DAT General Chemistry: Cell Potential, Free Energy, and the Equilibrium Constant

DAT High-Yield Concept

Spontaneity Trio: A spontaneous redox reaction under standard state conditions ( $\pu{25 degC}$ ) always has $E^\circ_{\text{cell}} > 0$ , $\Delta G^\circ < 0$ , and $K > 1$ .
Key Equation 1: $\Delta G^\circ = -nFE^\circ_{\text{cell}}$ , where $n$ is the number of moles of transferred electrons and $F = \pu{96485 C/mol e^-}$ .
Key Equation 2: $E^\circ_{\text{cell}} = \frac{RT}{nF} \ln K = \frac{0.0592\text{ V}}{n} \log K$ at $\pu{298 K}$ .
Intensive vs. Extensive: Standard cell potential $E^\circ_{\text{cell}}$ is an intensive property (multiplying reaction coefficients does not change $E^\circ_{\text{cell}}$ ), while free energy $\Delta G^\circ$ is an extensive property (scales directly with $n$ ).

Cell Potential and Free Energy

The electrical work performed by a galvanic cell operating at constant temperature and pressure equals the change in standard Gibbs free energy ( $\Delta G^\circ$ ). The maximum electrical work is given by the product of total charge transferred ( $nF$ ) and standard cell potential ( $E^\circ_{\text{cell}}$ ):

\Delta G^\circ = -nFE^\circ_{\text{cell}}

In this expression, $n$ represents the stoichiometric moles of electrons transferred in the balanced redox equation, $F$ is Faraday's constant ( $\pu{96485 C/mol e^-}$ or $\pu{96485 J/(V mol e^-)}$ ), and $E^\circ_{\text{cell}}$ is expressed in volts ( $\pu{1 V} = \pu{1 J/C}$ ).

Thermodynamic Sign Conventions:

A positive standard cell potential ( $E^\circ_{\text{cell}} > 0$ ) corresponds to a negative free energy change ( $\Delta G^\circ < 0$ ), indicating a thermodynamically favored (exergonic) forward reaction under standard conditions.

Variation in free energy during an exergonic reaction according to transition-state theory (Source: Microbialmatt · CC BY-SA 4.0 · Wikimedia Commons)

Relating Standard Cell Potential, Standard Free Energy, and K

From chemical thermodynamics, standard Gibbs free energy change is directly related to the reaction equilibrium constant ( $K$ ) by $\Delta G^\circ = -RT \ln K$ . Combining this with $\Delta G^\circ = -nFE^\circ_{\text{cell}}$ yields the master link connecting electrochemistry and chemical equilibrium:

-nFE^\circ_{\text{cell}} = -RT \ln K \implies E^\circ_{\text{cell}} = \frac{RT}{nF} \ln K

At standard temperature ( $T = \pu{298.15 K}$ ), substituting $R = \pu{8.314 J/(mol K)}$ and converting from natural logarithms to base-10 logarithms ( $\ln K = 2.303 \log K$ ) simplifies the equation to:

E^\circ_{\text{cell}} = \frac{0.0592\text{ V}}{n} \log K

Worked Example: Quantitative Link Between $E^\circ_{\text{cell}}$ , $\Delta G^\circ$ , and $K$

Problem: Calculate $\Delta G^\circ$ and $K$ at $\pu{25 degC}$ ( $\pu{298 K}$ ) for the standard Daniell cell reaction: $\ce{Zn(s) + Cu^{2+}(aq) <=> Zn^{2+}(aq) + Cu(s)}$ , given $E^\circ_{\text{cell}} = \pu{+1.10 V}$ .

Step 1: Determine $n$ and calculate $\Delta G^\circ$ .

Half-reactions: Oxidation is $\ce{Zn(s) -> Zn^{2+}(aq) + 2e^-}$ ; Reduction is $\ce{Cu^{2+}(aq) + 2e^- -> Cu(s)}$ . Thus, $n = 2\text{ mol e}^-$ .

\Delta G^\circ = -(2\text{ mol e}^-) \times \left(96485 \frac{\text{J}}{\text{V} \cdot \text{mol e}^-}\right) \times (+1.10\text{ V}) = -212267\text{ J/mol} = \pu{-212 kJ/mol}

Step 2: Calculate the equilibrium constant $K$ .

E^\circ_{\text{cell}} = \frac{0.0592\text{ V}}{n} \log K \implies 1.10\text{ V} = \frac{0.0592\text{ V}}{2} \log K = 0.0296\text{ V} \cdot \log K

\log K = \frac{1.10}{0.0296} = 37.16 \implies K = 10^{37.16} = 1.4 \times 10^{38}

DAT Exam Trap: Multiplying a chemical equation by a stoichiometric factor changes

\Delta G^\circ

because free energy is extensive, but it does not change

E^\circ_{\text{cell}}

because cell potential is intensive (joules per coulomb). Always keep

E^\circ_{\text{cell}}

unchanged while scaling

n

\Delta G^\circ = -nFE^\circ_{\text{cell}}

Summary of Thermodynamic Indicators

Reaction State	$E^\circ_{\text{cell}}$	$\Delta G^\circ$	Equilibrium Constant ( $K$ )
Product-favored / Spontaneous	$> 0$ (Positive)	$< 0$ (Negative)	$> 1$ (Log $K > 0$ )
Reactant-favored / Nonspontaneous	$< 0$ (Negative)	$> 0$ (Positive)	$< 1$ (Log $K < 0$ )
At Standard Equilibrium	$= 0$	$= 0$	$= 1$

Mnemonic:

References

Sources returned by Google Search grounding during lesson verification.

Chad's Prep - Cell Potential, Delta G, and the Equilibrium Constant — chadsprep.com
OpenStax Chemistry 2e - Potential, Free Energy, and Equilibrium — openstax.org

Cell Potential, Free Energy, and the Equilibrium Constant

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