Question

Ice starts forming in a lake with water at $0^{\circ}C$ when the atmosphere temperature is $-10^{\circ}C$. If the time taken for first 1 cm of ice be formed is 12 minutes, the time taken for the thickness of the ice to change from 1 cm to 2 cm will be:

Answer 1

36 minutes

Answer 2

The rate of heat transfer through ice is inversely proportional to its thickness ($dQ/dt \propto 1/x$). The heat required to freeze a layer of ice is proportional to the thickness of that layer ($dQ \propto dx$). Combining these, the time taken to form an additional layer is proportional to the thickness of the existing ice layer ($dt \propto x dx$). Integrating this relation shows that the total time $t$ to form a thickness $x$ is proportional to the square of the thickness ($t \propto x^2$). Given that the time to form the first 1 cm is 12 minutes, the total time to form 2 cm will be $12 \times (2/1)^2 = 48$ minutes. The time taken for the thickness to change from 1 cm to 2 cm is the difference between the total time to form 2 cm and the total time to form 1 cm, which is $48 - 12 = 36$ minutes.