI}}}$	0.8	20th	1	0.4896	(0.5104)
Lithium iodide	${\displaystyle {\ce {LiI}}}$	0.1	without specification	1	(0.318)	0.682
Silver acetate	${\displaystyle {\ce {AgAc}}}$ (Short name)	0.01	25th	1	(0.624)?	0.376
Lanthanum chloride	${\displaystyle {\ce {LaCl_3}}}$	0.01	25th	3	0.4625	(0.5375)
Zinc bromide	${\displaystyle {\ce {ZnBr_2}}}$	0.003-0.01	25th	2	0.402	(0.598)
Calcium sulfate	${\displaystyle {\ce {CaSO_4}}}$	0.0045	without specification	2	(0.441)	0.559
Thallium (I) chloride	${\displaystyle {\ce {TICl}}}$	0.01	22nd	1	(0.484)	0.516
Lead nitrate	${\displaystyle {\ce {Pb(NO_3)_2}}}$	0.03-0.1	25th	2	(0.487)	0.513
Hydrofluoric acid	${\displaystyle {\ce {HF}}}$	0.031	25th	1	(0.850)	0.150
Potassium acetate	${\displaystyle {\ce {KAc}}}$ (Short name)	0.02	14th	1	(0.668)?	0.332
Barium chloride	${\displaystyle {\ce {BaCl_2}}}$	0.001	25th	2	0.4444	(0.5556)
Uranyl nitrate	${\displaystyle {\ce {[UO_2](NO_3)_2}}}$	0.0024	25th	2	(0.19)	0.81

Practical use

In conductivity titration (conductometry) , the change in electrolytic conductivity due to consumption (precipitation of poorly soluble salts or formation of water from hydronium and hydroxide ions) and the addition (of the titrant) of various ions is decisive. If, for example, hydrochloric acid is introduced and sodium hydroxide solution is added (titrated), a V-shaped curve is shown in the diagram showing the electrolytic conductivity on the y-axis and the titrated volume of sodium hydroxide on the x-axis. A falling straight line intersects a rising straight line. The point of intersection (curve minimum) is close to the equivalence point . The falling straight line stands for the decreasing conductivity due to the decrease in the hydronium ion concentration (these form water with the OH ions in the added lye). The rising straight line shows the increasing conductivity due to the addition of "excess" hydroxide ions - the solution now becomes basic.

On the historical development of electrochemistry and transfer numbers

• 1797–1798, Johann Wilhelm Ritter , first galvanic experiments (with zinc-copper primary battery ), is therefore considered the founder of electrochemistry
• 1799 or 1798–1800, Alessandro Volta , based on Luigi Galvani's experiments, he builds the first technically useful primary battery " voltaic column " (silver-zinc primary battery) with voltages of up to 100 volts , which is later used by many researchers for electrical experiments
• 1800, Johann Wilhelm Ritter , electrolytic decomposition of water by means of a primary battery, later he recognizes the effect of the reduction and oxidation of the electrochemical process and arranges metals in an electrochemical series for the first time
• 1802–1803, Jöns Jakob Berzelius and Wilhelm von Hisinger jointly investigate the effect of electric current on salt solutions
• 1807–1808, Humphry Davy , first fused metal electrolysis of salts, first extraction of the metals sodium, potassium, calcium, strontium and barium
• 1826, Georg Simon Ohm , discovery of Ohm's law
• from 1832, Michael Faraday , he invents the terms cation , anion , ion , cathode , anode , electrolyte and electrolysis in his electrochemical experiments , but mainly deals with the processes at the electrodes
• 1833–1834, Michael Faraday determines the electrochemical equivalents and finds Faraday's First Law
• 1846, Johann Wilhelm Hittorf , basic idea of ​​ion migration
• 1853-1859, or 1853 Johann Wilhelm Hittorf , migration of ions in the electrolyte, determination of the migration speed and definition and determination of the transfer numbers by the Hittorf experiment, his results were initially hardly noticed, popularized only in 1891 in the "textbook of general chemistry" by Wilhelm Ostwald
• 1858–1859, Julius Plücker , discovery of cathode rays and their deflection by magnets
• after 1859 ?, William Crookes , assumption that cathode rays are charged particles (ions)
• Year ?, after Hittorf, William Crookes came to similar results with regard to the number of shipments independently of Hittorf
• before 1866, August Wilhelm von Hofmann , invention of Hofmann's water decomposition apparatus
• 1873, Friedrich Wilhelm Kohlrausch , Law of Independent Ion Migration (the equivalent conductivity of a salt is made up of that of the cation and that of the anion)
• Year ?, Friedrich Wilhelm Kohlrausch , introduces the terms ion mobility , equivalent conductivity and equivalent concentration a
