Question 1: Thermodynamics & KTG

Question

A metal rod AB of length 10x has its one end A in ice at 0$^\circ$C and the other end B in water at 100$^\circ$C. If a point P on the rod is maintained at 400$^\circ$C, 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 cal/g and latent heat of melting of ice is 80 cal/g. If the point P is at a distance of $\lambda x$ from the ice end A, find the value of $\lambda$. (Neglect any heat loss to the surrounding.)

Accepted Answer

The steady rate of flow of heat is
${{\Delta Q} \over {\Delta t}} = kA\left( {{{\Delta T} \over {\Delta x}}} ight)$
Therefore, ${{kA(400 - 0^\circ )} \over {\lambda x}} = {m_{ice}}\,{l_{ice}}$
${{kA(400 - 100)} \over {(10 - \lambda )x}} = {m_{water}}\,{l_{water}}$
where ${m_{ice}}$ is the mass of ice melted per unit times and ${m_{water}}$ is the mass of water evaporated per unit times. It is given that
${m_{ice}} = {m_{water}}$
Therefore, ${{kA 	imes 400} \over {\lambda x({l_{ice}})}} = {{kA 	imes 300} \over {(10 - \lambda )x\,{l_{water}}}}$
$ \Rightarrow 4(10 - \lambda ) 	imes 540 = 3\lambda  	imes 80$
$ \Rightarrow 5400 - 540\lambda  = 60\lambda $
$ \Rightarrow \lambda  = 9$
![](https://app-content.cdn.examgoal.net/fly/@width/image/1lft4qow9/5bdb7db8-f12a-4305-a6fa-3c3fc3247d7c/2242a790-cde2-11ed-9ce1-7bd2ccf24231/file-1lft4qowa.png?format=png)

Thermodynamics & KTG

Integer / Numeric Answer Type