A metal rod AB of length 10x has its one end A in ice at 0^oC and the other end ‘B’ in water at 100^oC. If a point P on the rod is maintained at 400^oC, then it is found that mass of ice melting per unit time equals the mass of water evaporated per unit time. The latent heat of evaporation of water is 540 cal/g and latent heat of melting of ice is 80 cal/g. 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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Detailed Solution

$\begin{array}{l} \frac{{dQ}_{1}}{dt} = \frac{400 - 0}{λx / kA} \\ \frac{{dQ}_{2}}{dt} = \frac{400 - 100}{10 x - λx / kA} \\ \frac{{dQ}_{1} / dt}{{dQ}_{2} / dt} = \frac{(\frac{dm}{dt}), L_{fision}}{(\frac{dm}{dt}) L_{vaporisation}} \Rightarrow \frac{L_{f}}{L_{v}} = \frac{\frac{400}{λx}}{\frac{300}{(10 - λ) x}} \end{array}$