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 $t_1$ whereas that in $Q$ takes a time $t_2$. The ratio of thermal conductivities of the materials of $P$ and $Q$ is

• A. $t_{2} : t_{1}$
• B. $t_{1} : t_{2}$
• C. $t_{1}^2 : t_{2}^2$
• D. $t_{2}^2 : t_{1}^2$

Answer 1

Answer: (A)

We know that
Relation between temperature gradient (TG) and thermal conductivity $(K)$
So, $\frac{d \theta}{d t}=-K A \frac{d \theta}{d x}=-K A \times( TG )$
i.e., $TG \propto \frac{1}{K}\left(\frac{d \theta}{d t}=\right.$ 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
$\frac{K_{1}}{K_{2}}=\frac{1 / t_{1}}{1 / t_{2}}$
$K_{1}: K_{2}=t_{2}: t_{1}$