Question

Two adiabatic vessels, each containing the same mass $m$ of water but at different temperatures, are connected by a rod of length $L$, cross-section $A$, and thermal conductivity $K$. the ends of the rod are inserted into the vessels, while the rest of the rod is insulated so that there is negligible loss of heat into the atmosphere. The specific heat capacity of water is $s$, while that of the rod is negligible. The temperature difference between the two vessels reduces to $1 / e$ of its original value after a time, $\Delta t$. The thermal conductivity $(K)$ of the rod may be expressed by :

• A. $\frac{m s L}{A \Delta t}$
• B. $\frac{e m s L}{A \Delta t}$
• C. $\frac{m s L}{2 e A \Delta t}$
• D. $\frac{m s L}{2 A \Delta t}$

Answer 1

Answer: (D)

Suppose that the temperature of the water in the first vessel is $\theta_{1}(t)$ and that of the second is $\theta_{2}(t)$, then
$m s \frac{d \theta_{1}}{d t}=-\frac{K A}{L}\left(\theta_{1}-\theta_{2}\right)$ ...(1)
and $m s \frac{d \theta_{2}}{d t}=\frac{K A}{L}\left(\theta_{1}-\theta_{2}\right)$ ...(2)
from (1) and (2), we get
$\frac{d \theta}{d t}=\frac{-2 K A}{m s L} \theta,$
where $\theta=\theta_{1}-\theta_{2},$
The time, in which the temperature difference reduces to $1/e$ of its initial value, is given by
$\Delta t=\frac{m s L}{2 K A}$
$\Rightarrow K=\frac{m s L}{2 A \Delta t}$