## Potential, Free Energy, and Equilibrium

### Learning Outcomes

• Explain the relations between potential, free energy change, and equilibrium constants
• Perform calculations involving the relations between cell potentials, free energy changes, and equilibrium
• Use the Nernst equation to determine cell potentials under nonstandard conditions

So far in this module, the relationship between the cell potential and reaction spontaneity has been described, suggesting a link to the free energy change for the reaction (see chapter on thermodynamics). The interpretation of potentials as measures of oxidant strength was presented, bringing to mind similar measures of acid-base strength as reflected in equilibrium constants (see the chapter on acid-base equilibria). This section provides a summary of the relationships between potential and the related thermodynamic properties $\Delta G$ and K.

### $E^\circ$ and $\Delta G^\circ$

The standard free energy change of a process, $\Delta G^\circ$, was defined in a previous chapter as the maximum work that could be performed by a system, $w_{max}$. In the case of a redox reaction taking place within a galvanic cell under standard state conditions, essentially all the work is associated with transferring the electrons from reductant-to-oxidant, $w_{elec}$:

$\Delta G^\circ = w_{max} = w_{elec}$

The work associated with transferring electrons is determined by the total amount of charge (coulombs) transferred and the cell potential:

$\Delta G^\circ = w_{elec} = -nFE_{cell}^\circ$

$\Delta G^\circ = -nFE_{cell}^\circ$

where n is the number of moles of electrons transferred, F is Faraday’s constant, and $E_{cell}^\circ$ is the standard cell potential. The relation between free energy change and standard cell potential confirms the sign conventions and spontaneity criteria previously discussed for both of these properties: spontaneous redox reactions exhibit positive potentials and negative free energy changes.

### $E^\circ$ and K

Combining a previously derived relation between $\Delta G^\circ$ and K (see the chapter on thermodynamics) and the equation above relating $\Delta G^\circ$ and $E_{cell}^\circ$ yields the following:

$\Delta G^\circ = -RT\text{ln}K = -nFE_{cell}^\circ$

$E_{cell}^\circ = (\dfrac{RT}{nF})\text{ln}K$

This equation indicates redox reactions with large (positive) standard cell potentials will proceed far towards completion, reaching equilibrium when the majority of reactant has been converted to product. A summary of the relations between $E^\circ$, $\Delta G^\circ$ and $K$ is depicted in Figure 1, and a table correlating reaction spontaneity to values of these properties is provided in Table 1.

Figure 1. Graphic depicting the relation between three important thermodynamic properties.

Table 1

 K ΔG° E°cell > 1 < 0 > 0 Reaction is spontaneous under standard conditions Products more abundant at equilibrium < 1 > 0 < 0 Reaction is non-spontaneous under standard conditions Reactants more abundant at equilibrium = 1 = 0 = 0 Reaction is at equilibrium under standard conditions Reactants and products equally abundant

### Example 1: Equilibrium Constants, Standard Cell Potentials, and Standard Free Energy Changes

What is the standard free energy change and equilibrium constant for the following reaction at 25 °C?

$2{\text{Ag}}^{\text{+}}\left(aq\right)+\text{Fe}\left(s\right)\rightleftharpoons \text{2Ag}\left(s\right)+{\text{Fe}}^{2+}\left(aq\right)$

What is the standard free energy change and the equilibrium constant for the following reaction at room temperature? Is the reaction spontaneous?

$\text{Sn}\left(s\right)+2{\text{Cu}}^{2+}\left(aq\right)\rightleftharpoons {\text{Sn}}^{2+}\left(aq\right)+2{\text{Cu}}^{\text{+}}\left(aq\right)$

### Potentials at Nonstandard Conditions: The Nernst Equation

Most of the redox processes that interest science and society do not occur under standard state conditions, and so the potentials of these systems under nonstandard conditions are a property worthy of attention. Having established the relationship between potential and free energy change in this section, the previously discussed relation between free energy change and reaction mixture composition can be used for this purpose.

$\Delta G = \Delta G^\circ + RT\text{ln}Q$

Notice the reaction quotient, $Q$, appears in this equation, making the free energy change dependent upon the composition of the reaction mixture. Substituting the equation relating free energy change to cell potential yields the Nernst equation:

$-nFE_{cell} = -nFE_{cell}^\circ + RT\text{ln}Q$

$E_{cell} = E_{cell}^\circ - \dfrac{RT}{nF}\text{ln}Q$

This equation describes how the potential of a redox system (such as a galvanic cell) varies from its standard state value, specifically, showing it to be a function of the number of electrons transferred, $n$ the temperature, $T$, and the reaction mixture composition as reflected in $Q$. A convenient form of the Nernst equation for most work is one in which values for the fundamental constants (R and F) and a factor converting from natural to base-10 logarithms have been included.

$E_{cell} = -E_{cell}^\circ - \dfrac{0.0592 \text{V}}{n}\text{log}Q$

### Example 2: Predicting Redox Spontaneity Under Nonstandard Conditions

Use the Nernst equation to predict the spontaneity of the redox reaction shown below.

$\text{Co}\left(s\right)+{\text{Fe}}^{2+}\left(aq,1.94M\right)\longrightarrow {\text{Co}}^{2+}\left(aq\text{, 0.15}M\right)+\text{Fe}\left(s\right)$

What is the cell potential for the following reaction at room temperature?

