Question

A layer of ice at $0^{\circ }\text{C}$ of thickness $x_1$ is floating on a surface of water. $L,\rho$ and $K$ respectively are the latent heat of fusion of ice, density of ice and thermal conductivity of ice. If the atmospheric temperature is $-T^{\circ }\text{C}$, the time taken for the thickeness of the layer of ice to increase from $x_1$ to $x_2$ is given by

• A. $\frac{\rho L}{kT}(x_1+x_2{)}^2$
• B. $\frac{\rho L}{2kT}(x_2^2+x_1^2)$
• C. $\frac{\rho L}{kT}(x_1-x_2{)}^2$
• D. $\frac{\rho L}{2KT}(x_2^2-x_1^2)$

Answer 1

The correct option is D $\frac{\rho L}{2KT}(x_2^2-x_1^2)$

We know that, Heat required to freeze $dx$ thickness of ice
$Q=mL=\left(Adx\right)\rho L......\left(1\right)$
where $A$ is the surface area.


Heat flow during the time interval $dt$ is given by
$Q=\frac{KA(0-(-T\left)\right)dt}{x}.....\left(2\right)$
Equating$\left(1\right)$ and $\left(2\right)$, we get
$\frac{KATdt}{x}=A\rho Ldx$
Integrating on both sides,
${\int }_0^{t}dt=\frac{\rho L}{kT}{\int }_{x_1}^{x_2}xdx$
$\Rightarrow t=\frac{\rho L}{2kT}(x_2^2-x_1^2)$
Thus, option (d) is the correct answer.