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Question

Ice starts forming in a lake with water at 0C when the atmospheric temperature is 10C.If the time taken for the first 1 cm of ice to be formed is 7 hours, then the time taken for the thickness of ice to change from 1 cm to 2 cm is


A
7 hours
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B
14 hours
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C
21 hours
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D
3.5 hours
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Solution

The correct option is C 21 hours
Let us assume that x length of ice has become formed, and to form a small strip of ice of length dx, dt time is required.
then
dQdt=KAΔTxd(mL)dt=KAΔTxAρwLdxdt=KAΔTxdt=ρwLxdxKΔT
where K is thermal conductivity of ice, A is area of the lake, L is latent heat of ice, m and ρw are mass density of ice and ΔT is the temperature difference.
by integrating the above equation on both sides
t0dt=ρwLKΔTl0xdx t=ρwLKΔTl22 t=Cl22
where C=ρwLKΔT
Given that to form 1cm thickness of ice, 7hours time has taken
then time required to form 2cm thickness of ice is given by
t1l21=t2l22t2=t1l21l22=71×4hours=28hours
time taken to form ice from 1cm to 2cm is
t2t1=287=21hours
option (C) is the correct answer

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