A metal rod AB of length $10 x$ has its one end $A$ in ice at 0⁰C and the other end B in water at 100⁰C. If a point $P$ on the rod is maintained at 400⁰C, then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is 540 calg^-1 and latent heat of melting of ice is 80 calg^-1. If the point $P$ is at a distance of $λ x$ from the ice end $A$ , find the value of $λ$ . (Neglect any heat loss to the surrounding.)

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answer is 9.

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Detailed Solution

Heat will flow both sides from point $P$ .

$\Rightarrow {(\frac{d Q}{d t})}_{c a l o r i m e t r y} = {(\frac{d Q}{d t})}_{c o n d u c t i o n} L_{1} \frac{d m_{1}}{d t} = {(\frac{T e m p e r a t u r e d i f f e r e n c e}{T h e r m a l r e s i s \tan c e})}_{1} = \frac{400}{(λ x) / K A}$

Similarly,

$L_{1} \frac{d m_{2}}{d t} = \frac{400 - 100}{(100 - λ) x / K A}$

In above tow equations, $\frac{d m_{1}}{d t} = \frac{d m_{2}}{d t} (g i v e n)$
$L_{1} = 80 c a \lg^{- 1}$ and $L_{2} = 540 c a \lg^{- 1}$
Solving these two equations, we get $λ = 9.$