17.4 Potential, Free Energy, and Equilibrium

Electrochemistry and Thermodynamics

Cell potential, free energy, and equilibrium constants are three ways of describing the same thing: whether a redox reaction will happen spontaneously. The powerful part is that if you know any one of these values, you can calculate the other two.

Cell Potential and Energy Relationships

Cell potential ($E_{cell}$) measures the voltage between two half-cells in a galvanic (voltaic) cell. Think of it as the "driving force" pushing electrons through the wire from anode to cathode.

This voltage connects directly to two thermodynamic quantities:

• Gibbs free energy ($\Delta G$), which tells you if a reaction is spontaneous:

$\Delta G = -nFE_{cell}$

where $n$ is the number of moles of electrons transferred and $F$ is Faraday's constant (96,485 C/mol).

• Equilibrium constant ($K$), which tells you how far a reaction goes toward products:

$\Delta G = -RT \ln K$

where $R$ is the gas constant (8.314 J/mol·K) and $T$ is temperature in Kelvin.

Notice the signs here. A positive $E_{cell}$ gives a negative $\Delta G$, which means the reaction is spontaneous. A large positive $E_{cell}$ also corresponds to a large $K$, meaning the reaction strongly favors products at equilibrium.

Standard cell potential ($E_{cell}^{\circ}$) is the cell potential measured under standard conditions: 1 M concentration for all dissolved species, 1 atm pressure for gases, and 25°C (298 K). The standard versions of the equations work the same way:

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

Since both expressions equal $\Delta G^{\circ}$, you can set them equal to link $E_{cell}^{\circ}$ directly to $K$.

Calculations for Electrochemical Systems

Calculating standard cell potential:

$E_{cell}^{\circ} = E_{cathode}^{\circ} - E_{anode}^{\circ}$

Both values come from a table of standard reduction potentials. The half-reaction with the higher (more positive) reduction potential becomes the cathode; the other becomes the anode.

Calculating free energy from cell potential:

$\Delta G = -nFE_{cell}$

• Negative $\Delta G$ → spontaneous (galvanic cell)
• Positive $\Delta G$ → non-spontaneous (requires external energy, i.e., electrolytic cell)

For example, if $E_{cell}^{\circ} = +1.10$ V and $n = 2$ (as in the Zn/Cu cell):

$\Delta G^{\circ} = -(2)(96{,}485)(1.10) = -212{,}267 \text{ J/mol} \approx -212 \text{ kJ/mol}$

That large negative value confirms the reaction is strongly spontaneous.

Calculating the equilibrium constant from cell potential:

Starting from $\Delta G^{\circ} = -nFE_{cell}^{\circ} = -RT \ln K$, you can rearrange to get:

$\ln K = \frac{nFE_{cell}^{\circ}}{RT}$

or equivalently:

$K = e^{\frac{nFE_{cell}^{\circ}}{RT}}$

Even a modest positive $E_{cell}^{\circ}$ produces a very large $K$. For the Zn/Cu cell above, $K$ is on the order of $10^{37}$, meaning the reaction essentially goes to completion.

The Nernst Equation

The Nernst equation lets you calculate cell potential under non-standard conditions (when concentrations aren't all 1 M):

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

At 25°C, this is often written using base-10 logarithms:

$E_{cell} = E_{cell}^{\circ} - \frac{0.0592}{n} \log Q$

The reaction quotient ($Q$) has the same form as the equilibrium expression. For a reaction $aA + bB \rightarrow cC + dD$:

$Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}$

How $Q$ affects cell potential:

• If $Q < K$ (reaction hasn't reached equilibrium yet), $E_{cell} > 0$ and the cell still produces voltage.
• If $Q = K$ (equilibrium), $E_{cell} = 0$. The battery is "dead."
• If $Q > K$, $E_{cell} < 0$, meaning the reverse reaction is favored.

At equilibrium, setting $E_{cell} = 0$ and $Q = K$ in the Nernst equation gives:

$0 = E_{cell}^{\circ} - \frac{RT}{nF} \ln K$

which rearranges to $E_{cell}^{\circ} = \frac{RT}{nF} \ln K$. This is the same relationship derived in the previous section, confirming that all three quantities are interconnected.

You can also use Le Chatelier's principle to predict the direction of change qualitatively. Increasing reactant concentration (lowering $Q$) increases $E_{cell}$, while increasing product concentration (raising $Q$) decreases it.