Activity (chemistry)

The activity is a thermodynamic quantity of the dimension number , which is used in physical chemistry instead of the substance concentration . With it, the same principles apply to real mixtures (with regard to physical measurement parameters such as freezing and boiling temperature, conductivity of electrolyte solutions, electrical voltage, vapor pressure of solvents) as with the concentration for ideal mixtures. ${\displaystyle a}$

The activity coefficient is the ratio of the activity to the mole fraction with the unit one . For ideal behavior that is observed at infinite dilution, it is 1 and deviates with increasing concentration. The activity coefficients can also be defined for molarities and molalities . The activity of a species is related to the chemical composition of the rest of the system (e.g. via the ionic strength ). ${\displaystyle \gamma }$

introduction

Activities can be used to precisely describe electrolyte solutions (conductivities), vapor pressures of a solvent system, gas quantities in liquids, freezing point and boiling point changes, osmotic pressures and also changes in volume and temperature changes in solvent mixtures. The actual concentration of the added substance results in a disproportionate change in the physical measured variable (e.g. conductivity), so that the product of the activity coefficient and concentration indicates the concentration (activity) that is correct for the measured value observed. The deviations regarding the measured values ​​are based on Coulomb interactions of salt ions, hydrogen bonds or van der Waals interactions in solvent systems. The word activity coefficient was first used by Svante Arrhenius to describe dissociation exactly .

In electrolyte systems, the respective activity coefficient can be calculated based on the charges of individual ions or determined very quickly by conductivity measurements. The activity coefficient in electrolyte solutions is almost always less than 1 and converges to 1 at very high dilution. In the case of very concentrated electrolyte solutions, the activity coefficient can also be greater than 1. It is not possible to determine an individual activity coefficient for one type of ion, since cations and anions are present together in the electrolyte. An average activity coefficient is therefore given for electrolyte solutions, which indicates the physical deviations of the two ions. Ions in water and other solvents attach to other ions with opposite charges. This shields the ions and leads to interactions, the stronger the higher the concentration and the higher the charge of the ion in question. In very high dilution (0.00001 molar) these interactions disappear. However, the activity coefficients of the Debye-Hückel theory must not be used to determine the equivalent conductivity from the limit conductivity of salts, acids and bases . The latter coefficients apply to the law of mass action and to the solubility product, the dependencies of the conductivity are described using Kohlrausch's law.

Even with mixtures of several solvents, there can be interactions that manifest themselves in changes in volume, temperature or vapor pressure. For a pure solvent, the vapor pressure can be precisely determined at a certain temperature. Therefore, with ideal behavior, the corresponding gas quantities of the molar fractions should also be found in a solvent mixture in the vapor phase according to their proportions in the solvent mixture ( Dalton's law ). In the case of many real solvent mixtures, however, a non-ideal behavior is observed. Van der Waals interactions can lead to higher or lower vapor pressures of the individual components. In this case, too, activity coefficients are used to describe the real situation. The activity coefficient can also be greater than 1. In the case of fractional distillation of mixtures of substances, knowledge of the activity coefficients or fugacity coefficients (at gas pressures) is therefore particularly important because the component enriched in the gas phase can be determined so precisely.

Even in the reverse case of the solution of a gas in a liquid, the gas solubility is often approximately proportional to the gas pressure ( Henry's law ). In this case, too, there are deviations from the ideal behavior, so the activity coefficient is used. In addition, when mixing solvents or when adding salts to solvents, changes in volume and temperature can occur which are also not proportional to the concentration of the added substance.

Ideal mixtures are characterized by the fact that no mixing effects occur, i.e. the volume (at constant pressure ) and the temperature do not change when the pure components are mixed and thermodynamically important variables such as enthalpy or heat capacity are multiplied by the values ​​of the individual components the respective mole fractions composed (mole fractions). This is only the case if the components are chemically very similar, so that the interactions between particles of different types are the same as the interactions between the same particles. Usually this will not be the case, so you are dealing with real rather than ideal mixes. In real mixtures, the thermodynamic quantities volume, internal energy and enthalpy are no longer put together additively from the values ​​of the individual components weighted with the proportions of the amount of substance. In order to be able to apply the equations for describing ideal mixtures to real mixtures as well, the activity is introduced, which is defined in such a way that when the mole fractions are exchanged by the activities, the additive relationship of the thermodynamic variables for the mixture of the corresponding variables for the individual components is also given for real mixtures. In ideal mixtures, the activity of a component corresponds exactly to the amount of substance of the component; in real mixtures, the two quantities will usually differ from one another.

definition

The absolute activity is defined by:

${\displaystyle \lambda =\exp \left({\frac {\mu }{RT}}\right)}$.

