Nernst Equation

Nernst Equation formula

What is the Nernst Equation?

The Nernst equation calculates electrochemical cell potential at any known temperature, pressure, and concentration. The equation relates the reduction potential of the cell at a non-standard condition to that of the standard conditions (298\text{K, }1 \text{atm, and }1 \text{M}).

     \begin{gather*} E_{\text{Cell}} = E^{0} - \left(\frac{RT}{nF}\right)lnQ \end{gather*}

  • E_{\text{Cell}} = cell potential
  • E^(0} = Cell potential under standard conditions
  • R = Universal gas constant (8.314 \frac{\text{J}}{\text{Kmol}})
  • T = Temperature
  • n = Number of electrons transferred in the reaction
  • F = Faraday constant (96485 \frac{\text{C}}{\text{mol}})
  • Q = Reaction Quotient

Q is the same as the equilibrium constant (K) except not necessarily at equilibrium. It makes sense then to use Q since we are trying to find non-standard cell potentials.

The equation can be used to find the the potential of half-cell reactions or cell potentials for various electrochemical cells.

The equation is also seen written in several other forms where the term \frac{RT}{nF} is simplified to a constant. For example, to make the equation more similar to the pH equation, the natural log can be changed to log.

     \begin{gather*} E_{\text{Cell}} = E^{0} - \left(\frac{2.303 \cdot RT}{nF}\right)lnQ \end{gather*}

For cells at 25 \degree C, the equation can be simplified to

     \begin{gather*} E_{\text{Cell}} = E^{0} - \left(\frac{0.0592 \text{V}}{n}\right)lnQ \end{gather*}

The Nernst equation is also useful for determining the increase in cell potential for an increase in the concentration of a reactant. If Q =1, than the cell potential will be the same as the standard cell potential. Any other value of Q will cause the cell potential to be different than the standard. A value of Q>1 means the cell potential will be less than the standard. A value of Q<1 means it will be higher than the standard.

When the cell potential becomes more positive, the reaction is more likely to take place.

Derivation of the Nernst Equation

The Nernst equation is derived from the Gibbs free energy.

     \begin{gather*} \Delta G = \Delta G\degree + RTlnQ \end{gather*}

We can rewrite this equation using the definitions of \Delta G = -nFE and  \Delta G \degree = -nFE \degree .

     \begin{gather*} -nFE = -nFE \degree + RTlnQ \end{gather*}

To simplify, we divide each side by -nF and arrive at the Nernst equation as it is commonly written.

     \begin{gather*} E_{\text{Cell}} = E^{0} - \left(\frac{RT}{nF}\right)lnQ \end{gather*}

An example of an electrochemical cell that you could analyze using the Nernst equation.
An example of an electrochemical cell that you could analyze using the Nernst equation.

Example Problem Using the Equation

Problem: You have an electrochemical cell with lead (Pb) and silver (Ag) running at 25 \degree C. The reaction of the cell is listed below.

     \begin{gather*} Pb(\text{s}) + 2Ag^{+} (\text{aq}) \rightarrow Pb^{2+}(\text{aq}) + 2Ag(\text{s}) \end{gather*}

Calculate the cell potential when [Ag^{+}] = 0.75 \text{M} and [Pb^{2+}] = 0.1 \text{M}. The standard cell potential is 0.93 \text{V}.

Worked Solution:

First, write out the reaction that is occurring and find the standard cell potential (E^{0}) if it is not already given. In this case, the reaction and cell potential are given in the problem. For a remind on how to find standard cell potential, see the tutorial on electrochemical cells.

Second, start plugging in given or known values into the Nernst equation. In this problem, we can plug in n, Q, and E^{0}. The gas constant (R) and Faraday constant (F) are constants that are simplified into the value of 0.0592 \text{V}. This simplification is possible because we are working at 25 \degree C.

     \begin{align*} {E_{Cell} &= E^{0} - \left( \frac{0.0592\text{V}}{n}\right)log\frac{[Pb^{2+}]}{[Ag^{+}]^{2}}} \\ {E_{Cell} &= (0.93\text{V}) - \left(\frac{0.0592\text{V}}{2}\right)log\left(\frac{0.1}{0.75^{2}}\right)} \\ {E_{Cell} &= 0.95\text{V}} \end{align*}

A reminder that for Q, solid components of the reaction are not included in the value. So, in this example, the Pb(s) and Ag(s) are not included. Our n value comes from seeing that in our reaction 2\text{mol} of electrons are transferred.

Check your units when you go through problems to make sure you end up with the correct units, especially if you can’t use the simplified coefficient for working at 25 \degree C.

Limitations of the Nernst Equation

Although the limitations of the Nernst equation do not come into play very often, there exist some assumptions made with the equation. Below are some situations in which the Nernst equation becomes inaccurate.

  1. When concentrations are super high the Nernst equation becomes inaccurate. When concentrations of reactants or products become extremely high experiments are used to determine potentials.
  2. In very very dilute solutions, the potential predicted can reach infinity, which does not reflect real conditions.
  3. The equation can’t be used when there is current flowing through the cell. The current changes the activity (effective concentration) of the ions. This effect causes the equation to then be inaccurate.

Fun Facts:

  1. Walther Hermann Nernst(1864-1941), a German chemist, developed the equation.
  2. Nernst received the Nobel Prize in chemistry in 1920 for work he did on thermochemistry.
  3. Another commonly known discovery by Nernst is the third law of thermodynamics.           
  4. Electrochemical cell potentials are one of the main applications of the Nernst equation. Electrochemical cells are what drive most types of batteries! Chemists also use these cells to plate thin layers of materials onto other surfaces, such as plating gold onto jewelry.

Nernst Equation Practice Problems

Problem 1

You have a 25 \degree C galvanic cell with gold and copper that involves the following redox reaction:

     \begin{gather*} {2Au^{3+}+3Cu\rightarrow 3Cu^{2+}+2Au} \\ {E^{0} = 1.16\text{V}} \end{gather*}

The actual potential of the cell is  E_{Cell} = 1.14\text{V} with a gold (III) concentration of [Ag^{3+}] = 1.0\text{M}. What is the concentration of copper (II)?

Problem 2

You have a galvanic cell with copper and zinc involving the following redox reaction:

     \begin{gather*} Zn + Cu^{2+} \rightarrow Zn^{2+} + Cu \end{gather*}

The cell is under standard conditions ( E_{Cell} = E^{0} ). If you added Zn(NO3)2 salt, would you therefore expect the cell’s potential to increase or decrease?

Nernst Equation Practice Problem Solutions

1:  [Cu^{2+}] = 0.1\text{M}

2:  E_{Cell} would decrease.

Further Reading