PH value

The pH value is a measure of the acidic or base character of an aqueous solution.

The pH value is a measure of the acidic or basic character of an aqueous solution . He is the opposite number of the logarithm (logarithm) of hydrogen ions - activity and size of the number of dimensions .

A dilute aqueous solution with a pH value of less than 7 is called acidic , with an H value of 7 neutral and with a pH value of more than 7 basic or alkaline .

definition

PH value

The pH value is defined as the opposite number of the decade logarithm (= logarithm ) of the hydrogen ion activity.

${\displaystyle \mathrm {pH} =-\log _{10}a\left(\mathrm {H} ^{+}\right)}$

The dimensionless, relative activity of the hydrogen ion a _{H ⁺} is the product of the molality of the hydrogen ion (m _{H ⁺} in mol / kg) and the activity coefficient of the hydrogen ion (γ _H ) divided by the unit of molality (m ⁰ in mol / kg).

To simplify the formulas, the H ⁺ (hydrogen ion) is usually used to define pH. In reality, these hydrogen ions (free protons) only exist in an associated form. In the first stage, the oxonium ion H ₃ O ⁺ forms in the water , which in turn accumulates further water molecules. The hydrated oxonium ion is referred to as the hydronium ion (H ₉ O ₄ ⁺ ).

However, the exact definition of the pH value is rarely used in simple calculations. Rather, for the sake of simplification, one is content with the approximation that the oxonium activity for dilute solutions is set equal to the measure of the oxonium ion concentration (in mol / dm³ or mol / l):

${\displaystyle \mathrm {pH} =-\log _{10}a\left(\mathrm {H} ^{+}\right)\approx -\log _{10}\left({\frac {c\left(\mathrm {H_{3}O^{+}} \right)}{c^{o}}}\right)}$.

It should also be noted that the individual ion activity of the hydrogen ion should actually be known in order to determine the pH value exactly as defined. However, it is controversial whether single ion activities can be determined.

pOH value

The pOH scale (top) and the pH scale (bottom) are opposite to each other.
(red: acidic area; blue: basic area)

A pOH value was also defined analogously to the pH value. It is the opposite number of the decadic logarithm of the measure of the hydroxide ion activity (in mol / dm³ or mol / l).

Both values ​​are related via the autoprotolysis equilibrium :

Chemical reaction equation :

${\displaystyle \mathrm {2\ H_{2}O\ \rightleftharpoons \ H_{3}O^{+}\ +\ OH^{-}} }$

Equilibrium constant of the reaction :

${\displaystyle K_{\mathrm {w} }={\frac {a\left(\mathrm {H_{3}O^{+}} \right)\cdot a\left(\mathrm {OH^{-}} \right)}{a^{2}\left(\mathrm {H_{2}O} \right)}}}$

${\displaystyle -\log _{10}K_{\mathrm {w} }=-\log _{10}a\left(\mathrm {H_{3}O^{+}} \right)-\log _{10}a\left(\mathrm {OH^{-}} \right)+\log _{10}a^{2}\left(\mathrm {H_{2}O} \right)=\mathrm {pH} +\mathrm {pOH} }$

The activity of water as a solvent for dilute systems is equal to one, especially at θ = 25 ° C (standard condition). So the logarithm of the activity of water is zero. The equilibrium constant is K _w = 10 ⁻¹⁴ under normal conditions . The relationship between pH and pOH of a dilute solution at room temperature is therefore a good approximation:

${\displaystyle \mathrm {pH} +\mathrm {pOH} =14}$

Further explanations can be found in the article Oxonium and Autoprotolysis .

The pH of other solvents

A measure comparable to the “pH value” is also defined for other amphiprotic solvents LH that can transfer protons . These are also based on the autoprotolysis of the respective solvent. The general response is:

2 LH LH ₂ ⁺ + L ^- , ${\displaystyle \leftrightharpoons }$

with the Lyonium ion LH ₂ ⁺ and the Lyat ion L ^- .

