# 20.6: Cell Potential Under Nonstandard Conditions

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The *Nernst Equation* enables the determination of cell potential under non-standard conditions. It relates the measured cell potential to the reaction quotient and allows the accurate determination of equilibrium constants (including solubility constants).

## The Effect of Concentration on Cell Potential: The Nernst Equation

Recall that the actual free-energy change for a reaction under nonstandard conditions, \(\Delta{G}\), is given as follows:

\[\Delta{G} = \Delta{G°} + RT \ln Q \label{Eq1}\]

We also know that ΔG = −nFE_{cell} and ΔG° = −nFE°_{cell}. Substituting these expressions into Equation \(\ref{Eq1}\), we obtain

\[−nFE_{cell} = −nFE^o_{cell} + RT \ln Q \label{Eq2}\]

Dividing both sides of this equation by \(−nF\),

\[E_\textrm{cell}=E^\circ_\textrm{cell}-\left(\dfrac{RT}{nF}\right)\ln Q \label{Eq3}\]

Equation \(\ref{Eq3}\) is called the **Nernst equation**, after the German physicist and chemist Walter Nernst (1864–1941), who first derived it. The Nernst equation is arguably the most important relationship in electrochemistry. When a redox reaction is at equilibrium (\(ΔG = 0\)), then Equation \(\ref{Eq3}\) reduces to Equation \(\ref{Eq31}\) and \(\ref{Eq32}\) because \(Q = K\), and there is no net transfer of electrons (i.e., E_{cell} = 0).

\[E_\textrm{cell}=E^\circ_\textrm{cell}-\left(\dfrac{RT}{nF}\right)\ln K = 0 \label{Eq31}\]

since

\[E^\circ_\textrm{cell}= \left(\dfrac{RT}{nF}\right)\ln K \label{Eq32}\]

Substituting the values of the constants into Equation \(\ref{Eq3}\) with T = 298 K and converting to base-10 logarithms give the relationship of the actual cell potential (E_{cell}), the standard cell potential (E°_{cell}), and the reactant and product concentrations at room temperature (contained in \(Q\)):

\[E_{\textrm{cell}}=E^\circ_\textrm{cell}-\left(\dfrac{\textrm{0.0591 V}}{n}\right)\log Q \label{Eq4}\]

Equation \(\ref{Eq4}\) allows us to calculate the potential associated with any electrochemical cell at 298 K for any combination of reactant and product concentrations under any conditions. We can therefore determine the spontaneous direction of any redox reaction under any conditions, as long as we have tabulated values for the relevant standard electrode potentials. Notice in Equation \(\ref{Eq4}\) that the cell potential changes by 0.0591/n V for each 10-fold change in the value of Q because log 10 = 1.

Applying the Nernst equation to a simple electrochemical cell such as the Zn/Cu cell allows us to see how the cell voltage varies as the reaction progresses and the concentrations of the dissolved ions change. Recall that the overall reaction for this cell is as follows:

\[Zn(s) + Cu^{2+}(aq) \rightarrow Zn^{2+}(aq) + Cu(s)\;\;\;E°cell = 1.10 V \label{Eq5}\]

The reaction quotient is therefore \(Q = [Zn^{2+}]/[Cu^{2+}]\). Suppose that the cell initially contains 1.0 M Cu^{2}^{+} and 1.0 × 10^{−6} M Zn^{2}^{+}. The initial voltage measured when the cell is connected can then be calculated from Equation \(\ref{Eq4}\):

\[\begin{align}E_\textrm{cell} & =E^\circ_\textrm{cell}-\left(\dfrac{\textrm{0.0591 V}}{n}\right)\log\dfrac{[\mathrm{Zn^{2+}}]}{[\mathrm{Cu^{2+}}]}\\

& =\textrm{1.10 V}-\left(\dfrac{\textrm{0.0591 V}}{2}\right)\log\left(\dfrac{1.0\times10^{-6}}{1.0}\right)=\textrm{1.28 V}\end{align} \label{Eq6}\]

