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Question

Ice starts forming in lake with water at 0. When the atmospheric temperature is 10 C, if the time taken to form 1 cm of ice be 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
3.5 hours
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D
21 hours
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Solution

The correct option is D 21 hours
Let us assume that x length of ice has formed and to form a small strip of ice of length dx, dt time is required, then dQdt=KAΔ Tx
d(mL)dt=KAΔ Tx
AρwLdxdt=KAΔ Tx
dt=ρxLxdxKΔ T
where,
K is thermal conductivity of ice,
A is area of the lake,
L is talent heat of ice,
ρw are mass density of ice and
Δ T is the temperature difference.
By integrating the above equation on both, sides
t0dt=ρwLKΔ Tl0xdx
t=ρwLKΔ Tl22
t=Cl22
where C=ρwLKΔ T
Given that to form ice of 1 cm thickness, time of 7 hours has been taken then, time required to form 2 cm thickness of ice will be given by
t1l21=t2l22
t2=t1l21l22=71×4 =28 hours
Time taken to increase thickness of ice from 1 cm to 2 cm is t2t1=287=21 hours

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