The equilibrium constant K is generally smaller here than for the ion product of water. The pH value is then defined as follows:

${\displaystyle \mathrm {pH} _{p}=-\log _{10}c\left(\mathrm {LH_{2}^{\,+}} \right)}$

Some examples of amphiprotic solvents
anhydrous formic acid	2 HCOOH HCOOH 2 + + HCOO - ${\displaystyle \leftrightharpoons }$
anhydrous ammonia	2 NH 3 NH 2 - + NH 4 + ${\displaystyle \leftrightharpoons }$
anhydrous acetic acid	2 CH 3 COOH CH 3 COO - + CH 3 COOH 2 + ${\displaystyle \leftrightharpoons }$
anhydrous ethanol	2 C 2 H 5 OH C 2 H 5 OH 2 + + C 2 H 5 O - ${\displaystyle \leftrightharpoons }$

Neutral value and classification

Average pH values ​​of some common solutions

substance	PH value	Art
Battery acid	<1	angry
Stomach acid (empty stomach)	1.0-1.5
Lemon juice	2.4
cola	2.0-3.0
vinegar	2.5
Juice of Schattenmorelle	2.7
Orange and apple juice	3.5
Wine	4.0
Sour milk	4.5
beer	4.5-5.0
Acid rain (from polluted air)	<5.0
coffee	5.0
tea	5.5
Human skin surface	5.5
Rain (precipitation with dissolved CO 2 )	5.6
Mineral water	6.0
milk	6.5
Human saliva	6.5 - 7.4	acidic to alkaline
Pure water (CO 2 -free)	7.0	neutral
blood	7.4	alkaline
Sea water	7.5-8.4
Pancreatic juice ( pancreas )	8.3
Soap	9.0-10.0
Household ammonia	11.5
Bleach	12.5
concrete	12.6
Caustic soda	13.5-14
Legend
highlighted in gray	Components of the human body
highlighted in color	Colors of the universal indicator

The ion product of the water at 25 ° C results from the autoprotolysis

${\displaystyle K_{\mathrm {w} }={\frac {c\left(\mathrm {H_{3}O^{+}} \right)}{c^{o}}}\cdot {\frac {c\left(\mathrm {OH^{-}} \right)}{c^{o}}}=10^{-14}}$

This size determines the scale and the neutral value of the pH value. The pH values ​​of dilute aqueous solutions are qualified as follows:

• pH <7 as an acidic aqueous solution, here c _{H 3 O ⁺} > c _{OH ^-}
• pH = 7 as a neutral aqueous solution, here c _{H 3 O ⁺} = c _{OH ^-} ; also a property of pure water
• pH> 7 as a basic (alkaline) aqueous solution, here c _{H 3 O ⁺} < c _{OH ^-}

Research history

In 1909, the Danish chemist Søren Sørensen introduced the hydrogen ion exponent in the notation p _{H ⁺} for the concentration of hydrogen ions C _p equal to 10 ^{−p _{H +}} . The p _{H ⁺} values were about electrometric determined measurements. The notation p _{H ⁺} later changed to the current notation pH. The letter H was used by Sørensen as a symbol for hydrogen ions, he arbitrarily chose the letter p as the index for his solutions to be measured (e.g. C _p ) and q as the index for his reference solutions (e.g. C _q ).

The letter p in pH was later assigned the meaning of potency or derived from the Neo-Latin of p otentia H ydrogenii   or from p ondus H ydrogenii   ( Latin pondus "weight"; potentia " strength "; hydrogenium " hydrogen ").

Later, the hydrogen ion activity has been associated with a conventional p H -scale introduced. It is based on a specified measuring method with specified standard solutions, from which an operational definition of the pH value was established. This definition serves the highest possible reproducibility and comparability of pH measurements .

To speak of hydrogen ions (H ⁺ ) or hydrogen ion exponent goes back to the acid-base concept according to Arrhenius . Today, the acid-base concept according to Brønsted is usually followed and the term oxonium ions (H ₃ O ⁺ ) is used, an ion that was formed from a water molecule by reaction with a proton donor and itself reacted as a proton acceptor .

Chemical-physical relationships

pH and acids and bases

If acids are dissolved in water, they dissociate hydrogen ions into the water and the pH value of the solution drops. If bases are dissolved, they release hydroxide ions which bind hydrogen ions from the dissociation of the water. You can also bind hydrogen ions yourself, as is the case with ammonia → ammonium. As a result, bases increase the pH value. PH is a measure of the amount of acids and bases in a solution. Depending on the strength, the acid or base dissociates to a greater or lesser extent and thus influences the pH value to different degrees.