Thus the initial voltage is greater than E° because \(Q<1\). As the reaction proceeds, [Zn^{2}^{+}] in the anode compartment increases as the zinc electrode dissolves, while [Cu^{2}^{+}] in the cathode compartment decreases as metallic copper is deposited on the electrode. During this process, the ratio Q = [Zn^{2}^{+}]/[Cu^{2}^{+}] steadily increases, and the cell voltage therefore steadily decreases. Eventually, [Zn^{2}^{+}] = [Cu^{2}^{+}], so Q = 1 and E_{cell} = E°_{cell}. Beyond this point, [Zn^{2}^{+}] will continue to increase in the anode compartment, and [Cu^{2}^{+}] will continue to decrease in the cathode compartment. Thus the value of Q will increase further, leading to a further decrease in E_{cell}. When the concentrations in the two compartments are the opposite of the initial concentrations (i.e., 1.0 M Zn^{2}^{+} and 1.0 × 10^{−6} M Cu^{2}^{+}), Q = 1.0 × 10^{6}, and the cell potential will be reduced to 0.92 V.

The variation of E_{cell} with \(\log{Q}\) over this range is linear with a slope of −0.0591/n, as illustrated in Figure \(\PageIndex{1}\). As the reaction proceeds still further, \(Q\) continues to increase, and E_{cell} continues to decrease. If neither of the electrodes dissolves completely, thereby breaking the electrical circuit, the cell voltage will eventually reach zero. This is the situation that occurs when a battery is “dead.” The value of \(Q\) when E_{cell} = 0 is calculated as follows:

E^\circ &=\left(\dfrac{\textrm{0.0591 V}}{n}\right)\log Q \\

\log Q &=\dfrac{E^\circ n}{\textrm{0.0591 V}}=\dfrac{(\textrm{1.10 V})(2)}{\textrm{0.0591 V}}=37.23 \\

Q &=10^{37.23}=1.7\times10^{37}\end{align} \label{Eq7}\]

Recall that at equilibrium, \(Q = K\). Thus the equilibrium constant for the reaction of Zn metal with Cu^{2}^{+} to give Cu metal and Zn^{2}^{+} is 1.7 × 10^{37} at 25°C.

## Concentration Cells

A voltage can also be generated by constructing an electrochemical cell in which each compartment contains the same redox active solution but at different concentrations. The voltage is produced as the concentrations equilibrate. Suppose, for example, we have a cell with 0.010 M AgNO_{3} in one compartment and 1.0 M AgNO_{3} in the other. The cell diagram and corresponding half-reactions are as follows:

\[Ag(s)\,|\,Ag^+(aq, 0.010 \;M)\,||\,Ag+(aq, 1.0 \;M)\,|\,Ag(s) \label{Eq8}\]

cathode:

\[Ag^+(aq, 1.0\; M) + e^− \rightarrow Ag(s) \label{Eq9}\]

anode:

\[Ag(s) \rightarrow Ag^+(aq, 0.010\; M) + e^− \label{Eq10}\]

Overall

\[Ag^+(aq, 1.0 \;M) \rightarrow Ag^+(aq, 0.010\; M) \label{Eq11}\]

As the reaction progresses, the concentration of \(Ag^+\) will increase in the left (oxidation) compartment as the silver electrode dissolves, while the \(Ag^+\) concentration in the right (reduction) compartment decreases as the electrode in that compartment gains mass. The total mass of \(Ag(s)\) in the cell will remain constant, however. We can calculate the potential of the cell using the Nernst equation, inserting 0 for E°_{cell} because E°_{cathode} = −E°_{anode}:

\[\begin{align} E_\textrm{cell}&=E^\circ_\textrm{cell}-\left(\dfrac{\textrm{0.0591 V}}{n}\right)\log Q \\[4pt] &=0-\left(\dfrac{\textrm{0.0591 V}}{1}\right)\log\left(\dfrac{0.010}{1.0}\right) \\[4pt] &=\textrm{0.12 V} \end{align}\]

An electrochemical cell of this type, in which the anode and cathode compartments are identical except for the concentration of a reactant, is called a **concentration cell**. As the reaction proceeds, the difference between the concentrations of Ag^{+} in the two compartments will decrease, as will E_{cell}. Finally, when the concentration of Ag^{+} is the same in both compartments, equilibrium will have been reached, and the measured potential difference between the two compartments will be zero (E_{cell} = 0).

