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Question

A layer of ice at 0C of thickness x1 is floating on a surface of water. L,ρ and K respectively are the latent heat of fusion of ice, density of ice and thermal conductivity of ice. If the atmospheric temperature is TC, the time taken for the thickeness of the layer of ice to increase from x1 to x2 is given by

A
ρLkT(x1+x2)2
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B
ρL2kT(x22+x21)
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C
ρLkT(x1x2)2
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D
ρL2KT(x22x21)
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Solution

The correct option is D ρL2KT(x22x21)
We know that, Heat required to freeze dx thickness of ice
Q=mL=(Adx)ρL ......(1)
where A is the surface area.


Heat flow during the time interval dt is given by
Q=KA(0(T))dtx .....(2)
Equating(1) and (2), we get
KATdtx=AρLdx
Integrating on both sides,
t0dt=ρLkTx2x1xdx
t=ρL2kT(x22x21)
Thus, option (d) is the correct answer.

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