**Problem:**
A metal rod AB of length 10x has its one end A in ice at 0°C and the other end B in water at 100°C. If a point P on the rod is maintained at 400°C, then it is found that equal amounts of water and ice evaporates and melts per unit time. The latent heat of evaporation of water is 540 cal/gram and latent heat of melting of ice is 80cal/gram. If the point P is at a distance of λx from the ice end A, find the value of λ. (Neglect any heat loss to the surroundings.)
(IIT JEE 2009)

**Answer:** The answer is λ = 9.

**Solution:**
The situation is shown in the figure.

Let $A$ be the cross-sectional area of the rod and $k$ be the thermal conductivity of the rod's material. The rates of heat flow from the point P towards the end A and the end B, are given by \begin{align} \frac{\mathrm{d}Q_A}{\mathrm{d}t}&=kA \frac{T_P-T_A}{\lambda x}\\ &=\frac{400kA}{\lambda x},\nonumber\\ \frac{\mathrm{d}Q_B}{\mathrm{d}t}&=kA \frac{T_P-T_B}{10x-\lambda x} \nonumber\\ &=\frac{300kA}{10 x-\lambda x}.\nonumber \end{align} Let the rate of melting of ice be $\mathrm{d}m_i/\mathrm{d}t$ and the rate of evaporation of water be $\mathrm{d}m_w/\mathrm{d}t$. The rates of heat required for the ice and the water are \begin{align} &\mathrm{d}Q_A/\mathrm{d}t=(\mathrm{d}m_i/\mathrm{d}t)L_i,\nonumber\\ &\mathrm{d}Q_B/\mathrm{d}t=(\mathrm{d}m_w/\mathrm{d}t) L_w, \nonumber \end{align} where $L_i=$ 80 cal/gram and $L_w=$ 540 cal/g are the latent heats of fusion and evaporation, respectively. Use $\mathrm{d}m_i/\mathrm{d}t=\mathrm{d}m_w/\mathrm{d}t$ and simplify to get, \begin{align} \left(\frac{400}{\lambda}\right)\cdot \left(\frac{10-\lambda}{300}\right)=\frac{80}{540},\nonumber \end{align} which gives, $\lambda=9$.

- Heat Conduction
- The end Q and R of two thin wires, PQ and RS, are soldered (IIT JEE 2016)
- Two blocks connected in series and parallel ( IIT JEE 2013)
- Two rods connected in series and parallel (IIT JEE 2014)
- Two conducting cylinders of equal length but different radii connected in series (IIT JEE 2018)
- Temperature at the junction of the three identical rods (IIT JEE 2001)
- Three rods forms an isosceles triangle. Temperature at the vertex? (IIT JEE 1995)
- Effective thermal conductivity of two concentric cylinders (IIT JEE 1988)
- MCQ on a composite block is made of slabs A, B, C, D, and E (IIT JEE 2011)
- A point source of heat of power P kept at the centre of a spherical shell (IIT JEE 1991)
- A double pane window used for insulating a room thermally from outside (IIT JEE 1997)
- An electric heater is used in a room of total wall area 137 m
^{2}( IIT JEE 1986)