Cell Potential and Free Energy

AP Chemistry· 45 min read

1. The Core Relationship Between E°cell and ΔG°★★☆☆☆⏱ 15 min

📘 Definition

ΔG° and E°cell Relationship

$Δ G^{\circ} = - n F E_{cell}^{\circ}$

The standard Gibbs free energy change of a redox reaction equals the negative product of moles of electrons transferred ( $n$ ), Faraday's constant ( $F$ ), and standard cell potential ( $E_{cell}^{\circ}$ ).

Example:

$E_{cell}^{\circ} = + 1.10$ V gives $Δ G^{\circ} < 0$ , so the reaction is spontaneous.

Each variable has a clear meaning: $n$ is the total number of moles of electrons transferred in the balanced full reaction, always a positive integer. Faraday's constant is approximated as $9.65 \times 1 0^{4}$ C/mol for AP Chemistry calculations. The negative sign creates the consistent spontaneity rule: positive $E_{cell}^{\circ}$ gives negative $Δ G^{\circ}$ (spontaneous), while negative $E_{cell}^{\circ}$ gives positive $Δ G^{\circ}$ (non-spontaneous).

📐 Worked Example

Calculate the standard Gibbs free energy change for the reaction: $Zn (s) + Cu^{2 +} (a q) \to Zn^{2 +} (a q) + Cu (s)$ . Given $E_{red}^{\circ} (Zn^{2 +} / Zn) = - 0.76$ V and $E_{red}^{\circ} (Cu^{2 +} / Cu) = + 0.34$ V.

1
First calculate $E_{cell}^{\circ}$ . Zn is oxidized (anode), Cu is reduced (cathode):
2
$E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} = 0.34 - (-0.76) = 1.10 \ \text{V}$
3
Identify $n$ : 2 moles of electrons are transferred in the balanced reaction, so $n = 2$ .
4
Substitute into the core equation, noting that 1 V·C = 1 J:
5
$\Delta G^\circ = -(2)(96500 \ \text{C/mol})(1.10 \ \text{V}) = -212300 \ \text{J/mol}$
6
Convert to kJ and confirm spontaneity:
7
$\Delta G^\circ = -212 \ \text{kJ/mol}$
8
$Δ G^{\circ}$ is negative, so the reaction is spontaneous under standard conditions.

2. Relating E°cell to the Equilibrium Constant K★★★☆☆⏱ 20 min

We can combine the relationship $Δ G^{\circ} = - R T ln K$ with $Δ G^{\circ} = - n F E_{cell}^{\circ}$ to get a direct connection between $E_{cell}^{\circ}$ and the equilibrium constant $K$ . This lets us predict how far a reaction will go at equilibrium directly from cell potential data.

📘 Definition

E°cell and K Relationship

$E_{cell}^{\circ} = \frac{0.0592 V}{n} lo g K (at 2 5^{\circ} C)$

The simplified base-10 log form is used almost exclusively for AP Chemistry calculations at 25°C, the standard temperature for most exam problems.

The same spontaneity rules apply here: $E_{cell}^{\circ} > 0$ means $lo g K > 0$ , so $K > 1$ , meaning products are favored at equilibrium. $E_{cell}^{\circ} < 0$ means $K < 1$ , so reactants are favored.

📐 Worked Example

Calculate $K$ for the reaction $Zn (s) + Cu^{2 +} (a q) \to Zn^{2 +} (a q) + Cu (s)$ at 25°C, given $E_{cell}^{\circ} = 1.10$ V.

1
Rearrange the simplified equation to solve for $lo g K$ :
2
$\log K = \frac{n E^\circ_{\text{cell}}}{0.0592 \ \text{V}}$
3
Substitute $n = 2$ and $E_{cell}^{\circ} = 1.10$ V:
4
$\log K = \frac{(2)(1.10)}{0.0592} \approx 37.16$
5
Solve for $K$ by taking the antilog:
6
$K = 10^{37.16} \approx 1.4 \times 10^{37}$
7
Interpret: $K ≫ 1$ , so the reaction goes almost to completion, consistent with the negative $Δ G^{\circ}$ calculated earlier.

