Question

On the surface of lake when the atmospheric temperature is $-{15}^{\circ }\text{C}$, $1.5cm$ thick layer of ice is formed in 20 minutes, time taken to change its thickness from $1.5cm$ to $3cm$ will be

• A. 20 minutes
• B. Less than 20 minutes
• C. Greater than 40 minutes
• D. Less than 40 minutes and greater than 20

Answer 1

The correct option is C Greater than 40 minutes

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

$\frac{dQ}{dt}=\frac{kA\Delta \theta }{x}$
$\left(\frac{dm}{dt}\right)L=\frac{kA\Delta \theta }{x}$,
where $k$ is thermal conductivity of ice, $A$ is area of the lake, $L$ is latent heat of ice, and $\Delta \theta$ is the temperature difference.

$A\rho \frac{dx}{dt}L=\frac{kA[0-(-15\left)\right]}{x}$
$\underset{x_1}{\overset{x_2}{\int }}xdx=\underset{0}{\overset{t}{\int }}\frac{15k}{\rho L}dt$
$t=\frac{\rho L}{30k}[x_2^2-x_1^2]$

Now if $t_2$ is the time taken to change ice thickness from $1.5cm$ to $3cm$ and if $t_1$ is the time taken to change ice thickness from $0cm$ to $1.5cm$, then

$\Rightarrow \frac{t_2}{t_1}=\frac{[{3.0}^2-{1.5}^2]}{{1.5}^2-0^2}\Rightarrow t_2=20\times 3=60\text{min}$