## Using Cell Potentials to Measure Solubility Products

Because voltages are relatively easy to measure accurately using a voltmeter, electrochemical methods provide a convenient way to determine the concentrations of very dilute solutions and the solubility products (K_{sp}) of sparingly soluble substances. As you learned previously, solubility products can be very small, with values of less than or equal to 10^{−30}. Equilibrium constants of this magnitude are virtually impossible to measure accurately by direct methods, so we must use alternative methods that are more sensitive, such as electrochemical methods.

To understand how an electrochemical cell is used to measure a solubility product, consider the cell shown in Figure \(\PageIndex{1}\), which is designed to measure the solubility product of silver chloride: K_{sp} = [Ag^{+}][Cl^{−}]. In one compartment, the cell contains a silver wire inserted into a 1.0 M solution of Ag^{+}; the other compartment contains a silver wire inserted into a 1.0 M Cl^{−} solution saturated with AgCl. In this system, the Ag^{+} ion concentration in the first compartment equals K_{sp}. We can see this by dividing both sides of the equation for K_{sp} by [Cl^{−}] and substituting: [Ag^{+}] = K_{sp}/[Cl^{−}] = K_{sp}/1.0 = K_{sp}. The overall cell reaction is as follows:

Ag^{+}(aq, concentrated) → Ag^{+}(aq, dilute)

Thus the voltage of the concentration cell due to the difference in [Ag^{+}] between the two cells is as follows:

\[E_\textrm{cell}=\textrm{0 V}-\left(\dfrac{\textrm{0.0591 V}}{1}\right)\log\left(\dfrac{[\mathrm{Ag^+}]_\textrm{dilute}}{[\mathrm{Ag^+}]_\textrm{concentrated}}\right)= \label{Eq121}\]

By closing the circuit, we can measure the potential caused by the difference in [Ag+] in the two cells. In this case, the experimentally measured voltage of the concentration cell at 25°C is 0.580 V. Solving Equation \(\ref{Eq1}\)2 for Ksp,

K_\textrm{sp} & =1.5\times10^{-10}\end{align} \label{Eq13}\]

Thus a single potential measurement can provide the information we need to determine the value of the solubility product of a sparingly soluble salt.+

## Using Cell Potentials to Measure Concentrations

Another use for the Nernst equation is to calculate the concentration of a species given a measured potential and the concentrations of all the other species. We saw an example of this in Example \(\PageIndex{3}\), in which the experimental conditions were defined in such a way that the concentration of the metal ion was equal to K_{sp}. Potential measurements can be used to obtain the concentrations of dissolved species under other conditions as well, which explains the widespread use of electrochemical cells in many analytical devices. Perhaps the most common application is in the determination of [H^{+}] using a pH meter, as illustrated in Example 20.5.6.

## Summary

The Nernst equation can be used to determine the direction of spontaneous reaction for any redox reaction in aqueous solution. The Nernst equation allows us to determine the spontaneous direction of any redox reaction under any reaction conditions from values of the relevant standard electrode potentials. Concentration cells consist of anode and cathode compartments that are identical except for the concentrations of the reactant. Because ΔG = 0 at equilibrium, the measured potential of a concentration cell is zero at equilibrium (the concentrations are equal). A galvanic cell can also be used to measure the solubility product of a sparingly soluble substance and calculate the concentration of a species given a measured potential and the concentrations of all the other species.