3. Summarizing Spontaneity Rules★★☆☆☆⏱ 10 min

All three thermodynamic quantities ( $E_{cell}^{\circ}$ , $Δ G^{\circ}$ , $K$ ) give consistent information about the spontaneity of a redox reaction under standard conditions. The table below summarizes all sign relationships:

$E_{cell}^{\circ}$	$Δ G^{\circ}$	$K$	Spontaneity (standard conditions)
$> 0$	$< 0$	$> 1$	Spontaneous
$= 0$	$= 0$	$= 1$	At equilibrium
$< 0$	$> 0$	$< 1$	Non-spontaneous

✓ Quick check

Test your understanding of the relationships:

A redox reaction has $K = 2.5 \times 1 0^{- 3}$ . What is true of $E_{cell}^{\circ}$ and $Δ G^{\circ}$ ?
- $E^{\circ}$ positive, $Δ G^{\circ}$ positive
- $E^{\circ}$ negative, $Δ G^{\circ}$ positive
- $E^{\circ}$ positive, $Δ G^{\circ}$ negative
- $E^{\circ}$ negative, $Δ G^{\circ}$ negative
Reveal answer
$E^\circ$ negative, $\Delta G^\circ$ positive —
Correct! $K < 1$ means the reaction favors reactants, so it is non-spontaneous under standard conditions, requiring $E^{\circ} < 0$ and $Δ G^{\circ} > 0$ .
What is $Δ G^{\circ}$ for a reaction with $E_{cell}^{\circ} = 0$ V?
- $Δ G^{\circ} > 0$
- $Δ G^{\circ} < 0$
- $Δ G^{\circ} = 0$
- Cannot be determined without $n$
Reveal answer
$\Delta G^\circ = 0$ —
Correct! $Δ G^{\circ} = - n F E_{cell}^{\circ}$ , so any value of $n$ and $F$ multiplied by $E_{cell}^{\circ} = 0$ gives $Δ G^{\circ} = 0$ .

4. Common Pitfalls

Wrong move:

Forgetting the negative sign in $Δ G^{\circ} = - n F E_{cell}^{\circ}$ , leading to reversed spontaneity

Why:

The negative sign is required to match the sign conventions for $Δ G$ (negative = spontaneous) and $E_{cell}^{\circ}$ (positive = spontaneous). Without it, all signs are flipped.

Correct move:

Always remember the negative sign: positive $E_{cell}^{\circ}$ → negative $Δ G^{\circ}$ → spontaneous reaction.

Wrong move:

Using $n$ equal to the electrons from only one half-reaction instead of the total balanced reaction

Why:

$n$ is the total moles of electrons transferred in the full balanced reaction, which must cancel out between oxidation and reduction half-reactions.

Correct move:

Balance the full redox reaction before identifying $n$ to get the correct total number of electrons transferred.

Wrong move:

Using the $0.0592$ V simplified equation for temperatures other than 25°C

Why:

The simplified log form is derived assuming 298 K (25°C), so it is not valid at other temperatures.

Correct move:

Use the full form $E_{cell}^{\circ} = \frac{R T}{n F} ln K$ if the reaction temperature is not 25°C.

Wrong move:

Assuming a non-spontaneous reaction ( $E_{cell}^{\circ}$ negative) can never occur

Why:

$E_{cell}^{\circ}$ only describes spontaneity under standard conditions. Non-standard conditions can make the reaction spontaneous.

Correct move:

Remember that $E_{cell}^{\circ}$ only tells you about spontaneity under standard state conditions, not all possible conditions.

5. Quick Reference Cheatsheet

Relationship	Formula (25°C)	Spontaneity Rule
$Δ G^{\circ}$ from $E_{cell}^{\circ}$	$Δ G^{\circ} = - n F E_{cell}^{\circ}$	$E^{\circ} > 0 \to Δ G^{\circ} < 0 \to$ Spontaneous
$K$ from $E_{cell}^{\circ}$	$lo g K = \frac{n E _{cell}^{\circ}}{0.0592 V}$	$E^{\circ} > 0 \to K > 1 \to$ Favors products
All Sign Rules	N/A	$E^{\circ} > 0 = Δ G^{\circ} < 0 = K > 1 \to$ Spontaneous
		$E^{\circ} < 0 = Δ G^{\circ} > 0 = K < 1 \to$ Non-spontaneous

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · MCQ
Calculate ΔG from given E°cell
2022 · FRQ
Relate E° to spontaneity and K
2021 · MCQ
Identify spontaneous reaction from E° values

Going deeper

College Board SyllabusAP Chemistry Course and Exam DescriptionSee Unit 9, topic 9.10

What's Next

Now that you understand how cell potential relates to Gibbs free energy and equilibrium, you can extend this knowledge to non-standard conditions, where cell potential changes as reaction concentrations shift. This relationship is described by the Nernst equation, which is critical for solving problems involving concentration cells and batteries that are not at standard state. You can also apply these concepts to electrolysis, where non-spontaneous redox reactions are driven by an external voltage, a common topic in AP Chemistry FRQs. These concepts build directly on the relationships you learned here, connecting all of thermodynamics and electrochemistry into a unified framework.