Question

Consider the shown case of a freezing lake due to negative environmental temperature $(–{\theta }^{\circ }C)$. Thickness $\left(x\right)$ of ice layer is small in comparison to depth of lake. Rate of increase in $x$ will be greater

• A. if environmental temperature increases
• B. for larger thickness of ice layer
• C. if environmental temperature decreases
• D. for smaller thickness of ice layer

Answer 1

Answer: (C), (D)

The correct options are
C if environmental temperature decreases
D for smaller thickness of ice layer

Let the air temperature be $-{\theta }^{\circ }C$ and water temperature just below ice be $-0^{\circ }C$. At a certain time let the thickness of ice be $x$ and let it increase $dy$ in time $dt$.
The latent heat released on melting has to be conducted away through the ice layers as the water freezes and therefore
$Q=\frac{KA\left(\Delta T\right)}{x}=\frac{KA(0-(\theta \left)\right)}{x};\Rightarrow LA\rho \frac{dy}{dt}=\frac{KA\theta }{x}$
Here $L$ is Specific latent heat of fusion of water,
$\rho$ is density of ice
$A$ is area of cross section

$\Rightarrow \frac{dy}{dt}=\frac{K\theta }{L\rho x}$
From the equation we can see that Rate is directly proportional to $\theta$ while inversely proportional to the thickness of ice layer.