In most aqueous solutions, the pH values ​​are between 0 (strongly acidic) and 14 (strongly alkaline). Nevertheless, even in one-molar solutions of strong acids and bases, these limits can be exceeded by one unit, i.e. from −1 to 15. The pH scale is only limited by the solubility of acids or bases in water. At very high or very low pH values ​​and in concentrated salt solutions, it is not the concentrations that are decisive for the pH value, but the activities of the ions. Activities are not linearly dependent on ion concentrations.

Most pH electrodes behave almost linearly in the measuring range between 0 and 14. Approximately constant differences in the measured electrode potential therefore correspond to the same differences in the pH value. According to international conventions, pH values ​​can only be measured directly in this range.

Solutions of a weak acid and one of its salts or of a weak base and one of its salts give buffer solutions . Here equilibria are established which, close to the logarithmized value of their acid constants or base constants multiplied by −1 , result in almost identical pH values. The pH value of these solutions changes significantly less when adding strong acids or bases than when adding acids and bases to pure, salt-free, "unbuffered" water. These buffer solutions have a certain buffer capacity ; the effect lasts as long as the amount added does not exceed the supply of the used buffer component.

Pure water absorbs carbon dioxide from the air, depending on the temperature about 0.3 to 1 mg / l. This is how carbonic acid (H ₂ CO ₃ ) is formed, which dissociates into hydrogen carbonate and hydrogen ions:

${\displaystyle \mathrm {CO_{2}+H_{2}O\to H_{2}CO_{3}\to HCO_{3}^{-}+H^{+}} }$

If the entry of carbon dioxide is not prevented in “chemically pure water”, a pH value of just under 5 is established. A strong influence on the pH value of pure, distilled or deionized water with a calculated pH value close to 7 by very small traces of proton donors or proton acceptors says nothing about the effect on chemical reactions or living beings.

Temperature dependence

The equilibrium constant of water dissociation K _w is temperature dependent:

At 0 ° C it is 0.115 · 10 ⁻¹⁴ (pK _w = 14.939),

at 25 ° C: 1.009 · 10 ⁻¹⁴ (pK _w = 13.996),

at 60 ° C: 9.61 · 10 ⁻¹⁴ (pK _w = 13.017).

The sum of pH + pOH behaves accordingly (14.939, 13.996 and 13.017, respectively).

The pH values ​​of solutions are temperature dependent. For example, a molar phenol solution has a pK at a solution temperature of 30 ° C _s value of the phenol as phenyl-OH of 10. The solution has a pH of about 4.5. If the temperature changes, three coupled effects occur. The first is by far the most important.

1. The equilibrium constant K for the dissociation of phenol increases with increasing temperature, and so does the dissociation of the acid. If K increases, the pH value decreases, and vice versa: ${\displaystyle \mathrm {PhOH\,{\stackrel {K\gg }{\longrightarrow }}\,PhO^{-}+H^{+}} }$
2. When the temperature drops from 30 ° C to 20 ° C, phenol has a lower solubility in water. Only about 0.9 mol / l dissolve. This increases the pH to around 4.55. This effect only plays a role for solutions close to solubility saturation.
3. When the temperature increases, the volume of the solution increases slightly and the molar concentration of phenol decreases (mol per volume). Thus the pH rises differentially. The pH value falls analogously when the temperature drops.

Determination of the pH value

calculation

A common problem is that the pH value should be calculated while the concentration is known and the value (which represents the strength of the acid or base) can be taken from tables. An example from practice is the preparation of solutions with a given pH value. There are formulas that can be used to approximate the pH value. Despite approximations, the results are usually accurate enough. ${\displaystyle \mathrm {p} K_{\mathrm {s} }}$

The formulas are derived from the

Law of mass action: ${\displaystyle K_{\mathrm {s} }={\frac {c\left(\mathrm {A^{-}} \right)\cdot c\left(\mathrm {H_{3}O^{+}} \right)}{c\left(\mathrm {HA} \right)\cdot c^{o}}}}$

Ion product of water ${\displaystyle K_{\mathrm {w} }={\frac {c\left(\mathrm {H_{3}O^{+}} \right)}{c^{o}}}\cdot {\frac {c\left(\mathrm {OH^{-}} \right)}{c^{o}}}}$

Conservation of mass law ${\displaystyle c_{0}=c\left(\mathrm {HA} \right)+c\left(\mathrm {A^{-}} \right)}$

Conservation of charge law ${\displaystyle c\left(\mathrm {H_{3}O^{+}} \right)=c\left(\mathrm {A^{-}} \right)+c\left(\mathrm {OH^{-}} \right)}$

