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Same quantity of ice is filled in each of the two metal containers P and Q having the same size, shape and wall thickness but made of different materials. The containers are kept in identical surroundings. The ice in P melts completely in time t₁, whereas in Q takes a time t₂. The ratio of thermal conductivities of the materials of P and Q is

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t₁ : t₂

$t_{2}^{2} : t_{1}^{2}$

t₂ : t₁

$t_{1}^{2} : t_{2}^{2}$

answer is A.

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Detailed Solution

We know that, relation between the temperature gradient (TG) and thermal conductivity (K),

$\frac{d Q}{d t} = - K A \frac{d θ}{d x} = - K A \times (TG)$

i.e. $TG \propto \frac{1}{K}$ $(∵ \frac{d Q}{d t} = constant)$

or $K \propto \frac{1}{t}$

$\Rightarrow$ $K_{1} \propto \frac{1}{t_{1}}$ ... (i)

and $K_{2} \propto \frac{1}{t_{2}}$ ... (ii)

From Eqs. (i) and (ii), we get

$\frac{K_{1}}{K_{2}} = \frac{1 / t_{1}}{1 / t_{2}}$

$\Rightarrow K_{1} : K_{2} = t_{2} : t_{1}$

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