Question

A rectangular ice box of total surface area of 1000 cm$^2$ initially contains 1.5 kg of ice at 0 $^\circ$C. If the thickness of the walls of the box is 2 mm and the temperature outside the box is 42 $^\circ$C, then the mass of the ice remaining in the box after 160 minutes is (Thermal conductivity of the material of the box = $10^{-2}$ Wm$^{-1}$K$^{-1}$ and latent heat of the fusion of ice = $336\times10^3$ Jkg$^{-1}$)

• A. 0.6 kg
• B. 0.9 kg
• C. 0.8 kg
• D. 0.7 kg

Hint:

This is a two-part problem. First, use the heat conduction formula ($\frac{dQ}{dt} = \frac{kA\Delta T}{d}$) to find the rate of heat flow. Second, use the latent heat formula ($Q = mL_f$) to relate the total heat transferred to the mass of substance that undergoes a phase change.

Answer 1

The Correct Option is B

Step 1: Calculate the rate of heat flow into the box.
The rate of heat conduction ($\frac{dQ}{dt}$) through the walls is given by:
$\frac{dQ}{dt} = \frac{kA\Delta T}{d}$.
Where:
$k$ = thermal conductivity = $10^{-2}$ W/(m$\cdot$K).
$A$ = total surface area = 1000 cm$^2$ = $1000 \times 10^{-4}$ m$^2$ = $0.1$ m$^2$.
$\Delta T$ = temperature difference = $42^\circ\text{C} - 0^\circ\text{C} = 42$ K.
$d$ = thickness of the walls = 2 mm = $0.002$ m.
$\frac{dQ}{dt} = \frac{(10^{-2})(0.1)(42)}{0.002} = \frac{0.042}{0.002} = 21$ J/s.
Step 2: Calculate the total heat that flows into the box in the given time.
Time $t = 160$ minutes = $160 \times 60$ seconds = 9600 s.
Total heat $Q = \left(\frac{dQ}{dt}\right) \times t = 21 \text{ J/s} \times 9600 \text{ s} = 201600$ J.
Step 3: Calculate the mass of ice that melts due to this heat.
The heat required to melt a mass $m_{melted}$ of ice is $Q = m_{melted} L_f$, where $L_f$ is the latent heat of fusion.
$L_f = 336 \times 10^3$ J/kg.
$m_{melted} = \frac{Q}{L_f} = \frac{201600}{336000} = \frac{2016}{3360}$.
$m_{melted} = \frac{1008}{1680} = \frac{504}{840} = \frac{252}{420} = \frac{126}{210} = \frac{63}{105} = \frac{21}{35} = \frac{3}{5} = 0.6$ kg.
Step 4: Calculate the mass of ice remaining.
Initial mass of ice $m_{initial} = 1.5$ kg.
Mass remaining = $m_{initial} - m_{melted} = 1.5 \text{ kg} - 0.6 \text{ kg} = 0.9$ kg.