15.0 Thermodynamics of the Cells

15.1 Condition of Equilibrium
15.2 Spontaneity of the Reaction

Emf of the cell is related to free energy by the following equation

$$\Delta G = - nFE\ \ \ \ ...(i)$$

And we know that $$\Delta S = - \left( {\frac{{\partial \Delta G}}{{\partial T}}} \right)$$

differentiating equation (i) wrt. $T$, we get

$$\begin{equation} \begin{aligned} \left( {\frac{{\partial \Delta G}}{{\partial T}}} \right) = - nF{\left( {\frac{{\partial E}}{{\partial T}}} \right)_P} = - \Delta S \\ \Delta S = nF{\left( {\frac{{\partial E}}{{\partial T}}} \right)_P}...(ii) \\\end{aligned} \end{equation} $$

So enthalpy of the reaction is given as $$\Delta G = \Delta H - T\Delta S\ \ \ \ ...(ii)$$

Putting the value of $\Delta S$ in $(ii)$ $$\Delta S = - \left( {\frac{{\partial \Delta G}}{{\partial T}}} \right)$$

$$\begin{equation} \begin{aligned} - nFE = \Delta H - T{\left( { - \frac{{\partial \Delta G}}{{\partial T}}} \right)_P} \\ - nFE = \Delta H + T{\left( {\frac{{\partial \Delta G}}{{\partial T}}} \right)_P} \\\end{aligned} \end{equation} $$

Now using $(i)$, we get

$$\begin{equation} \begin{aligned} - nFE = \Delta H + T{\left( { - \frac{{\partial nFE}}{{\partial T}}} \right)_P} \\ - nFE = \Delta H + nFT{\left( { - \frac{{\partial E}}{{\partial T}}} \right)_P} \\ \Delta H = - nFE + nFT{\left( {\frac{{\partial E}}{{\partial T}}} \right)_P} \\\end{aligned} \end{equation} $$

So we can find out the thermodynamic quantities using the above relations, depending on the temperature.

The heat effects can also be calculated by using following relations.

If reaction is irreversible, so the heat flow to the system can be given by the reaction, $$\Delta H = {Q_P}$$

And if the reaction is reversible, the heat flow to the system is given by $${Q_P} = T\Delta S$$