The relative activity is defined as the quotient of the absolute activities of the state and the reference state: ${\displaystyle a}$

${\displaystyle a={\frac {\lambda }{\lambda ^{o}}}=\exp \left({\frac {\mu -\mu ^{o}}{RT}}\right)}$

Here stands for the absolute temperature, for the chemical potential and for the chemical potential of the reference state, typically the standard state (see standard conditions ). is the general gas constant . ${\displaystyle T}$ ${\displaystyle \mu }$ ${\displaystyle \mu ^{o}}$ ${\displaystyle R}$

When using the term activity, it should always be stated whether the relative or absolute activity is meant.

In an ideal mixture, the chemical potential of the components results from the equation

${\displaystyle \mu _{i}(T,p)=\mu _{i}^{o}+RT\ln(x_{i})}$,

where the mole fraction (mole fraction) of component i is. ${\displaystyle x_{i}}$

A correction must be made for a real mix:

${\displaystyle \mu _{i}(T,p)=\mu _{i}^{o}+RT\ln(a_{i})}$

The activity is related to the mole fraction by the following relationship:

${\displaystyle a_{i}=\gamma _{i}\cdot x_{i}}$

The dimensionless factor is called the activity coefficient. While in an ideal mixture there are no intermolecular interactions whatsoever and thus the chemical potentials and all associated quantities can be traced back to the molar proportions of the components, in a real mixture there are interactions between the particles. These interactions can e.g. B. be electrostatic. The activity coefficient describes precisely these deviations of the mixture from ideal behavior. It was introduced by Gilbert Newton Lewis in 1907 as a purely empirical quantity for strong electrolytes . Svante Arrhenius and Jacobus Henricus van't Hoff used the expression activity coefficient somewhat earlier for the degree of dissociation in the lowering of the freezing point by salts. ${\displaystyle \gamma }$

One approach is to choose very small amounts of fabric. This leads (of course only in theory) to the following considerations / results:

• few particles means a large distance between the particles and thus no interactions
• interactions with the vessel walls can be neglected

In the case of solutions, it is still assumed that there is no significant interaction between the solvent and the dissolved substance. Mathematically, the approximations can be expressed as follows:

${\displaystyle a_{i}=x_{i}}$ (general)

${\displaystyle a_{i}={\frac {c_{i}}{c^{o}}}}$ (in solutions)

${\displaystyle a_{i}={\frac {p_{i}}{p^{o}}}}$ (in gas mixtures)

The approximation for solutions is relatively good for concentrations in the range of approx. , Whereby in practice the measurement method used also matters. The divisor follows from the dimension of the chemical potential: If it is to be dimensionless, the corresponding concentration must be divided by a concentration. The values ​​for c ° and p ° are arbitrary, generally accepted and recommended by IUPAC since 1982 are the standard concentration and the standard pressure . To go back to real mixtures, the required terms are simply multiplied by the activity coefficient. ${\displaystyle 10^{-3}\,\mathrm {mol} /\mathrm {l} }$ ${\displaystyle a}$ ${\displaystyle c^{o}=1\,\mathrm {mol} /\mathrm {l} }$ ${\displaystyle p^{o}=100000\,\mathrm {Pa} }$

Activity of a solvent

In solutions, the activity of a substance is defined via ${\displaystyle a_{B}}$ ${\displaystyle B}$

${\displaystyle a_{B}={\frac {p_{B}}{p_{B}^{*}}}}$

where is the vapor pressure of the pure solvent (which is chosen as the standard fugacity) , the vapor pressure of the solvent in the solution (actually the fugacity). In this case, a is to be seen analogously to the mole fraction : This is the mole fraction, which is decisive when considering the system. The Raoult's law is even more solvents, the closer the molar fraction approaches 1, d. H. the purer the solvent is. In order to clearly describe the behavior, a correction factor was also introduced here, which is now defined as follows: ${\displaystyle p_{B}^{*}}$ ${\displaystyle p_{B}}$ ${\displaystyle a_{B}}$

${\displaystyle a_{B}=\gamma _{B}x_{B}}$ with for ${\displaystyle \gamma _{B}\rightarrow 1}$ ${\displaystyle x_{B}\rightarrow 1}$

This relationship can be derived from the chemical potential and Raoult's law.

