Question

The temperature of the water of a pond is $0^{o}C$ while that of the surrounding atmosphere is $-{20}^{o}C$. If the density of ice is $\rho$, coefficient of thermal conductivity is $k$ and latent heat of melting is $L$ then the thickness $Z$ of ice layer formed increases as a function of time $t$ as :

• A. $Z^2=\frac{60k}{\rho L}t$
• B. $Z=\sqrt{\frac{40k}{\rho L}t}$
• C. $Z^2=\frac{40k}{\rho L}\sqrt{t}$
• D. $Z^2=\frac{40k}{\rho L}t$

Answer 1

The correct option is D $Z^2=\frac{40k}{\rho L}t$

According to the question, the above diagram can be drawn.

So, $H=\frac{dQ}{dT}=\frac{kA(0-(-20\left)\right)}{x}$

$⟹\frac{dm}{dt}L=\frac{kA\left(20\right)}{x}$

$⟹\rho A\frac{dx}{dt}L=\frac{kA\left(20\right)}{x}$

On integrating both sides, we get:

${\int }_0^{z}xdx=\frac{20k}{\rho L}{\int }_0^{t}dt$

$⟹z^2=\frac{40k}{\rho L}t$