ΔG, ΔG° and Equilibrium Constant

CHENernst Equation & Thermodynamics

This subtopic links cell potential, Gibbs energy and the equilibrium constant. It tests the sign of $\Delta G$ for spontaneity and the conversion between $E^\circ_{cell}$ , $\Delta G^\circ$ and $K$ , including back-calculating $E^\circ_{cell}$ from a given $\Delta G^\circ$ .

Core relations

\Delta G^\circ = -nFE^\circ_{cell} = -RT\ln K = -2.303RT\log K

F = 96500\,\text{C}

R = 8.314\,\text{J K}^{-1}\text{mol}^{-1}

T

in K.

EMF and K (298 K)

E^\circ_{cell} = \dfrac{0.0591}{n}\log K \qquad \log K = \dfrac{nE^\circ_{cell}}{0.0591}

n

= electrons exchanged in the balanced reaction.

Non-standard Gibbs energy

\Delta G = \Delta G^\circ + RT\ln Q

Q

= reaction quotient; at equilibrium

Q = K

and

\Delta G = 0

Core relations: $\Delta G^\circ = -nFE^\circ_{cell}$ and $\Delta G^\circ = -RT\ln K$ , so $-nFE^\circ_{cell} = -RT\ln K$ .
Spontaneity: a reaction is spontaneous when $\Delta G < 0$ , which corresponds to $E_{cell} > 0$ ; at equilibrium $\Delta G = 0$ and $E_{cell} = 0$ .
Use $F = 96500\,\text{C mol}^{-1}$ ; $\Delta G^\circ$ comes out in joules, so divide by $1000$ to report $\text{kJ mol}^{-1}$ .
At $298\,\text{K}$ : $\log K = \dfrac{nE^\circ_{cell}}{0.0591}$ (equivalently $E^\circ_{cell} = \dfrac{0.0591}{n}\log K$ ).
Reverse direction: from a given $\Delta G^\circ$ , $E^\circ_{cell} = \dfrac{-\Delta G^\circ}{nF}$ (watch the sign — a negative $\Delta G^\circ$ gives a positive $E^\circ_{cell}$ ).
Magnitude check: positive $E^\circ_{cell}$ gives negative $\Delta G^\circ$ and $K > 1$ (products favoured); negative $E^\circ_{cell}$ gives positive $\Delta G^\circ$ and $K < 1$ .
Distinguish $\Delta G^\circ$ (standard, fixed for the reaction) from $\Delta G$ (depends on actual concentrations via $\Delta G = \Delta G^\circ + RT\ln Q$ ).
$\Delta G^\circ$ also equals the maximum non-expansion (electrical) work obtainable: $w_{max} = \Delta G^\circ = -nFE^\circ_{cell}$ .
Worked check ( $E^\circ_{cell}=0.46\,\text{V}$ , $n=2$ ): $\Delta G^\circ = -2\times96500\times0.46 \approx -88.8\,\text{kJ mol}^{-1}$ ; $\log K = \dfrac{2\times0.46}{0.0591} \approx 15.6$ , so $K \approx 4\times10^{15}$ .
Worked check ( $\Delta G^\circ=-200\,\text{kJ}$ , $n=2$ ): $E^\circ_{cell} = \dfrac{-(-200000)}{2\times96500} \approx 1.04\,\text{V}$ .
Because $\Delta G^\circ$ scales with $n$ but $E^\circ_{cell}$ does not, always read $n$ from the balanced equation before converting between the two.

Where the marks go

Sign slip in $\Delta G^\circ = -nFE^\circ_{cell}$ — a spontaneous cell ( $E^\circ_{cell}>0$ ) must give NEGATIVE $\Delta G^\circ$ ; a positive answer signals a sign error.
Reporting $\Delta G^\circ$ in joules as if it were kilojoules (or forgetting to multiply $\text{kJ}$ back to $\text{J}$ before dividing by $nF$ ).
Confusing $\Delta G$ (concentration-dependent) with $\Delta G^\circ$ (standard, fixed) — only $\Delta G$ uses $\ln Q$ .
Using $0.0591$ at a temperature other than $298\,\text{K}$ when relating $E^\circ_{cell}$ and $K$ .
Forgetting that $\Delta G^\circ$ depends on $n$ , so the same $E^\circ_{cell}$ gives different $\Delta G^\circ$ values for differently scaled equations.

How the board asks it

Numericalgibbs energy from cell potential
For a cell with $E^\circ_{cell} = 0.46\,\text{V}$ and $n = 2$ , calculate the standard Gibbs energy change $\Delta G^\circ$ for the reaction (take $F = 96500\,\text{C mol}^{-1}$ ). Express your answer in $\text{kJ mol}^{-1}$ .
Numericalequilibrium constant from cell potential
The standard EMF of a cell is $E^\circ_{cell} = 0.295\,\text{V}$ at $298\,\text{K}$ for a reaction involving $n = 2$ electrons. Calculate the equilibrium constant $K$ for the cell reaction using $\log K = \frac{nE^\circ_{cell}}{0.0591}$ .
Numericalback-calculating cell potential from gibbs energy
Given $\Delta G^\circ = -200\,\text{kJ mol}^{-1}$ for a cell reaction in which $n = 2$ , calculate the standard cell potential $E^\circ_{cell}$ (take $F = 96500\,\text{C mol}^{-1}$ ).
Give reasonssign of gibbs energy and spontaneity
For a particular cell, $E^\circ_{cell}$ is found to be positive. Give reasons to state whether $\Delta G^\circ$ is positive or negative, and whether the equilibrium constant $K$ is greater than or less than $1$ .
Define / statemaximum electrical work of a cell
State the relationship between the standard Gibbs energy change of a cell reaction and the maximum electrical work obtainable from the cell, and write the expression connecting $\Delta G^\circ$ , $n$ , $F$ and $E^\circ_{cell}$ .
Distinguishgibbs energy versus standard gibbs energy
Distinguish between $\Delta G$ and $\Delta G^\circ$ for a cell reaction, stating clearly which one depends on the reaction quotient $Q$ and which remains fixed for a given reaction.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.