$\text{Al}\left(s\right)\mid {\text{Al}}^{3+}\left(aq,0.15M\right)\parallel {\text{Cu}}^{2+}\left(aq,0.025M\right)\mid \text{Cu}\left(s\right)$

What are the values of n and Q for the overall reaction? Is the reaction spontaneous under these conditions?

concentration cell is constructed by connecting two nearly identical half-cells, each based on the same half-reaction and using the same electrode, varying only in the concentration of one redox species. The potential of a concentration cell, therefore, is determined only by the difference in concentration of the chosen redox species. The example problem below illustrates the use of the Nernst equation in calculations involving concentration cells.

### Example 3: Concentration Cells

What is the cell potential of the concentration cell described by

$\text{Zn}\left(s\right)\mid {\text{Zn}}^{2+}\left(aq\text{, 0.10}M\right)\parallel {\text{Zn}}^{2+}\left(aq\text{, 0.50}M\right)\mid \text{Zn}\left(s\right)$

The concentration cell above was allowed to operate until the cell reaction reached equilibrium. What are the cell potential and the concentrations of zinc(II) in each half-cell for the cell now?

### Key Concepts and Summary

Potential is a thermodynamic quantity reflecting the intrinsic driving force of a redox process, and it is directly related to the free energy change and equilibrium constant for the process. For redox processes taking place in electrochemical cells, the maximum (electrical) work done by the system is easily computed from the cell potential and the reaction stoichiometry and is equal to the free energy change for the process. The equilibrium constant for a redox reaction is logarithmically related to the reaction’s cell potential, with larger (more positive) potentials indicating reactions with greater driving force that equilibrate when the reaction has proceeded far towards completion (large value of K). Finally, the potential of a redox process varies with the composition of the reaction mixture, being related to the reactions standard potential and the value of its reaction quotient, Q, as described by the Nernst equation.

#### Key Equations

• ${E}_{\text{cell}}^{\circ }=\dfrac{RT}{nF}\ln{K}$
• ${E}_{\text{cell}}^{\circ }=\dfrac{0.02\text{57 V}}{n}\ln{K}=\dfrac{0.0\text{592 V}}{n}\log{K}\left(\text{at 298.15}K\right)$
• ${E}_{\text{cell}}={E}_{\text{cell}}^{\circ }-\dfrac{RT}{nF}\ln{Q}\text{(Nernst equation)}$
• ${E}_{\text{cell}}={E}_{\text{cell}}^{\circ }-\dfrac{0.02\text{57 V}}{n}\ln{Q}={E}_{\text{cell}}^{\circ }-\dfrac{0.05\text{92 V}}{n}\log{Q}\left(\text{at 298.15}K\right)$
• $\Delta{G}=-nF{E}_{\text{cell}}$
• $\Delta {G}^{\circ }=-nF{E}_{\text{cell}}^{\circ }$
• ${w}_{\text{ele}}={w}_{\text{max}}=-nF{E}_{\text{cell}}$

### Try It

1. For the standard cell potentials given here, determine the ΔG° for the cell in kJ.
1. 0.000 V, n = 2
2. +0.434 V, n = 2
3. −2.439 V, n = 1
2. For the ΔG° values given here, determine the standard cell potential for the cell.
1. 12 kJ/mol, n = 3
2. −45 kJ/mol, n = 1
3. Determine the standard cell potential and the cell potential under the stated conditions for the electrochemical reactions described here. State whether each is spontaneous or nonspontaneous under each set of conditions at 298.15 K.
1. $\text{Hg}\left(l\right)+{\text{S}}^{2-}\left(aq\text{, 0.10}M\right)+2{\text{Ag}}^{\text{+}}\left(aq\text{, 0.25}M\right)\longrightarrow 2\text{Ag}\left(s\right)+\text{HgS}\left(s\right)$
2. The galvanic cell made from a half-cell consisting of an aluminum electrode in 0.015 M aluminum nitrate solution and a half-cell consisting of a nickel electrode in 0.25 M nickel(II) nitrate solution.
3. The cell made of a half-cell in which 1.0 M aqueous bromine is oxidized to 0.11 M bromide ion and a half-cell in which aluminum ion at 0.023 M is reduced to aluminum metal. Assume the standard reduction potential for Br2(l) is the same as that of Br2(aq).
4. Determine ΔG and ΔG° for each of the reactions in the previous problem.
5. Use the data in Standard Electrode (Half-Cell) Potentials to determine the equilibrium constant for the following reactions. Assume 298.15 K if no temperature is given.
1. $\text{AgCl}\left(s\right)\rightleftharpoons {\text{Ag}}^{\text{+}}\left(aq\right)+{\text{Cl}}^{-}\left(aq\right)$
2. $\text{CdS}\left(s\right)\rightleftharpoons {\text{Cd}}^{2+}\left(aq\right)+{\text{S}}^{2-}\left(aq\right)\text{at 377 K}$
3. ${\text{Hg}}^{2+}\left(aq\right)+4{\text{Br}}^{-}\left(aq\right)\rightleftharpoons {\left[{\text{HgBr}}_{4}\right]}^{2-}\left(aq\right)$
4. ${\text{H}}_{2}\text{O}\left(l\right)\rightleftharpoons {\text{H}}^{\text{+}}\left(aq\right)+{\text{OH}}^{-}\left(aq\right)\text{at 25}^\circ C$

## Glossary

concentration cell: galvanic cell in which the two half-cells are the same except for the concentration of the solutes; spontaneous when the overall reaction is the dilution of the solute

electrical work (wele): negative of total charge times the cell potential; equal to wmax for the system, and so equals the free energy change (ΔG)

Faraday’s constant (F): charge on 1 mol of electrons; F = 96,485 C/mol e

Nernst equation: equation that relates the logarithm of the reaction quotient (Q) to nonstandard cell potentials; can be used to relate equilibrium constants to standard cell potentials