We would like to know how the cell voltage E depends upon the cell voltage at standard state and the concentrations and pressures of reactants and products. It turns out that it is easy to answer that question by making use of the principal result of our study of chemical thermodynamics

ΔG = Δ G^o + RT ln Q

As the reaction approaches the state of equilibrium, the reaction quotient becomes the equilibrium constant, K, and ΔG=0. That leads to the relation between the equilibrium constant, K, and the standard state free energy change, ΔG^o, which is

ΔG = 0 = ΔG^o + RT ln K

To determine the dependence of the cell voltage on concentrations, pressure and temperature, all we need to do is substitute our relation between the Gibbs free energy change and the cell voltage

ΔG = -n F E

-n F E = -n F E^o + RT ln Q

E = E^o - (RT/nF) ln Q       (at any temperature)

It turns out that we have 10 fingers, and so we like to think in terms of powers of 10, and not powers of e. If we had e fingers, things would be different! So it is convenient to replace the natural logarithm with the base 10 logarithm

ln Q = ln 10^log₁₀Q = ln 10 x log₁₀ Q = 2.3026 log₁₀ Q

ln(10) RT/nF = 2.3026 x 8.314 CV/(mol K) x 298.15K / (n 96,485 C/mol) = 0.0592V /n

E = E^o - (0.0592V/n) log₁₀ Q       (at 25C)

The cell voltage at standard state and the equilibrium constant

At the point of equilibrium, there is no driving force in the reaction, the voltage is zero, an there is no net flow of electrons. At that point

E = 0 = E^o - (0.0592V/n) log₁₀ Q

so that

K = 10^nEo/0.0592V       (at 25C)

which relates the cell voltage measured at standard state to the equilibrium constant! We see that if we create an electrochemical cell at standard state conditions and 25C, we can measure the cell voltage and make a very accurate estimate of the equilibrium constant for the reaction.

Analyzing the concentration dependence of the Zn/Zn²⁺||Cu²⁺|Cu galvanic cell

For the galvanic cell at 25C we find

E_{Zn/Zn²⁺||Cu²⁺|Cu} = E^o_{Zn/Zn²⁺||Cu²⁺|Cu} - (0.0592V/n) log₁₀ Q       (at 25C)

Using the fact that for this reaction the standard state cell voltage is E^o_{Zn/Zn²⁺||Cu²⁺|Cu}=1.10V, Q=[Zn²⁺]/[Cu²⁺], and n=2 mol

E_{Zn/Zn²⁺||Cu²⁺|Cu} = 1.10 V - 0.0296V log₁₀ [Zn²⁺]/[Cu²⁺]       (at 25C)

This is a very interesting result! We can see that for every change in the reaction quotient by a power of 10, we will change the cell voltage by 0.0296V, up or down. For example, at standard state conditions [Zn²⁺]=[Cu²⁺]=1.0M so that Q=1, log₁₀ 1 = 0, and

E_{Zn/Zn²⁺||Cu²⁺|Cu} = 1.10V

the cell voltage at standard state.

If we increase the concentration of zinc relative to that of copper so that [Zn²⁺]=1.0M and [Cu²⁺]=0.1M, Q=10, log₁₀Q=1 and

E_{Zn/Zn²⁺||Cu²⁺|Cu} = 1.07V

and we have moved closer to equilibrium (where the cell voltage would be zero).

If we decrease the concentration of zinc relative to that of copper so that [Zn²⁺]=0.01M and [Cu²⁺]=0.1M, Q=0.1, log₁₀Q=-1 and

E_{Zn/Zn²⁺||Cu²⁺|Cu} = 1.13V

and we have moved farther from equilibrium.

Finally, using the standard state cell voltage we can determine the equilibrium constant for this reaction

K = [Zn²⁺]/[Cu²⁺] = 10^nEo/0.0592V = 10^37.2 = 1 x 10³⁷       (at 25C)

where n=2 mol and E^o=1.10V. In thinking about the number of significant figures, recall the rules that we learned for significant figures and pH.

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Predicting the change in cell voltage as a function of solution concentrations

Question: Determine the cell voltage for the galvanic cell Co/Co²⁺||Ni²⁺|Ni for the following concentration ratios at 25C.

[Co²⁺]     [Ni²⁺]
1.0M        1.0M
0.1M        1.0M
0.01M        1.0M
0.1M        0.1M
0.1M        0.001M

You will find that the system can attain the standard state cell voltage at conditions other than standard state.

You can check your answers here.

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