Marginal activity coefficient

The limit activity coefficient is the activity coefficient of a component at infinite dilution ( ) in a solvent or solvent mixture . ${\displaystyle \gamma ^{\infty }}$ ${\displaystyle x\rightarrow 0}$

Activity of ions and electrolytes

The determination of the activities was based on the dissociation theory of electrolytes. Van't Hoff determined the decrease in vapor pressure, osmotic pressures, increase in boiling point and lowering of the freezing point of electrolyte solutions and was able to determine considerable deviations from the determination of the molecular weight in very dilute salt solutions. Conductivity measurements revealed almost identical deviations as with the lowering of the freezing point. Activity coefficients were later also determined by potential measurements. There are slight deviations between the specific activity coefficients between the various methods, so that information about the type of detection should not be missing. The determination of activity coefficients in aqueous solutions has become very simple and uncomplicated today thanks to very inexpensive electronic measuring devices.

Around 1920, however, it was a very arduous undertaking for chemists to determine the activity coefficients from electrolyte solutions. Debye and Hückel ( Debye-Hückel theory ) were able to present a theory on the behavior of electrolytes in the electrolyte solution in 1923, which, however, only delivers reliable values ​​at low ionic strengths.

For dilute electrolytes, the activity coefficient can be calculated from the Debye-Hückel theory. However, this purely mathematical method is only halfway accurate for solutions with ionic strengths below 0.01 mol / l. It should be noted that the definition equation given above applies to each individual type of ion individually ${\displaystyle i}$

${\displaystyle a_{i}=\gamma _{i}\cdot c_{i}}$

At higher ionic strengths (I> 0.1 M), the Davies equation can be used to determine the activity coefficient: at I = 0.5 M, however, the relative error of the Davies equation is already 10%. No good approximations are known for even higher ionic strengths and gE models have to be used to obtain reliable values ​​for the activity coefficients .

Since every solution has to be electroneutral, i.e. it has to have the same number of positive and negative charges, individual activities could be calculated usefully, but not measured. This prevailing opinion goes back to Edward Guggenheim in the 1920s . However, this would not, for example, fully define the pH value, which describes the activity of hydrogen ions in water. Recently, some authors report to have measured individual ion activities experimentally. Before individual activity coefficients could be measured, the concept of mean activity coefficients was used.

Mean activity coefficient

For one according to the equation

${\displaystyle A_{\nu _{+}}B_{\nu _{-}}\longrightarrow \nu _{+}A^{z_{+}}+\nu _{-}B^{z_{-}}}$

${\displaystyle \nu }$ stoichiometric coefficients

${\displaystyle z}$ Charge numbers (negative for anions)

completely dissociating electrolytes, if we use the indices "+" for the cations and "-" for the anions , ${\displaystyle A^{z_{+}}}$ ${\displaystyle B^{z_{-}}}$

${\displaystyle a\left(A_{\nu _{+}}B_{\nu _{-}}\right)=a_{+}^{\nu _{+}}\cdot a_{-}^{\nu _{-}}=\gamma _{+}^{\nu _{+}}c_{+}^{\nu _{+}}\cdot \gamma _{-}^{\nu _{-}}c_{-}^{\nu _{-}}=\gamma _{+}^{\nu _{+}}\gamma _{-}^{\nu _{-}}\cdot c_{+}^{\nu _{+}}c_{-}^{\nu _{-}}}$

To simplify this equation, the mean activity

${\displaystyle a_{\pm }=a\left(A_{\nu _{+}}B_{\nu _{-}}\right)^{\frac {1}{\nu _{+}+\nu _{-}}}}$

the factor taking into account the stoichiometric coefficients

${\displaystyle Q=\left(\nu _{+}^{\nu _{+}}\,\cdot \,\nu _{-}^{\nu _{-}}\right)^{\frac {1}{\nu _{+}+\nu _{-}}}}$

as well as the z. B. mean activity coefficient measurable in galvanic cells

${\displaystyle \gamma _{\pm }=\left(\gamma _{+}^{\nu _{+}}\cdot \gamma _{-}^{\nu _{-}}\right)^{\frac {1}{\nu _{+}+\nu _{-}}}}$

Are defined. The concentration of the electrolyte is then also used for the mean activity of the electrolyte ${\displaystyle c\left(A_{\nu _{+}}B_{\nu _{-}}\right)}$

${\displaystyle a_{\pm }=Q\,\cdot \,\gamma _{\pm }\,\cdot \,c\left(A_{\nu _{+}}B_{\nu _{-}}\right)}$

Examples:

For an aqueous solution containing 10 ⁻³ mol · dm ⁻³ each of potassium chloride (KCl) and sodium sulfate (Na ₂ SO ₄ ), the Debye-Hückel limit law is used. It is so ${\displaystyle \gamma _{\pm }\left(\mathrm {KCl} \right)=0{,}929}$ ${\displaystyle Q=1}$

