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Question

A lake is covered with ice and the atmospheric temperature above the ice is 10C. Find the rate of formation of ice (in cm/hr) when its thickness is 5 cm.
[Thermal conductivity of ice =0.005 cgs units, density of ice =0.9 g/cc and latent heat of fusion of ice =80 cal/g].
  1. 50 cm hr1
  2. 25 cm hr1
  3. 0.5 cm hr1
  4. 0.25 cm hr1


Solution

The correct option is C 0.5 cm hr1
Consider a layer of ice that is x cm thick. Just below the ice, temperature is 0C and above it, the temperature is θ=10C.
Let A be the area of the ice layer.
Heat conducted through ice in next dt seconds will be 
dQ=kAθxdt
If ice layer thickens by dx, the mass of ice frozen in time dt will be ρAdx
dQ=mL=(ρAdx)L
kAθxdt=ρALdx
dxdt=kθxρL
is the rate of formation of ice, when thickness is x cm.
From the data given in the question,
dxdt=(0.005 cal cm1s1C1)×(10 C)(5 cm)(0.9 g cm3)(80 cal g1)=0.00014 cm s1 
=0.00014×3600 cm hr10.5 cm hr1
Thus, option (c) is the correct answer.

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