Very strong acids

The calculation assumes that strong acids are completely deprotonated. This applies to acids with a pKa <1. In this case, the calculation is independent of the respective acid constant, so the pKa is not required for the calculation. The decoupling from is based on the leveling effect of the water. The autoprotolysis of the water only plays a role with very dilute, strong acids (from concentrations ). Thus, the concentration of the protons in solution results from the concentration of the acid, described by the formula: ${\displaystyle K_{\mathrm {s} }}$ ${\displaystyle \leq 10^{-6}\mathrm {mol/l} }$

${\displaystyle c\left(\mathrm {H_{3}O^{+}} \right)=c_{0}=c\left(\mathrm {A^{-}} \right)}$

Strong acids

Acids with a 4.5> pKa> 1 are no longer described precisely enough as completely deprotonated. However, the autoprotolysis of the water can also be neglected here. According to the principles of mass equality and electronic neutrality, the equation results:

${\displaystyle c\left(\mathrm {H_{3}O^{+}} \right)=-{\frac {K_{\mathrm {s} }c^{o}}{2}}+c^{o}\cdot {\sqrt {{\frac {K_{\mathrm {s} }^{2}}{4}}+K_{\mathrm {s} }\cdot c_{0}/c^{o}}}}$

The formula can also be used for weaker acids, which is particularly recommended for low-concentration solutions. Only when the pK _s exceeds 9 or the concentration is below 10 ⁻⁶ mol / l does the formula become imprecise, since the autoprotolysis of the water must then be taken into account.

Weak acids

With weak acids (4.5 <pKa <9.5) the proportion of dissociated acid molecules is small compared to the proportion of undissociated ones. As a simplification, it can therefore be assumed that there are still as many protonated acid molecules in the solution as were originally added. This simplifies the equation for weak acids to:

${\displaystyle c\left(\mathrm {H_{3}O^{+}} \right)=c^{o}\cdot {\sqrt {K_{\mathrm {s} }\cdot c_{0}/c^{o}}}}$

The resulting error decreases with increasing concentration and the pK _s value. In case of doubt, the formula for strong acids can also be used.

Very weak acids

With very weak acids, the protons generated by the auto-dissociation of the water must be taken into account. This gives the equation:

${\displaystyle c\left(\mathrm {H_{3}O^{+}} \right)=c^{o}\cdot {\sqrt {K_{\mathrm {s} }\cdot c_{0}/c^{o}+K_{\mathrm {w} }}}}$

This formula for very weak acids (but also bases!) Has to be taken into account every time the product is made of and not significantly larger than that . ${\displaystyle K_{\mathrm {s} }}$ ${\displaystyle c_{0}}$ ${\displaystyle K_{\mathrm {w} }}$

Bases

The same formulas are used to calculate the pH of a basic solution. However, instead of is K _s of the K _b used and the result does not provide the proton concentration c (H ₃ O ⁺ ), but the hydroxide ion concentration c (OH ^- ). This can be converted into the pOH and the pH follows from this.

Other calculations

For solutions of an acid and its corresponding salt (a buffer, see above), the pH value can be calculated using the Henderson-Hasselbalch equation .

For polybasic acids can be calculated (approximately) only the value of the first protolysis stage, ie for the lowest pK _s value. The dissociation of the second stage is usually much less. An exact calculation is extremely complex, since it is a system of coupled equilibria. The oxonium ions from the first protolysis stage influence the second and vice versa.

The same applies to mixtures of several acids and / or bases. An exact algebraic solution is usually no longer possible; the equations have to be solved numerically using iterative methods. In the case of very high concentrations of acids or bases, the concentration in mol / dm ^{3 must be} replaced by the activity of the oxonium ions.

Measurement

The pH value of a solution can be determined using different methods:

Determination by the reaction of indicator dyes

The pH value is easily determined by visual or colorimetric evaluation of the color changes of indicator dyes . The evaluation is usually carried out using color comparison scales.

Within a narrow measuring range (two to three pH levels), the color change of a single dye is sufficient. Universal indicators are used for larger measuring ranges . These are mixtures of dyes that show different colors over a wide range of pH values. An alternative to universal indicators are measuring strips that have fields with different colorants arranged next to one another, each of which can be optimally read in a different value range. For special purposes, the color display of an indicator dye can be measured with a photometer and thus evaluated more precisely.

