# Nernst equation

The Nernst equation (named after Walther Nernst ) describes the concentration dependence of the electrode potential of a redox pair (Ox + z e - Red) ${\ displaystyle \ rightleftharpoons}$

${\ displaystyle E = E ^ {0} + {\ frac {RT} {z_ {e} F}} \ ln {\ frac {a _ {\ mathrm {Ox}}} {a _ {\ mathrm {Red}}} }}$

By combining the natural constants R and F , the conversion factor for the natural logarithm to the decadic logarithm and a fixed temperature of 25 ° C (298 K), this form is simplified to:

${\ displaystyle E = E ^ {0} + {\ frac {8 {,} 314 \ \ mathrm {\ frac {J} {mol \ cdot K}} \ cdot 298 \ \ mathrm {K} \ cdot 2 {, } 303} {z_ {e} \ cdot 96485 {,} 336 \ \ mathrm {\ frac {C} {mol}}}} \ log {\ frac {a _ {\ mathrm {Ox}}} {a _ {\ mathrm {Red}}}} = E ^ {0} + {\ frac {8 {,} 314 \ cdot 298 \ cdot 2 {,} 303 \ \ mathrm {J}} {z_ {e} \ cdot 96485 {,} 336 \ \ mathrm {C}}} \ log {\ frac {a _ {\ mathrm {Ox}}} {a _ {\ mathrm {Red}}}} \ approx E ^ {0} + {\ frac {0 {, } 059 \ \ mathrm {V}} {z_ {e}}} \ log {\ frac {a _ {\ mathrm {Ox}}} {a _ {\ mathrm {Red}}}}}$
 ${\ displaystyle E}$ Electrode potential ${\ displaystyle E ^ {0}}$ Standard electrode potential ${\ displaystyle R}$ Universal or molar gas constant , R = 8.31447 J mol −1 K −1 = 8.31447 CV mol −1 K −1 ${\ displaystyle T}$ absolute temperature (= temperature in Kelvin ) ${\ displaystyle z_ {e}}$ Number of electrons transferred (also equivalent number ) ${\ displaystyle F}$ Faraday's constant , F = 96485.34 C mol −1 = 96485.34 JV −1 mol −1 ${\ displaystyle a}$ Activity of the redox partner concerned

## Interpretation and meaning

Galvanic cell

Any combination of two electrodes is called a galvanic cell (e.g. batteries , accumulators or biological cells ). Their open circuit voltage U 0 (historically: electromotive force ) is equal to the potential difference Δ E of the electrodes, which can be calculated as U 0 = Δ E = E acceptor - E donator when applying the Nernst equation to the half-cells . Similarly, it allows the calculation of the equilibrium activities that arise when a voltage is applied to the half-cells.

The Nernst equation is of central importance in electrochemistry , electroplating and electroanalysis because it connects the electrical variable voltage (or electrode potential ) with the chemical variable concentration . Strictly speaking, it is only valid for cells without transfer and currentless processes, but offers a starting point for the derivation of equations in electrochemical systems through which current flows. The Nernst potential U 0 multiplied by the charge z · F for a molar turnover z · F · U 0 gives the Gibbs energy. ΔG = - z · F · U 0 . The Nernst potential indicates the chemical energy of the electrochemical reaction, divided by the charge involved.

## Alternative formulations

The term Nernst equation is used for various derived or extended equations, depending on the application.

### Special (historical) Nernst equation

The German physicist and chemist Walther Nernst derived the original form in 1889 using concentrations c .

${\ displaystyle E = E ^ {0} + {\ frac {RT} {z _ {\ mathrm {e}} F}} \ ln {\ frac {c _ {\ mathrm {Ox}}} {c _ {\ mathrm { Red}}}}}$

The factor R T / z e F is called the Nernst factor or electrode slope; a table of values ​​for R T / z e F at different temperatures can be found in the article on electrode slope.

