Using the relationship between potential difference and Gibbs energy to calculate a value for the equilibrium constant, K

E_cell^⦵ (the maximum potential difference between two half cells) is measured when the reactants are under standard conditions (1.00 mol dm^-3, 298K, 100kPa) and no current flows.

Under exactly the same conditions, we can calculate the change in Gibbs energy for the reaction, which also represents the maximum work the cell can perform. 

Gibbs energy change, Δ_rG^⦵, and E_cell^⦵ are related by the equation:

Δ_rG^⦵ = -zF E_cell^⦵

Z = number of moles of electrons flowing and F = Faraday’s constant, 96500 Cmol^-1

If we replace the high resistance voltmeter with a motor or a lamp, for example, current is drawn from the cell as electrons flow from the negative half cell to the positive. 

These reactions will continue until equilibrium is reached, at which point the concentrations stop changing and no current flows. The cell has run out / gone flat. 

At equilibrium, there is no more change in Gibbs energy, Δ_rG = 0, and the system has reached its minimum Gibbs energy. 

The concentrations (or pressures) of reactants and products at equilibrium are related by the equilibrium constant, K.

If K > 1, the equilibrium position lies to the right and the bigger the value, the closer the reaction will be to completion.

 Δ_rG^⦵ and K are related by the equation:

And as we know that Δ_rG^⦵ = -zF E_cell^⦵ , we can combine these two equations to give:

Δ_rG^⦵ = -RT lnK
R = gas constant 8.314 JK^-1mol^-1
T = 289K

And as we know that Δ_rG^⦵ = -zF E_cell^⦵, we can combine these two equations to give:

 -zF E_cell^⦵=-RT lnK      
lnK =   zF / RT . E_cell^⦵

Example:

Calculate the thermodynamic equilibrium constant, K, for the following reaction at 298K and under standard conditions, given that E_cell^⦵ = +1.10V

Cu²⁺_(aq) + Zn_(s)  ⇾ Zn²⁺_(aq) + Cu_(s)

Practice questions

Calculate the thermodynamic equilibrium constant, K, for the following reactions at 298K and under standard conditions, using standard electrode potential data.

1.  Ag⁺_(aq) + Cu_(s)  ⇾ Cu²⁺_(aq) + Ag_(s)

2.  Sn⁴⁺_(aq) + Ni_(s)  ⇾ Ni²⁺_(aq) + Sn²⁺_(aq)

Answers

1. E_cell^⦵ =+ 0.46V;   lnK = 35.8;   K = 3.65 x 10¹⁵

2.    E_cell^⦵ =+ 0.40V;   lnK = 31.2;  K = 3.41 x 10¹³

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