• from 1875, Friedrich Wilhelm Kohlrausch , empirically finds Kohlrausch's square root law for the dependence of the equivalent conductivity on the concentration, construction of the first conductometer for measuring the conductivity of liquids, calculation rule for determining the migration speeds of ions from the limit conductivity
• 1884, Svante August Arrhenius , research on the conductivity of acids, definition of the degree of dissociation still valid today
• 1886, Oliver Lodge , developed the moving boundary method to determine the speed of migration of colored (?) Ions
• 1887, Svante August Arrhenius , publication of his dissociation theory "About the dissociation of substances dissolved in water"
• 1887, Paul Walden , discovery of the Waldens rule for the equivalent conductivity of an ion in different solvents
• 1888, Wilhelm Ostwald , finds Ostwald's law of dilution , the influence of the degree of dissociation in weak electrolytes on the equilibrium concentration that is established
• 1888, Walther Nernst , research on the transport of electricity in liquids, publication "The electromotive effectiveness of ions"
• 1893, William Cecil Dampier , with / further development of the moving boundary method for determining the speed of migration of colored (?) Ions
• 1900, Walther Nernst proposed as the origin of the electrochemical series the potential of the normal hydrogen electrode ago
• Jahr ?, Walther Nernst determines the speed of migration of colored ions by observing a migrating interface between two different salt solutions in an electric field
• 1923–1924, Duncan Arthur MacInnes (1885–1965) - Author of: "The Principles of Electrochemistry" - and Edgar Reynolds Smith - Author of: "A Method for Determining the Change in Transference Number of a Salt with Change in Concentration: A Modification of the Moving Boundary Method "-, improve Nernst's method for determining the migration speed of ions, now named after Mac Innes and also applicable to colorless ions
• Year ?, Walther Nernst is the first person to determine the true transfer figures by means of an experiment according to Hittorf with the addition of urea or sugar (non-electrolyte) from the changes in the concentration of the non-electrolyte in the cathode and anode compartment
• 1921, GN Lewis and M. Randall define the ionic strength as a measure of the effective concentration of solutions with polyvalent ions and confirm Kohlrausch's empirical square root law (concentration replaced by ionic strength) also for solutions with polyvalent ions
• 1922–1925, Peter Debye and Erich Hückel develop the Debye-Hückel theory for conductivity in real-concentrated solutions using activity
• 1927, Peter Debye , Erich Hückel and Lars Onsager , Debye-Hückel-Onsager law for the equivalent conductivity of ions

literature

• Tables with conversion numbers of different aqueous electrolytes at different concentrations and temperatures and for solid electrolytes (non-molten salts) as well as different methods for determining the conversion numbers. In: Giulio Milazzo: Electrochemistry-Theoretical Basics and Applications. Springer-Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , pp. 21-33.
• Example for the determination of the transfer number and negative transfer number for complex ions. In: Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , pp. 340–341.
• GA Lonergan, DC Pepper: Transport numbers and ionic mobilities by the moving boundary method. In: Journal of Chemical Education. 42 (2), February 1965, p. 82, doi: 10.1021 / ed042p82

Individual evidence

1. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstoffindustrie Leipzig, 1972, transfer figures with symbol “t” on p. 140.
2. ↑ Udo R. Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. Thieme Verlag, Stuttgart 1996, ISBN 3-13-585804-9 , p. 268.
3. ↑ Udo R. Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. Thieme Verlag, Stuttgart 1996, ISBN 3-13-585804-9 , p. 268.
4. ↑ Lewis Longworth, Theodore Shedlovsky: Duncan Arthure Macinnes. A Biographical Memoir. P. 302.
5. ↑ Udo R. Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. Thieme Verlag, Stuttgart 1996, ISBN 3-13-585804-9 , p. 265.
6. ↑ Karl-Heinz Näser, Gerd Peschel: Physico-chemical measuring methods. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00371-5 , p. 62.
7. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstoffindindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 370 u. 374
8. ^ Lothar Kolditz: Inorganikum. VEB Deutscher Verlag der Wissenschaften, Berlin, GDR, 1977. Chapter "Mobility, transfer number and chains with transfer", p. 328.
9. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , pp. 371, 374.
10. ↑ Hübschmann: Tables for Chemistry, Verlag Handwerk und Technik Hamburg 1991, p. 61.
11. ↑ Milazzo: Elektrochemie, Springer Verlag, Vienna 1951, p. 36.
12. ↑ Milazzo: Elektrochemie, Springer Verlag, Vienna 1951, p. 30.
13. ↑ Hübschmann: Tables for Chemistry, Verlag Handwerk und Technik Hamburg 1991, p. 62.
14. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 371.
15. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstofftindustrie Leipzig, 1972, p. 140.
16. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 28.
17. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 340.
18. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstoffindindustrie Leipzig, 1972, item 8.1.8, p. 140.
19. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 341.
20. ^ Brockhaus ABC chemistry. Brockhaus Verlag Leipzig, GDR, 1965, p. 543, entry: Hittorfsche transfer figures
21. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstoffindindustrie Leipzig, 1972, section 8.17 on p. 140.
22. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , Gl. 7.88, p. 339, expanded by the quotient of the equivalent conductivities.
23. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstoffindindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , example on p. 339.
24. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 27.
25. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 46.
26. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 27.
27. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstofftindustrie Leipzig, 1972, transfer figures p. 140.
28. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 . Chapter Concentration Elements, p. 100.
29. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 374.
30. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstoffindindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 374, equation below Eq. 7.145
31. ^ Lothar Kolditz: Inorganikum. VEB Verlag Deutscher Wissenschaften Berlin, GDR 1985, p. 328.
32. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstoffindindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 370, level 7.137.
33. ↑ Hans Keune: chimica, a knowledge store. VEB German publishing house for primary industry Leipzig, 1972, p. 155, Gl. 8.71b
34. ↑ Lothar Kolditz: Inorganikum, VEB Verlag Deutscher Wissenschaften Berlin, GDR 1985, diffusion stresses and transfer numbers, p. 331, Eq. 481.
35. ↑ Lothar Kolditz: Anorganikum, VEB Verlag Deutscher Wissenschaften Berlin, GDR 1985, diffusion stresses and transfer numbers, pp. 328–331.
36. ↑ ABC Chemie , Brockhaus-Verlag Leipzig, GDR, 1965, keyword diffusion potential. P. 294.
37. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 25, Jahnscher apparatus with mercury cathode.
38. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 . Description of the Mc Innes method with an illustration of an apparatus, pp. 26–28.
39. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 . Tab. 2, p. 28., comparison of the accuracies according to Hittorf and Mac Innes
40. ^ Giulio Milazzo: Elektrochemie, Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 42.
41. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 339.
42. ^ Giulio Milazzo: Elektrochemie, Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 . Table with transfer figures for aqueous salt solutions, p. 30.
43. ↑ Udo R. Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. 4th edition. Thieme Verlag, Stuttgart, 1996, ISBN 3-13-585804-9 , pp. 320-322 and p. 47. Table and calculation of individual activity coefficients fi.
44. ^ Giulio Milazzo: Elektrochemie, Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 29.
45. ^ ^{A b c} Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 32 (books.google.de)
46. ^ ABC Chemie , Brockhaus-Verlag Leipzig, GDR, 1965, keyword ammonium tetrachlorozincate .
47. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 47 with table no.10.
48. ^ ^{A b c} Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 29.
49. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 339.
50. ↑ Udo R. Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. 4th edition. Thieme Verlag, Stuttgart, 1996, ISBN 3-13-585804-9 , p. 47 and p. 270.
51. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB German publishing house for primary industry Leipzig, 1972, p. 148, Gl.8.57.
52. ^ Lothar Kolditz: Inorganikum. Volume 1, VEB Deutscher Verlag der Wissenschaften, Berlin, GDR, 1985, p. 342.
53. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 28.
54. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 340.
55. ^ ^{A b} Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , pp. 333, 334.
56. ↑ Ulrich Hübschmann, Erwin Links: Tables on chemistry. Verlag Handwerk und Technik Hamburg, 1991, limit conductivity values ​​and their temperature coefficients alpha, p. 62.
57. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstoffindindustrie Leipzig, 1972, Chapter Dependence of the conductivity of the electrolytes on the temperature, Eq. 8.23, p. 141.
58. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstoffindindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , entry: Temperature coefficient of electrolytic conductivity.
59. ↑ Ulrich Hübschmann, Erwin Links: Tables on chemistry. Verlag Handwerk und Technik Hamburg, 1991, limit conductivity values ​​and their temperature coefficients alpha, p. 62.
60. ^ Giulio Milazzo: Electrochemistry. Springer-Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 21.
61. ↑ ABC Chemie , Brockhaus-Verlag Leipzig, GDR, 1965, calls the ionic radius an "apparent ionic radius" because it is difficult to determine and depends on the type of definition / method used.
62. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstoffindindustrie Leipzig, 1972, Chapter Dependence of the conductivity of the electrolytes on the temperature, Eq. 8.8, 8.9 and 8.10, p. 139.
63. ^ Hollemann-Wiberg: Textbook of inorganic chemistry . 90th edition.
64. ↑ ^{a b} Ulrich Hübschmann, Erwin Links: Tables on chemistry. P. 62, Limit equivalent conductivities of ions.
65. ↑ Giulio Mlazzo: electrochemistry. Springer-Verlag 1952, ISBN 978-3-211-80268-7 , p. 55, table of the equivalent conductivity of some melts.
66. ^ Hollemann-Wiberg: Textbook of inorganic chemistry. 90th edition, p. 99; Quote: "With isoelectronic ions, as expected, the ion radius increases from the cations to the anions, namely with the cations with a decreasing number of charges (valence), with the anions with an increasing number of charges (valence)".
67. ^ Giulio Milazzo: Electrochemistry. Springer-Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 42.
68. ↑ Udo R. Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. 4th edition. Thieme Verlag, Stuttgart, 1996, ISBN 3-13-585804-9 . Chapter: "Polarization methods" with the limit current on pp. 285–286.
69. ↑ Udo R. Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. 4th edition. Thieme Verlag, Stuttgart, 1996, ISBN 3-13-585804-9 , Formula (188), p. 191.
70. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , pp. 148, 137, 153, overvoltages of metals, oxygen and hydrogen.
71. ^ Giulio Milazzo: Electrochemistry. Springer-Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , pp. 145–150.
72. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB German publishing house for basic industry Leipzig, 1972, production of hydrogen peroxide on p. 253.
73. ↑ Hollemann, Wiberg: Textbook of Inorganic Chemistry, 90th Edition, pp. 202-207.
74. ^ Giulio Milazzo: Electrochemistry. Springer-Verlag, Vienna 1952, ISBN 978-3-211-80268-7 . Chapter Depolarization, pp. 161-163.
75. ↑ ABC Chemie , Brockhaus-Verlag Leipzig, GDR, 1965, entries: “Zwitterionen”, “Amino Acids”, “Betaine”, “Hydrochloride”.
76. ↑ Udo R. Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. 4th edition. Thieme Verlag, Stuttgart, 1996, ISBN 3-13-585804-9 . Table 10, p. 95, dissociation numbers (and dielectric constants) of various solvents.
77. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 109, electrochemical series of metals / cations.
78. ^ ABC Chemie , Brockhaus-Verlag Leipzig, GDR, 1965, keyword overvoltage .
79. ^ ABC Chemie , Brockhaus-Verlag Leipzig, GDR, 1965, keyword tritium .
80. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 43, table of the equivalent conductivity of ions.
81. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 306, table of the electrolytic separation factors of some isotopes.
82. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 151.
83. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstoffindindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , section: Electrolysis, p. 391.
84. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstoffindindustrie Leipzig, 1972, field strength effect p. 148.
85. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstofftindustrie Leipzig, 1972, Dispersionseffekt p. 148.
86. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , pp. 340–341.
87. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstofftindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 341.
88. ^ Giulio Milazzo: Electrochemistry. Springer Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , p. 33.
89. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstoffindindustrie Leipzig, 1972, chapter conductivity of melts and solid electrolytes, p. 141.
90. ^ Giulio Milazzo: Electrochemistry. Springer-Verlag, Vienna 1952, ISBN 978-3-211-80268-7 , pp. 383-385.