### General Nernst equation (derivation)

For the change in the Gibbs energy (free enthalpy ) of a chemical reaction in which substances according to ${\ displaystyle \ Delta _ {\ mathrm {R}} G}$${\ displaystyle k}$${\ displaystyle A_ {1}, \ dots, A_ {k}}$

${\ displaystyle \ nu _ {1} A_ {1} + \ nu _ {2} A_ {2} + \ dots \ longrightarrow \ dots + \ nu _ {k-1} A_ {k-1} + \ nu _ {k} A_ {k}}$

are involved

${\ displaystyle \ Delta _ {\ mathrm {R}} G = \ Delta _ {\ mathrm {R}} G ^ {0} + RT \, \ ln \ prod _ {i = 1} ^ {k} a_ { i} ^ {\ nu _ {i}}}$
 ${\ displaystyle a_ {i}}$ activity of the substance related to the standard activity${\ displaystyle i}$ ${\ displaystyle \ nu _ {i}}$ stoichiometric coefficient of the substance in the reaction equation (negative for educts) ${\ displaystyle i}$

The relationship between and the logarithm is plausible because on the one hand it is proportional to the number of particles (or the notation of the chemical equation), on the other hand the individual activities are included in the activity quotient with the power of the stoichiometric coefficients. The logarithm converts the exponent into a factor. ${\ displaystyle \ Delta _ {\ mathrm {R}} G}$${\ displaystyle \ Delta _ {\ mathrm {R}} G}$

${\ displaystyle \ mathrm {2 \, H_ {2} + O_ {2} \ longrightarrow 2 \, H_ {2} O}}$results in the same (electro) chemical potential as per . ${\ displaystyle \ mathrm {H_ {2} O}}$${\ displaystyle \ mathrm {H_ {2} + {1 \ over 2} \ O_ {2} \ longrightarrow H_ {2} O} \ qquad}$

${\ displaystyle \ Delta _ {\ mathrm {R}} G}$is at constant pressure and constant temperature from the reaction recoverable maximum work that fully into useful electrical work can be converted. Due to the law of conservation of energy, the following applies

${\ displaystyle \ Delta _ {\ mathrm {R}} G = -z_ {e} \, F \, \ Delta E}$

which results in the general Nernst equation

${\ displaystyle \ Delta E = \ Delta E ^ {0} - {\ frac {RT} {z_ {e} F}} \, \ ln \ prod _ {i = 1} ^ {k} a_ {i} ^ {\ nu _ {i}}}$

The general Nernst equation allows the calculation of the equilibrium constants (for ), direction (voluntary for , forced for ) and voltage that the reaction delivers when its redox partial reactions are allowed to run in separate half-cells for the reaction under consideration. ${\ displaystyle \ Delta E = 0}$${\ displaystyle \ Delta E> 0}$${\ displaystyle \ Delta E <0}$${\ displaystyle \ Delta E}$

## application

The Nernst equation is used in potentiometric titration . For example, a measuring electrode is immersed in a sample solution and must react to the ion to be determined, that is, the potential of this electrode must be dependent on the concentration of the ion to be determined. This dependency is described by the Nernst equation. During the experiment, it must be ensured that the measurement is carried out without current, as otherwise the potentials would be falsified by electrolysis . A voltage compensation circuit was therefore used for the measurement .

### reduction

For the reduction

${\ displaystyle \ mathrm {Ox + {\ mathit {z}} _ {\ mathit {e}} e ^ {-} \ longrightarrow Red}}$

the general form goes directly into the first-mentioned equation. This identity has two practical meanings:

• The electrochemical series basically lists reductions.
• Since every chemical reaction can be broken down into oxidation and reduction partial reactions of redox pairs, Δ E is the sum of the Nernst equations for the partial reactions multiplied by the associated stoichiometric coefficients. The partial oxidation reactions enter into a negative stoichiometric coefficient.

### Oxyhydrogen reaction

The partial reactions of the so-called oxyhydrogen reaction

${\ displaystyle \ mathrm {2 \ H_ {2} + O_ {2} \ longrightarrow 2 \ H_ {2} O}}$

run as oxidation

${\ displaystyle \ mathrm {2 \ H_ {2} \ longrightarrow 4 \ H ^ {+} + 4 \ e ^ {-}}}$

or reduction

${\ displaystyle \ mathrm {O_ {2} +4 \ e ^ {-} \ longrightarrow 2 \ O ^ {2-}}}$

spatially separated in hydrogen-oxygen fuel cells . The voltage that can be achieved with this can be calculated using the Nernst equation and under standard conditions is Δ E 0  = 1.23 V.

### Concentration elements

Copper concentration element

A concentration element consists of two half-cells that contain electrolytes with the same components but with different ion concentrations. It is therefore particularly suitable for demonstrating the Nernst equation.

