The resistance of an electrolyte varies with temperature T as RT =  R01+αT, where R0 and α 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 T0 it the temperature of the surroundings, the loss of heat to the surroundings per unit surface area per unit time is governed by the same relation H=kT−T0A, then the steady state temperature is

# The resistance of an electrolyte varies with temperature T as ${\mathrm{R}}_{\mathrm{T}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}}\frac{{\mathrm{R}}_{0}}{1+\mathrm{\alpha T}}$, where ${R}_{0}$ and $\alpha$ 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 T0 it the temperature of the surroundings, the loss of heat to the surroundings per unit surface area per unit time is governed by the same relation $H=k\left(T-{T}_{0}\right)A,$ then the steady state temperature is

1. A

$\frac{\left({\mathrm{V}}^{2}+{\mathrm{kAR}}_{0}\mathrm{T}\right)}{\left({\mathrm{kR}}_{0}\mathrm{A}-{\mathrm{\alpha V}}^{2}\right)}$

2. B

$\frac{{\mathrm{V}}^{2}}{{\mathrm{kR}}_{0}\mathrm{A}-{\mathrm{\alpha V}}^{2}}$

3. C

$\frac{{V}^{2}}{RkA}$

4. D

None of these

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### Solution:

Power generated in the resistance is being lost in the environment.

At steady state, $\frac{{\mathrm{V}}^{2}}{{\mathrm{R}}_{\mathrm{T}}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{k}\left(\mathrm{T}-{\mathrm{T}}_{0}\right)\mathrm{A}$

Given $\mathrm{R}\text{\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}}\frac{{\mathrm{R}}_{0}}{1+\mathrm{\alpha T}},\text{\hspace{0.17em}\hspace{0.17em}}\frac{{\mathrm{V}}^{2}}{{\mathrm{R}}_{0}}\left(1+\mathrm{\alpha T}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{k}\left(\mathrm{T}-{\mathrm{T}}_{0}\right)\mathrm{A},$

$\mathrm{T}\left(-\frac{{\mathrm{V}}^{2}\mathrm{\alpha }}{{\mathrm{R}}_{0}}\text{\hspace{0.17em}}+\mathrm{kA}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\left(\frac{{\mathrm{V}}^{2}\mathrm{\alpha }}{{\mathrm{R}}_{0}}\text{\hspace{0.17em}}+{\mathrm{kAT}}_{0}\right),\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{T}=\left(\frac{{\mathrm{V}}^{2}+{\mathrm{kAR}}_{0}\mathrm{T}}{{\mathrm{kR}}_{0}\mathrm{A}-{\mathrm{\alpha V}}^{2}}\right)$

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