Consider an electrochemical cell made of two half-cells that are identical except that they are maintained at different temperatures. This is called a thermogalvanic cell. Determine the thermogalvanic coefficient,
\alpha = \frac{\partial \Delta \varphi }{\partial T} = \frac{\Delta _\alpha S}{zF_F} .
using the definition of the battery potential introduced in exercise (8.12).
According to the definition of the battery potential introduced in exercise (8.12), the thermogalvanic coefficient is given by,
\alpha = \frac{\partial \Delta \varphi }{\partial T} = -\frac{1}{zF_F} \frac{\partial}{\partial T} (\Delta _\alpha H – T \Delta _\alpha S) = \frac{\Delta _\alpha S}{zF_F} .
Note that a typical value for the Seebeck coefficient of a metal (equation 11.86) is much smaller than a typical value of the thermogalvanic coefficient. Thermogalavanic cells have been considered to convert heat to energy.
∇ \varphi = −ε (s, n_e, q) ∇ T . (11.86)