The resistance of an electrolyte varies with temperature `T` as `R_(T) = (R_(O))/(1 + alpha T)`, where `R_(0)` and are constants. A constant potential difference `V` is applied to the two electrodes (each of surface area `A`) which are dipped in this electrolyte. If `T_(0)` is the temperature of the surroundings, the loss of heat to the surroundings per unit surface area put unit time is governed by the same relation `H = k (T - T_(0)) A`, the steady state temperature is

Options

A. `((V^(2) + k A R_(O) T))/((k R_(O) A - alpha V^(2)))`
B. `(V^(2))/(kR_(O) A - alpha V^(2))`
C. `(V^(2))/(RkA)`
D. None of these

Answer

A

Explanation

(a) Power generated in the resistance is being lost in the environment.
At steady state `(V^(2))/(R ) = k (T - T_(0)) A`
Given `R = (R_(0))/(1 + alpha T) , (V^(2))/(R_(0)) (1 + alpha T) = k (T - T_(0)) A`,
`T (- (V^(2) alpha)/(R_(0)) + k A) = (- (V^(2))/(R_(0)) + k AT_(0)), T = ((V^(2) + k AR_(0) T)/(kR_(0) A - alpha V^(2)))`