Various substances are used to color the universal indicator, which change color with different pH values. Such pH indicators are for example

• Litmus : pH <4.5 = red, pH> 8.3 = blue
• Phenolphthalein : pH <8.2 = colorless, pH> 10.0 = red-violet
• Methyl orange pH <3.1 = red, pH> 4.4 = yellow
• Bromothymol blue : pH <6.0 = yellow, pH> 7.6 = blue

Potentiometry

→ Main article : pH combination electrode

Most commercially available pH meters are based on the principle of potentiometry . A glass membrane ball filled with buffer solution is immersed in the liquid to be measured. Due to the tendency of the hydrogen ions to accumulate in thin layers on silicate groups on the glass surface, a galvanic voltage builds up between the inside and the outside of the sphere , depending on the pH difference . This electromotive force is measured by means of two reference electrodes , one of which is located in the glass sphere and the other in a reference electrolyte. For detailed information see pH electrode .

Measurement by ion-sensitive field effect transistors

Similar to the glass electrode, hydrogen ions build up a potential on the sensitive gate membrane of an ion-sensitive field effect transistor (ISFET), which influences the current permeability of the transistor. Using suitable measurement technology, this signal can then be displayed as a pH value.

Importance of the pH value

Effects of pH in chemistry

Some chemical compounds change their chemical structure depending on the pH value and thus also their color, as is the case for pH indicators such as phenolphthalein from colorless to red.

The hydrogen ions play a role in many reactions, either directly in aqueous solution or as a “ catalyst ”. The pH value influences the reaction rate , as in the example of the hardening of aminoplasts .

Effects of pH on the growth of plants

The pH value of the soil influences the (biological) availability of nutrient salts . If the soil pH is neutral and alkaline, iron oxide hydroxides are formed which cannot be absorbed, resulting in an iron deficiency . If there are strong pH changes, the plant organs can also be directly affected.

In addition to some other elements, nitrogen is also important for the nutrient balance of plants. It is absorbed in the form of water-soluble ammonium ions (NH ₄ ⁺ ) or more often as nitrate ions (NO ₃ ^- ). Ammonium and nitrate are in equilibrium in soils with a pH value of 7. In acidic soils, the predominate NH ₄ ⁺ ions, in alkaline soils, the NO ₃ ^- ions. If plants can only absorb NH ₄ ⁺ due to the permeability of the root membranes , they are dependent on acidic soils, i.e. acidophilic (acid-loving). When absorbing nitrate NO ₃ ^- , they can only grow on soils rich in bases (“obligate basophil”). The demands on the soil pH are lower when the membranes allow both ammonium and nitrate to pass through. Ammonium nitrate (NH ₄ NO ₃ ) is used in mineral fertilizers , which means that both ammonium and nitrate ions are present. The reactions in the soil lead to transformations.

If the pH value is high or low, the nutrients in the soil are fixed and the plants have insufficient access to them. At a low pH value, aluminum or manganese ions become soluble and are accessible to plants in harmful quantities.

See also : Soil pH  and Kalkstet

See also : Calcareous soil  and soil acidification

See also : Acidification of the Seas

The importance of pH in humans

The range of the pH value of blood and cell fluid that is tolerable for humans is limited to narrow ranges. So are z. B. normal pH values ​​of arterial blood in the narrow pH range of 7.35-7.45. The prevailing pH value is set by the so-called blood buffer , a complex buffer system formed by the carbon dioxide dissolved in the blood , the anions of dissolved salts and the dissolved proteins . The pH value of the blood influences the ability of hemoglobin to bind oxygen: the lower the pH value of the blood, the less oxygen the hemoglobin can bind ( Bohr effect ). If carbon dioxide is exhaled in the lungs, the pH value of the blood rises there and thus the ability of the hemoglobin in the blood to absorb oxygen also increases. If, conversely, the pH value of the blood is lowered in the tissue of a body cell by the carbonic acid, the hemoglobin therefore releases the bound oxygen again. The pH value also plays an important role in human reproduction. While the vaginal environment is weakly acidic to ward off pathogens , the man's sperm is weakly basic. The neutralization reaction that sets in after the sexual act then leads to pH environments in which sperm move optimally. The human skin has a pH of ≈ 5.5. slightly acidic, forms a protective acid mantle and protects against pathogens in this way. When used on the skin, curd soaps create a slightly alkaline environment, but the protective acid mantle has built up again just 30 minutes after washing. In addition to dirt, surfactants also partially remove the skin's natural grease layer (hydro-lipid film) , "dry out" the skin and in this way impair the protective effect of the grease layer, especially when used frequently. Today's washing lotions, which consist of a mixture of a carrier substance, water, glycerine , sodium chloride, sodium thiosulfate , sodium hydrogen carbonate , distearates and a small proportion of synthetic surfactants , are adjusted to a pH of around 5.

