The cell potential in the relationships shown previously was determined at 25 °C, where the standard values have been tabulated. However, it is often desirable to calculate the cell voltage at a different temperature. To do so, we need the standard potential at that temperature. This section describes how to correct the standard potential to the temperature of interest. Beginning with one of the fundamental relations from thermodynamics,

(2.15)

we find that

(2.16a)

If we further assume that  does not change significantly over the temperature range of interest, it follows that

(2.17)

and

(2.18)

Tabulated values of  are available in the literature. If the entropy change varies significantly with temperature over the range of interest, then integration of the temperature-dependent entropy term may be required as follows:

(2.19)

Once the standard potential is known at the new temperature, activity corrections can be made as usual. This process is illustrated in the example at the end of the next section.

The following alternative to Equation 2.16 is equally rigorous:

(2.16b)

This approach has an advantage when the temperature range is sufficiently large that the assumption of constant ΔS or ΔH is not valid. If constant pressure heat capacity data (C_p) are available, then the change in enthalpy can be calculated as a function of temperature,

(2.20)

See Problem 2.29 to explore further this method.

ILLUSTRATION 2.3

A chlor-alkali cell is used in industry to produce chlorine, hydrogen, and sodium hydroxide. The cell is divided into two sides, the anodic side and the cathodic side. The anodic reaction is the oxidation of chloride ion, which takes place under slightly acidic conditions at a pH of 4. The cathodic reaction is the evolution of hydrogen under basic conditions (pH∼14). The salt solution on the anodic side is 5 M NaCl. Please determine the standard potential for the cell at 65 °C.

SOLUTION:

We begin by writing the equations for the anodic and cathodic reactions; that is, the half-cell reactions. As usual, we put the positive electrode on the right side (see Figure 2.1b).

Note that if the circuit were closed for this cell, chlorine, hydrogen, and hydroxyl would spontaneously be consumed to generate water and chloride. This is opposite the desired direction. Therefore, in practice, the potential of the cell is raised above the equilibrium potential, and work is added to the system in order to force the reaction in the reverse direction in order to produce hydrogen and chlorine.

To correct the standard potential for temperature: