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 that in Q takes a time t2. The ratio of thermal conductivities of the materials of P and Q is

Question

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 that in Q takes a time t2. The ratio of thermal conductivities of the materials of P and Q is (A) t2: t1 (B) t1: t2 (C) t12: t22 (D) t22: t12

Tardigrade Admin · Accepted Answer

We know that
Relation between temperature gradient (TG) and thermal conductivity

(K)

So,

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

i.e.,

TG \propto \frac{1}{K} (\frac{d θ}{d t} =

constant

)

or

K \propto \frac{1}{T}

K_{1} \propto \frac{1}{t _{1}} ... (i)

K_{2} \propto \frac{1}{t _{2}} ... (ii)

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

\frac{K _{1}}{K _{2}} = \frac{1/ t _{1}}{1/ t _{2}}

K_{1} : K_{2} = t_{2} : t_{1}

$t_{2} : t_{1}$

$t_{1} : t_{2}$

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

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

t2​:t1​

t1​:t2​

t12​:t22​

t22​:t12​

$t_{2} : t_{1}$

$t_{1} : t_{2}$

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

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