${\displaystyle a_{\pm }\left(\mathrm {KCl} \right)=0{,}929\cdot 10^{-3}\,\mathrm {mol\cdot dm^{-3}} }$

For a 0.1 molar magnesium chloride solution (MgCl ₂ ) is

${\displaystyle \gamma _{\pm }\left(\mathrm {MgCl_{2}} \right)=0{,}577}$. With results ${\displaystyle Q=\left(1^{1}\cdot 2^{2}\right)^{\frac {1}{1+2}}={\sqrt[{3}]{4}}}$

${\displaystyle a_{\pm }\left(\mathrm {MgCl_{2}} \right)=0{,}916\cdot 10^{-3}\,\mathrm {mol\cdot dm^{-3}} }$.

Activity coefficient

Activity coefficient regression with UNIQUAC ( chloroform / methanol mixture)

Engineering area

In the technical field, so-called models such as NRTL (Non-Random-Two-Liquid), UNIQUAC (Universal Quasichemical) and UNIFAC (Universal Quasichemical Functional Group Activity Coefficients) are used to estimate the activity coefficient. ${\displaystyle g^{E}}$

Electrochemistry

According to the interpretation of Debye and Hückel is the electrostatic potential energy that arises when 1 mole of ion type A is charged from the fictitious uncharged state to its real amount of charge within its oppositely charged ion cloud. This potential energy is therefore negative (“You have to put in energy to remove ion type A from your ion cloud”). ${\displaystyle RT\cdot \ln \gamma _{A}}$

Debye-Hückel: ${\displaystyle W=RT\cdot \ln \gamma _{A}}$

Therein is the universal gas constant and the temperature. W is negative; thus, for systems that are described by the Debye-Hückel approximation, between 0 and 1. The Debye-Hückel theory requires a strong dilution of A. In general, however, activity coefficients greater than one are also possible for any system. ${\displaystyle R}$ ${\displaystyle T}$ ${\displaystyle \gamma }$

Experimental measurements revealed a relationship between the ionic concentration of A and its activity coefficient. For large concentrations of A becomes greater than one (“I gain energy when I remove A”). Large ion concentrations of A are not taken into account by the Debye-Hückel theory. ${\displaystyle \gamma }$

For example, for a 1 molar acetic acid is  = 0.8 and 0.1 molar acetic acid 0.96. ${\displaystyle \gamma }$

One method for estimating activity coefficients is regular solution theory . The activity coefficient was written a = f × c. Here, c is the concentration, f is the activity coefficient and a is the activity.

literature

• Gustav Kortüm: Textbook of Electrochemistry , 5th Edition, Verlag Chemie GmbH, Weinheim 1972.
• AF Holleman , E. Wiberg , N. Wiberg : Textbook of Inorganic Chemistry . 81–90. Edition. Walter de Gruyter, Berlin 1976, ISBN 3-11-005962-2 , pp. 179-180.

Individual evidence

1. ↑ Gerd Wedler: Textbook of Physical Chemistry. John Wiley & Sons, 2012, ISBN 978-3-527-32909-0 . P. 359.
2. ^ Walter J. Moore: Physical chemistry. Walter de Gruyter, 1986, ISBN 978-3-110-10979-5 , p. 539.
3. ↑ Journal f. Physics. Chemistry I, 631 (1887)
4. ↑ Entry on absolute activity . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.A00019 Version: 2.3.2.
5. ↑ Entry on activity (relative activity) . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.A00115 Version: 2.3.2.
6. ↑ Entry on standard state . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.S05925 Version: 2.3.2.
7. ↑ Entry on standard chemical potential . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.S05908 Version: 2.3.2.
8. ^ Max Le Blanc: Textbook of Electrochemistry, Oskar Leiner Verlag Leipzig, 1922, p. 56.
9. ^ Gustav Kortüm: Textbook of Electrochemistry, 5th Edition, Verlag Chemie GmbH, Weinheim 1972, p. 146.
10. ^ P. Debye and E. Hückel: Physik Z., 24 , 185 (1923).
11. ^ Ionic Equilibrium: Solubility and PH Calculations, ISBN 0471585262 , page 49 ff.
12. ^ Alan L. Rockwood: Meaning and Measurability of Single-Ion Activities, the Thermodynamic Foundations of pH, and the Gibbs Free Energy for the Transfer of Ions between Dissimilar Materials. In: ChemPhysChem. 16, 2015, S. 1978, doi : 10.1002 / cphc.201500044 .

See also

• Gibbs energy
• Helmholtz energy
• Nernst equation
• Margules equation
• Group contribution methods