9.4: The Nernst Equation

We will now extend electrochemistry by determining the relationship between [latex]E^\circ_\ce{cell}[/latex] and the thermodynamics quantities such as ΔG° (Gibbs free energy) and K (the equilibrium constant). In galvanic cells, chemical energy is converted into electrical energy, which can do work. The electrical work is the product of the charge transferred multiplied by the potential difference (voltage):

[latex]\mathrm{electrical\: work=volts \times (charge\: in\: coulombs)=J} \nonumber[/latex]

The charge on 1 mole of electrons is given by Faraday’s constant (F)

[latex]\begin{align*} F &=\dfrac{6.022 \times 10^{23}\:e^-}{mol} \times \dfrac{1.602 \times 10^{−19}\:C}{e^-} \[latex]4pt] &=9.648 \times 10^4\:\dfrac{C}{mol} \[latex]4pt] &=9.648 \times 10^4\:\dfrac{J}{V⋅mol} \end{align*} \nonumber[/latex]

Therefore

[latex]\mathrm{total\: charge=(number\: of\: moles\: of\: e^-)} \times F=nF \nonumber[/latex]

In this equation, [latex]n[/latex] is the number of moles of electrons for the balanced oxidation-reduction reaction. The measured cell potential is the maximum potential the cell can produce and is related to the electrical work (w_ele) by

[latex]E_\ce{cell}=\dfrac{−w_\ce{ele}}{nF}\hspace{40px}\ce{or}\hspace{40px}w_\ce{ele}=−nFE_\ce{cell} \nonumber[/latex]

The negative sign for the work indicates that the electrical work is done by the system (the galvanic cell) on the surroundings. In an earlier chapter, the free energy was defined as the energy that was available to do work. In particular, the change in free energy was defined in terms of the maximum work ([latex]w_{max}[/latex]), which, for electrochemical systems, is [latex]w_{ele}[/latex].

We can verify the signs are correct when we realize that [latex]n[/latex] and [latex]F[/latex] are positive constants and that galvanic cells, which have positive cell potentials, involve spontaneous reactions. Thus, spontaneous reactions, which have [latex]ΔG < 0[/latex], must have [latex]E_{cell} > 0[/latex]. If all the reactants and products are in their standard states, this becomes

This provides a way to relate standard cell potentials to equilibrium constants, since

[latex]ΔG°=−RT\ln K \nonumber[/latex]

[latex]−nFE^\circ_\ce{cell}=−RT\ln K \nonumber[/latex]

or

[latex]E^\circ_\ce{cell}=\dfrac{RT}{nF}\ln K \nonumber[/latex]

Most of the time, the electrochemical reactions are run at standard temperature (298.15 K). Collecting terms at this temperature yields

[latex]\begin{align*} E^\circ_\ce{cell}&=\dfrac{RT}{nF}\:\ln K \[latex]4pt] &=\dfrac{\left(8.314\:\dfrac{\ce{J}}{\textrm{K⋅mol}}\right)(298.15\:K)}{n \times 96,485\: \textrm{C/V⋅mol}}\:\ln K \[latex]4pt] &=\dfrac{\mathrm{0.0257\: V}}{n}\:\ln K \end{align*} \nonumber[/latex]

where [latex]n[/latex] is the number of moles of electrons. The logarithm in equations involving cell potentials is often expressed using base 10 logarithms (i.e., [latex]\log_{10}[/latex] or just [latex]\log[/latex]), which changes the constant by a factor of 2.303:

[latex]E^\circ_\ce{cell}=\dfrac{\mathrm{0.0592\: V}}{n}\:\log K \nonumber[/latex]

Thus, if ΔG°, K, or [latex]E^\circ_\ce{cell}[/latex] is known or can be calculated, the other two quantities can be readily determined. The relationships are shown graphically in Figure 9.4.1.

Given any one of the quantities, the other two can be calculated.

Now that the connection has been made between the free energy and cell potentials, nonstandard concentrations follow. Recall that

[latex]ΔG=ΔG°+RT\ln Q \nonumber[/latex]

where [latex]Q[/latex] is the reaction quotient (see the chapter on equilibrium fundamentals). Converting to cell potentials:

[latex]−nFE_\ce{cell}=−nFE^\circ_\ce{cell}+RT\ln Q \label{nernst1A}[/latex]

or

[latex]E_\ce{cell}=E^\circ_\ce{cell}−\dfrac{RT}{nF}\:\ln Q \label{nernst1B}[/latex]

Equation \ref{nernst1B} is the generalized Nernst equation that is applicable at any temperature. However, is can be simplified for reactions occuring at 25 °C (298.15 K) by rewriting it as

[latex]E_\ce{cell}=E^\circ_\ce{cell}−\dfrac{\mathrm{0.0257\: V}}{n}\:\ln Q \label{nernst2A}[/latex]

or

[latex]E_\ce{cell}=E^\circ_\ce{cell}−\dfrac{\mathrm{0.0592\: V}}{n}\log_{10} Q \label{nernst2B}[/latex]

If the temperature is not 298.15 K, it is necessary to recalculate the potential with Equation\ref{nernst1B}. With the Nernst equation, it is possible to calculate the cell potential at nonstandard conditions. This adjustment is necessary because potentials determined under different conditions will have different values.

Finally, we will take a brief look at a special type of cell called a concentration cell. In a concentration cell, the electrodes are the same material and the half-cells differ only in concentration. Since one or both compartments is not standard, the cell potentials will be unequal; therefore, there will be a potential difference, which can be determined with the aid of the Nernst equation.