Question

The temperature coefficient of the e.m.f. Of the cell is given by:
$\begin{array}{rl}& a){\displaystyle \frac{nF}{\mathrm{\Delta }S}}\\ & b){\displaystyle \frac{\mathrm{\Delta }S}{nF}}\\ & c){\displaystyle \frac{\mathrm{\Delta }S}{nFT}}\\ & d)-nFE\end{array}$

Answer 1

Hint: Write down the formula for e.m.f. Of the cell in terms of Gibbs free energy. We know that Gibbs energy can be written in terms of differential of entropy in the system. Temperature coefficient can be easily calculated now using these two formulas.

Formula used: $E={\displaystyle \frac{\mathrm{\Delta }G}{nF}}$
 $\mathrm{\Delta }G={\displaystyle \frac{\mathrm{\Delta }S}{dT}}$

Complete step by step answer:
Temperature coefficient of the emf is given by,
${\displaystyle \frac{dE}{dT}}$.
The emf of the cell can be written as $E={\displaystyle \frac{\mathrm{\Delta }G}{nF}}$
We know, the Gibbs free energy can be written as $\begin{array}{rl}& \mathrm{\Delta }G={\displaystyle \frac{\mathrm{\Delta }S}{dT}}\end{array}$
Therefore, we can write the above two equations together as,
$\begin{array}{rl}& E={\displaystyle \frac{\mathrm{\Delta }S}{nFdT}}\\ & {\displaystyle \frac{dE}{dT}}={\displaystyle \frac{\mathrm{\Delta }S}{nFdT}}(dT)\\ & {\displaystyle \frac{dE}{dT}}={\displaystyle \frac{\mathrm{\Delta }S}{nF}}\end{array}$

So, the correct answer is “Option B ”.

Additional Information: The relation between the temperature and the electromotive force is monotonic and non-linear, that is, temperature increases when the emf of the battery increases. It also doesn’t mean that the emf increases at a constant rate. It doesn’t increase at a constant rate. The above relation between emf of the battery and the Gibbs free energy term can be derived from the Gibbs Helmholtz equation. Electromotive force is the energy per unit electric charge that is produced by an energy source, such as electric generator or batteries. The emf force is also described using the analogy of water pressure. Devices produce emf by converting other forms of energy into electrical energy from external sources.

Note: The temperature coefficient of the emf of the cell is derived from Gibbs Helmholtz law. The emf value decreases with increase in temperature. Temperature also affects the internal resistance. High temperature lowers the resistance and low temperatures increase the resistance.