Question

A carbon ball at ${40}^{\circ }C$ is dropped from a height of $5\text{km}$. The ball is heated due to air resistance and it completely melts just before reaching the ground. The specific heat capacity of carbon is $125{\text{J kg}}^{-1}{{}^{\circ }C}^{-1}$ and melting point is ${340}^{\circ }\text{C}$. Then, the latent heat of the fusion of the ball (in ${\text{kJ kg}}^{-1}$) is

[Take $g=10{\text{m/s}}^2$]

Answer 1

Let the mass of the ball be $m\text{kg}$.
Gravitational potential energy $Q=mgh$
$\Rightarrow Q=m\times 5\times {10}^4\text{J}$
Energy required to heat the ball from ${40}^{\circ }\text{C}$ to ${340}^{\circ }\text{C}$ is given by
$Q_1=ms\Delta T$
$Q_1=m\times 125\times [340-40]=m\times 37500\text{J}$
Energy required to melt the ball completely
$Q_2=mL$, where $L$ = Latent heat of fusion of the ball

For the ball to melt completely, gravitational P.E must be converted to the required heat.
From the data given in the question,
$Q=Q_1+Q_2$
$\Rightarrow m\times 5\times {10}^4=m\times 37500+mL$
$\Rightarrow L=12500{\text{J kg}}^{-1}$
The latent heat of the fusion of carbon ball is $12500{\text{J kg}}^{-1}=12.5\text{kJ/kg}$