Question

Ice starts forming in a lake with water at $0^{\circ }C$ when the atmospheric temperature is $-{10}^{\circ }C$.If the time taken for the first $1$ cm of ice to be formed is $7$ hours, then the time taken for the thickness of ice to change from $1$ cm to $2$ cm is

• A. 7 hours
• B. 14 hours
• C. 21 hours
• D. 3.5 hours

Answer 1

The correct option is C 21 hours

Let us assume that $x$ length of ice has become formed, and to form a small strip of ice of length $dx$, $dt$ time is required.

then

$\frac{dQ}{dt}=\frac{KA\Delta T}{x}\frac{d\left(mL\right)}{dt}=\frac{KA\Delta T}{x}\frac{A{\rho }_{w}Ldx}{dt}=\frac{KA\Delta T}{x}dt=\frac{{\rho }_{w}Lxdx}{K\Delta T}$

where $K$ is thermal conductivity of ice, $A$ is area of the lake, $L$ is latent heat of ice, $m$ and ${\rho }_w$ are mass density of ice and $\Delta T$ is the temperature difference.

by integrating the above equation on both sides

${\int }_0^{t}dt=\frac{{\rho }_{w}L}{K\Delta T}{\int }_0^{l}xdxt=\frac{{\rho }_{w}L}{K\Delta T}\frac{l^2}{2}t=C\frac{l^2}{2}$

where $C=\frac{{\rho }_{w}L}{K\Delta T}$

Given that to form $1cm$ thickness of ice, $7hours$ time has taken

then time required to form $2cm$ thickness of ice is given by

$\frac{t_1}{l_1^2}=\frac{t_2}{l_2^2}{t}_2=\frac{t_1}{l_1^2}{l}_2^2=\frac{7}{1}\times 4hours=28hours$

time taken to form ice from $1cm$ to $2cm$ is

$t_2-t_1=28-7=21hours$

option (C) is the correct answer