# How to Calculate Nernst Equations

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The Nernst equation, named after chemist Walther Nernst, is used in electrochemistry to determine the point at which a redox reaction in an electrochemical half-cell reaches equilibrium – this also determines when the instantaneous cell potential has reached zero. The cell potential might sound complicated, but it essentially just describes how the cell reaction is proceeding. If we imagine a ball rolling down a hill, it starts with a lot of potential energy (a large cell potential). As it rolls down the hill and the slope starts to level out, the ball still continues to roll, but it starts to gain less energy because the hill is less steep (the cell potential is getting smaller). Finally the ball comes to a stop when the ground is flat (the cell potential is zero, and the reaction is at equilibrium). Also look out for any sources that refer to the cell potential as an electromotive force (EMF), this is just another way of referencing the cell potential.

#### TL;DR (Too Long; Didn't Read)

Cell potential is measured in volts which is equivalent to joules per coulomb (energy per charge).

## The General Form

The general form of the Nernst equation can be applied to an electrochemical cell at any temperature. It uses the ideal gas constant (R – 8.3145 J⋅mol−1⋅Kelvin−1), Faraday’s constant (F – 9.6485×104 C⋅mol−1), the number of electrons transferred (n), the absolute temperature (T – in kelvin), and the standard reduction potential (Eo) of the cell as constant values to solve for the instantaneous reduction potential. The reaction quotient (Q) will be the input that varies with time. It is the ratio between the concentration of products and the concentration of reactants, so it allows us to solve for the cell potential at an point in the reaction – as long as we know these two concentrations. The general Nernst equation follows from these inputs to solve for the cell potential E:

E_{cell} = E^{o}_{cell} - \frac{RT}{nF}\ln Q \\ \text{} \\ \text{Where } Q = \frac{\text{[conc. of products]}}{\text{[conc. of reactants]}}

The standard cell potential (Eo) is a known value for specific reactions. Because we can account for any temperature, this general form of the Nernst equation works under non-standard conditions to find the electric potential in the cell. When dealing with non-standard cell potentials, this equation can prove invaluable, however when looking for a more general equation to represent the relationships in a standard electrode potential we can use a simplified equation.

## Standard Conditions

When dealing with the Nernst equation under standard conditions, we are not necessarily dealing with the concentration of solutions being held constant because this is actually what we are representing the variation of. Instead, we simply hold the temperature constant at 298 kelvin (25 degrees celsius – just above room temperature). When we substitute this value for T and plug in the universal gas constant and Faraday’s constant, we can swap out the natural logarithm for a standard logarithm and receive the much simpler formula:

E_{cell} = E^{o}_{cell} - \frac{0.0592\text{V}}n\log{Q}

An example can be very useful to demonstrate what each of these values will look like. Drawing from the Zn-Cu redox reaction, we can use the given standard potential of Eo = +1.10 V (see Example 1 in the Chemistry LibreTexts online book for the original demonstration). We will assume standard conditions. The full calculation is shown below.

\text{Given} \\ \text{} \\ \\Zn_{(s)} + Cu^{2+}_{(aq)} \rightleftharpoons Zn^{2+}_{(aq)} + Cu_{(s)} \\ \text{} \\ \text{and \ } E^o_{cell} = 1.10\text{ V} \\ \text{} \\ \text{with concentrations after one minute of} \\ \text{} \\ [Cu^{2+}] = 0.05 \text{ M} \text{ and } [Zn^{2+}] = 1.95 \text{ M} \\ \text{} \\ \text{we solve for }E: \\ \text{} \\ E = E^o - \frac{0.0592 \text{ V}}{n}\log{Q} \\ \text{} \\ n = 2 \\\text{ (we can see 2 electrons transferred between } Zn^{2+} \text{ and } Cu^{2+}) \\ \text{} \\ Q = \frac{[Zn^{2+}]}{[Cu^{2+}]} = \frac{1.95 \text{ M}}{0.05 \text{ M}} = 39 \\ \text{} \\ \text{plugging all of these values in:} \\ \text{} \\ E = 1.10 \text{ V} - \frac{0.0592 \text{ V}}{2} \log39 = \textbf{$1.053$ V}

#### Tips

• The (aq) stands for aqueous meaning the element is dissolved in solution.

## What does this mean?

Now that we have established the Nernst equation we can find out what it actually tells us. The key value here is the reaction quotient Q. The cell potential during an oxidation-reduction reaction is closely related to the Gibbs Free Energy equation (often called delta G) which describes how an electrochemical reaction is proceeding (see the full derivation here). Gibbs free energy discusses thermodynamic favorability and equilibrium, and we can relate this to the reaction quotient:

• When Q = 1, the instantaneous cell potential is the same as the standard cell potential. Nothing has changed from the general initial state, so the reaction is favorable.
• When Q < 1, the instantaneous cell potential is greater than the standard cell potential. The reaction is more favorable than the standard situation.
• When Q > 1, the instantaneous cell potential is less than the standard cell potential. The equation is progressing towards equilibrium, but it is still favorable.
• When Q = K, the instantaneous cell potential is 0; the reaction is at equilibrium.

K is the equilibrium constant, a value that can be used to mark the concentrations of reactants and products that have reached a state of equilibrium where the reaction stops progressing.

#### Tips

• The above situations assume that the cell potential is positive meaning that the reaction wants to proceed in the given direction. If this is not the case, then the situations of thermodynamic favorability are the opposite.

## Other applications

In cellular biology the membrane potential describes the concentration of ions on each side of a membrane. The Goldman Equation includes other factors like the permeability of the cell membrane and the type of transfer across the surface to calculate this potential. When these cells reach equilibrium then the it’s membrane potential is equal to the Nernst potential.

Galvanic cells convert chemical energy to electrical energy using these relationships of concentrations and equilibriums. They work by have a potential difference between the anode and cathode, operating as a chemical battery.