91. ^ ABC chemistry. Brockhaus-Verlag, GDR, 1965, keyword aluminum halides .
92. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstoffindindustrie Leipzig, 1972, chapter conductivity of melts and solid electrolytes, p. 141.
93. ^ Lothar Kolditz: Inorganikum. VEB Verlag der Wissenschaften Berlin, GDR, 1980, p. 341.
94. ↑ Udo R. Kunze, Georg Schwedt: Fundamentals of qualitative and quantitative analysis. 4th edition. Thieme Verlag, Stuttgart, 1996, ISBN 3-13-585804-9 , p. 272, Fig. No. 55, conductivity titration diagram.
95. ↑ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, p. 1169.
96. ^ Brockhaus Lexicon "ABC researcher and inventor", Brockhaus Verlag Leipzig, GDR, 1990, pp. 492–493, entry: JW Ritter
97. ^ Brockhaus Lexicon "ABC researcher and inventor", Brockhaus Verlag Leipzig, GDR, 1990, pp. 584-585, entry: Alessandro Volta
98. ↑ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, p. 1169.
99. ↑ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, p. 1169.
100. ↑ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, p. 1129.
101. ^ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, p. 1170.
102. ↑ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, pp. 1131-1132.
103. ^ Brockhaus Lexicon "ABC Researcher and Inventor", Brockhaus Verlag Leipzig, GDR, 1990, p.?, Entry: Michael Faraday
104. ^ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, p. 1170.
105. ^ Brockhaus Lexikon "ABC researcher and inventor", Brockhaus Verlag Leipzig, GDR, 1990, p. 280, entry: Johann Wilhelm Hittorf
106. ^ Brockhaus Lexikon "ABC researcher and inventor", Brockhaus Verlag Leipzig, GDR, 1990, p. 280, entry: Johann Wilhelm Hittorf
107. ↑ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, p. 1139.
108. ^ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, p. 1128.
109. ↑ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, p. 1154.
110. ^ Hollemann, Wiberg: Textbook of Inorganic Chemistry, Edition 90, p. 1128.
111. ^ Brockhaus Lexicon "ABC researcher and inventor", Brockhaus Verlag Leipzig, GDR, 1990, p. 280, comments on William Crookes under entry: Johann Wilhelm Hittorf
112. ^ Brockhaus ABC chemistry. Brockhaus Verlag Leipzig, GDR, 1965, p. 470.
113. ^ Brockhaus Lexicon "ABC Researcher and Inventor", Brockhaus Verlag Leipzig, GDR, 1990, pp. 29–30, entry: Svante August Arrhenius
114. ^ Laidler, Keith James and Meiser, JH, "Physical Chemistry" (Benjamin / Cummings 1982) pp. 276-280, ISBN 0-8053-5682-7 .
115. ^ Brockhaus Lexicon "ABC Researcher and Inventor", Brockhaus Verlag Leipzig, GDR, 1990, pp. 29–30, entry: Svante August Arrhenius
116. ^ Brockhaus Lexicon "ABC Researcher and Inventor", Brockhaus Verlag Leipzig, GDR, 1990, pp. 426–427, entry: Walther Nernst
117. ^ Laidler, Keith James and Meiser, JH, "Physical Chemistry" (Benjamin / Cummings 1982) pp. 276-280, ISBN 0-8053-5682-7 .
118. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstofftindustrie Leipzig, 1972, transfer figures p. 140.
119. ↑ Lewis Longworth, Theodore Shedlovsky: Duncan Arthure Macinnes. A Biographical Memoir. P. 310, entry on Edgar Reynolds Smith
120. ^ Karl-Heinz Näser, Dieter Lempe, Otfried Regen: Physical chemistry for technicians and engineers. Verlag für Grundstoffindindustrie Leipzig, GDR, 1990, ISBN 3-342-00545-9 , p. 340 ?, entry on Smith
121. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB Deutscher Verlag für Grundstofftindustrie Leipzig, 1972, transfer figures p. 140.
122. ^ Brockhaus ABC chemistry. Brockhaus Verlag Leipzig, GDR, 1965, p. 543, entry: Hittorfsche transfer figures
123. ↑ Hans Keune: chimica, a knowledge store. Volume II, VEB German publishing house for primary industry Leipzig, 1972, p. 146, Gl.8.46.