Importance of the pH value in drinking water

According to the Drinking Water Ordinance , the drinking water from the tap should have a pH value between 6.5 and 9.5. Tap water with a low pH value loosens metal ions from the pipe wall, which can lead to the poisoning of living beings if lead and copper pipes are used. If you do not know the pH value and the pipe material, you can first let process water run out of the pipe (e.g. for cleaning purposes) before drawing off drinking water.

Importance of the pH value for aquariums

Plants and fish in aquariums require specific pH ranges. Living things have a pH tolerance range and outside this range cannot survive in the long term.

Guide values ​​for freshwater aquarium fish:

• acidic water (pH ≈ 6):
  • South Americans (neon, angelfish, discus, L-catfish and others)
  • Asians (guaramis, gouramis and others)
• neutral water (pH ≈ 7)
  • Central American (fire mouth cichlid and others)
• alkaline water (pH ≈ 8)
  • East African grave lakes (cichlids from Lake Tanganyika and Lake Malawi and others)

See also

• The SH value (degree of acidity) records all acidic components of the sample, while the pH value only indicates the H ₃ O ⁺ ion concentration.
• For super acids , the Hammett acidity function is used to determine the acid strength.

literature

• RP Buck, S. Rondinini, AK Covington et al: Measurement of pH. Definition, standards, and procedures (IUPAC Recommendations 2002). In: Pure and Applied Chemistry . Volume 74 (11), 2002, pp. 2169-2200. Facsimile (PDF; 317 kB).
• Gerhart Jander, Karl Friedrich year: measurement analysis. 17th edition. De Gruyter, Berlin 2009, ISBN 978-3-11-019447-0 , p. 99: Indicators.
• Willy W. Wirz: pH and. pCI values. Handbook with interpretations and an introduction to pX measurement technology; Measurement tables according to electronic (electrometric) pH and pCI measurements; with 22 special tables. Chemie-Verlag, Solothurn 1974, ISBN 3-85962-020-7 .

Web links

Commons : pH  - Collection of pictures, videos and audio files

Wiktionary: pH value  - explanations of meanings, word origins, synonyms, translations

Wikibooks: Chemistry formula collection / calculation of the pH value  - learning and teaching materials

• Rob Beynon: Buffers for pH control. University of Liverpool
• Average value calculation

Individual evidence

1. ↑ ^{a b} ÖNORM M 6201-2006 - pH measurement - terms
2. ↑ ^{a b} entry on pH . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.P04524 Version: 2.3.3.
3. ^ 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 . tape 16 , no. 9 , 2015, p. 1978-1991 , doi : 10.1002 / cphc.201500044 , PMID 25919971 .
4. ↑ SPL Sörensen: About the measurement and importance of the hydrogen ion concentration in enzymatic processes. In: Biochem. Magazine 21, 1909, pp. 131-304.
5. ^ ^{A b} Jens G. Nørby: The origin and the meaning of the little p in pH. In: Trends in Biochemical Sciences . 25, 2000, pp. 36-37.
6. ^ Pschyrembel Clinical Dictionary. 258th edition. de Gruyter, Berlin 1998.
7. ↑ Duden - German Universal Dictionary. 4th edition. Duden, Mannheim 2001.
8. ↑ ^{a b} Gerhard Schulze, Jürgen Simon: Jander year measurement analysis. 17th edition. de Gruyter, Berlin 2009, p. 77.
9. ↑ JA Campbell: General chemistry - energetics, dynamics and structure of chemical systems. 2nd Edition. Verlag Chemie, Weinheim et al. O. 1980, ISBN 3-527-25856-6 .
10. ^ Matthias Otto: Analytical Chemistry. 3. Edition. Wiley-VCH, 2006, ISBN 3-527-31416-4 , p. 53 f.
11. ↑ Gerhard Schulze, Jürgen Simon: Jander · year measurement analysis. 17th edition. de Gruyter, Berlin 2009, pp. 83-89.