Question

Two vessels of different materials are identical in size and wall-thickness. They are filled with equal quantities of ice at $0^{o}C$. If the ice melts completely in $10$ and $25$ minutes respectively then comparison of the coefficient of thermal conductivity of the materials of the vessels.

Answer 1

Let the vessel be cubic of side $a$ and thickness be $d$

$t_1=10min$

$t_2=25min$

Let the room temperature be $T_o$

Thermal conductivity of the first and the second vessel be $K_1$ and $K_2$ respectively.

Let $△Q$ amount of heat is required to melt volume of ice; $△Q=\frac{6K_1{a}^2(T_o-0)t_1}{d}$

( factor of 6 is for 6 sides of cube through which heat is flowing in)

$\Rightarrow △Q_1=\frac{6K_1{a}^2(T_o-0)10}{d}$

For second container, $△Q_2=\frac{6K_2{a}^2(T_o-0)25}{d}$

$\because △Q_1=△Q_2\Rightarrow \frac{6K_1{a}^2(T_o-0)10}{d}=\frac{6K_2{a}^2(T_o-0)25}{d}\Rightarrow 10K_1=25K_2\Rightarrow K_1:K_2=25:10=5:2\Rightarrow K_1:K_2=5:2$