Gibbs Free Energy and Standard Cell Potentials
The Equation: $ΔG^\circ = −nFE^\circ_{\text{cell}}$
- The equation $$ΔG^\circ = −nFE^\circ_{\text{cell}}$$ links two fundamental concepts in chemistry: Gibbs free energy change ($ΔG^\circ$) and standard cell potential ($E^\circ_{\text{cell}}$).
- Let’s break it down step by step to understand how it works.
$ΔG^\circ$ (Standard Gibbs Free Energy Change):
- $ΔG^\circ$ represents the maximum amount of energy available to do useful work from a chemical reaction under standard conditions (298 K, 1 atm, and 1 M concentrations for all solutions).
- The sign of $ΔG^\circ$ determines spontaneity:
- Negative $ΔG^\circ$:The reaction is spontaneous under standard conditions.
- Positive $ΔG^\circ$:The reaction is non-spontaneous under standard conditions.
- $ΔG^\circ=0$: The system is at equilibrium.
$E^\circ_{\text{cell}}$ (Standard Cell Potential):
- $E^\circ_{\text{cell}}$ measures the voltage of an electrochemical cell under standard conditions.
- It is calculated as the difference between the standard electrode potentials of the cathode (reduction) and the anode (oxidation): $$E^{\ominus}{\text{cell}} = E^{\ominus}_{\text{cathode}} - E^{\ominus}_{\text{anode}}$$
- A positive $E^\circ_{\text{cell}}$ corresponds to a negative $ΔG^\circ$, indicating a spontaneous reaction.
- Conversely, a negative $E^\circ_{\text{cell}}$ means $ΔG^\circ$ is positive, and the reaction is non-spontaneous.
n (Number of Electrons Transferred):
$n$ represents the number of moles of electrons transferred in the balanced redox reaction.
This value is determined from the half-equations of the reaction.
F (Faraday’s Constant):
$F$ = 96500 C mol⁻¹, which is the charge of one mole of electrons.
By combining these terms, the equation $ΔG^\circ = −nFE^\circ_{\text{cell}}$ allows you to calculate how much energy ( $ΔG^\circ$) is available for a given electrochemical reaction based on its cell potential ($E^\circ_{\text{cell}}$).
- Ensure that $E^\ominus_\text{cell}$ is in volts (V) and $\Delta G^\ominus$ is in joules (J) when using the equation.
- To convert $\Delta G^\ominus$ to kilojoules (kJ), divide by 1,000.
Spontaneity and the Sign of $ΔG^\circ$
The relationship between $ΔG^\circ$ and $E^\circ_{\text{cell}}$ is straightforward:
- If $ΔG^\circ < 0$: The reaction is spontaneous under standard conditions.
- If $ΔG^\circ >0$: The reaction is non-spontaneous and requires energy input.
- If $ΔG^\circ = 0$: The reaction is at equilibrium, meaning no net energy is available.
Since $ΔG^\circ$ is directly proportional to $E^\circ_{\text{cell}}$, a positive $E^\circ_{\text{cell}}$ indicates a spontaneous reaction, while a negative $E^\circ_{\text{cell}}$ indicates a non-spontaneous reaction.
Calculating $ΔG^\circ$
Let’s calculate $ΔG^\circ$ for the reaction in a Daniell cell:
Reaction: Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s)
Step 1: Identify the components.
- From the data booklet:
- Zn²⁺/Zn: $E^\circ$ = −0.76 V
- Cu²⁺/Cu: $E^\circ$ = +0.34 V
- Calculate $E^\circ_{\text{cell}}$: $$ E^{\ominus}{\text{cell}} = E^{\ominus}{\text{cathode}} - E^{\ominus}{\text{anode}}$$ $$E^{\ominus}{\text{cell}} = (+0.34 \, \text{V}) - (-0.76 \, \text{V}) = +1.10 \, \text{V}$$
- $n = 2$ (from the balanced redox equation).
- F = 96500 C mol⁻¹.
Step 2: Apply the equation.
$$
\Delta G^{\ominus} = -nFE^{\ominus}_{\text{cell}}
$$
$$
\Delta G^{\ominus} = -(2)(96500 \, \text{C mol}^{-1})(1.10 \, \text{V})
$$
$$
\Delta G^{\ominus} = -212,267 \, \text{J mol}^{-1}
$$
Step 3: Convert to kilojoules.
$$
\Delta G^{\ominus} = -212.3 \, \text{kJ mol}^{-1}
$$
Interpretation: The negative value of $ΔG^\circ$ indicates that the reaction is spontaneous under standard conditions.
In this example, the Daniell cell produces 212.3 kJ of energy per mole of reaction, which can be harnessed to perform electrical work, such as powering a device.
- Forgetting to match units: Always ensure that $E^\circ_{\text{cell}}$ is in volts (V) and $ΔG^\circ$ is in joules (J) when using the equation.
- Misidentifying $n$: Double-check the balanced redox equation to determine the correct number of electrons transferred.
- If $E^\circ_{\text{cell}}$ = −0.45 V for a reaction with $n = 3$, calculate $ΔG^\circ$. Is the reaction spontaneous?
- A reaction has $ΔG^\circ$ = +75 kJ mol⁻¹. What does this tell you about the spontaneity of the reaction and the sign of $E^\circ_{\text{cell}}$?
