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Question

A metal rod AB of length 10x has its one end A in ice at 0C and the other end (B) in water at 100C. If a point P on the rod is maintained at 400C, then it is found that equal amounts of water and ice, evaporate and melt respectively per unit time. The latent heat of vaporization of water is 540cal g1 and the latent heat of fusion of ice is 80cal g1. If the point P is at a distance of λx from the end A, find the value of λ.
(Neglect any heat loss to the surroundings)


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Solution

The rate of steady state heat transfer is given by
ΔQΔt=kA(ΔTΔx)
The amount of heat flow from P to A
ΔQΔt=kA(4000)λx=Lfdm1dt ....(i)
where m1 is the amount of ice melted.
The amount of heat flow from P to B
ΔQΔt=kA(400100)(10λ)x=Lvdm2dt ...(ii)
where m2 is the amount of water evaporated.
Given, dm1dt=dm2dt
kA(4000)λxLf=kA(400100)(10λ)xLv
400λ×80=300(10λ)540
216(10λ)=24λ
λ=9

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