An example is a copper concentration element made up of two copper electrodes and two copper sulphate solutions that differ only in their concentration. When the current flows, the concentrations in the cells equalize because the following reactions then take place:

The reduction in the half-cell with the greater copper ion concentration c g :

${\ displaystyle \ mathrm {Cu ^ {2 +} ({\ mathit {c}} _ {g}) + 2 \ e ^ {-} \ longrightarrow Cu}}$

The oxidation in the half-cell with the smaller copper ion concentration c k :

${\ displaystyle \ mathrm {Cu \ longrightarrow Cu ^ {2 +} ({\ mathit {c}} _ {k}) + 2 \ e ^ {-}}}$

Using the Nernst equation for the partial reactions or the general Nernst equation for the overall reaction, the voltage Δ E of the copper concentration element is obtained:

${\ displaystyle \ Delta E = {\ frac {RT} {2F}} \ ln {\ frac {c _ {\ mathrm {g}} (\ mathrm {Cu} ^ {2 +})} {c _ {\ mathrm { k}} (\ mathrm {Cu} ^ {2+})}}}$

In general, the following applies to the tension of a concentration element :

${\ displaystyle \ Delta E = {\ frac {RT} {z_ {e} F}} \ ln {\ frac {a _ {\ mathrm {g}}} {a _ {\ mathrm {k}}}}}$.

In the temperature range from 22 to 26 ° C the following applies:

${\ displaystyle \ Delta E = {\ frac {0 {,} 059V} {z_ {e}}} \ lg {\ frac {a _ {\ mathrm {g}}}} {a _ {\ mathrm {k}}}} }$.

### Concentration elements with different elements

For concentration elements with different elements and concentrations that deviate from the normal conditions, the following formula is used:

${\ displaystyle E = E ^ {0} + {\ frac {0 {,} 059V} {z_ {e}}} \ lg {\ frac {(c _ {\ mathrm {ox}}) ^ {a}} { (c _ {\ mathrm {red}}) ^ {b}}}}$.
• a = factor of the oxidation side (example: 4 )${\ displaystyle H ^ {+}}$
• b = factor of the reduction side (example: 2 )${\ displaystyle {H_ {2}} O}$

### PH value

If we consider H + concentration elements ( ), then the Nernst equation works at room temperature ( T  = 298.15 K ≙ 25 ° C), converting the natural logarithm into the decadic logarithm (lg a (H + ) = ln a (H + ) / ln 10) and taking into account the definition of the pH value (pH = −lg a (H + )) into the form ${\ displaystyle z_ {e} = 1}$

${\ displaystyle \ Delta E = -0 {,} 059 \, \ mathrm {V} \, (\ mathrm {pH} _ {1} - \ mathrm {pH} _ {2})}$

over. In principle, glass electrodes for pH measurement represent such H + concentration elements. They contain a solution with a known pH value. If contact is made with a solution with an unknown pH value, the associated measuring device measures a voltage that is converted directly into a pH value with a factor of 0.059 V and displayed. The factor can vary depending on the manufacturing process and must be calibrated before use, but is always close to 0.059 V.

### Lambda sensors

In the case of a lambda probe , the sensor element of which is conductive for oxygen ions, a voltage is established due to the concentration gradient of the oxygen between air and exhaust gas, which is used to set a desired mixture with the lambda control .

### Nernst equation in biology and physiology

In biological systems, cell membranes separate areas with different ion concentrations. If the membrane is selectively permeable for a certain ion, it will diffuse along the concentration gradient, but at the same time, since the ion is charged, a voltage ( resting membrane potential ) arises . The equilibrium position of this process can be described with the Nernst equation.

A simplified form of the equation is common, in which R , F and T (310 K) as well as the conversion factor for the decadic logarithm are put into a constant:

${\ displaystyle E = {\ frac {61 {,} 51 ​​\, \ mathrm {mV}} {z_ {e}}} \ lg {\ frac {c _ {\ mathrm {outside}}} {c _ {\ mathrm { Inside} }}}}$

See also : Goldman equation

## literature

• Gerold Adam, Peter Läuger, Günther Stark: Physical chemistry and biophysics . 5th, revised. Edition. Springer, Berlin, Heidelberg 2009, ISBN 978-3-642-00423-0 .
• PW Atkins, Julio De Paula (Ed.): Physical chemistry . 5th edition. Wiley-VCH, Weinheim 2013, ISBN 978-3-527-33247-2 .