Question

A system receives heat continuously at the rate of 10 W. The temperature of the system becomes constant at 70°C when the temperature of the surrounding is 30°C. After the heat is switched off, the system cools from 50°C to 49.9°C in 1 minute. Find the heat capacity of the system (in kJ/°C).

Answer 1

2.9925

Answer 2

In steady state, heat supplied = heat lost: $10 \mathrm{~W} = k(70^{\circ} \mathrm{C} - 30^{\circ} \mathrm{C})$ $k = 0.25 \mathrm{~W} /{ }^{\circ} \mathrm{C}$.

During cooling, $dQ = C \Delta T = C(50^{\circ} \mathrm{C} - 49.9^{\circ} \mathrm{C}) = 0.1C$. Average temperature $T_{avg} = \frac{50 + 49.9}{2} = 49.95^{\circ} \mathrm{C}$. Average rate of heat loss $P_{avg} = k(T_{avg} - T_{surrounding}) = 0.25(49.95 - 30) = 4.9875 \mathrm{~W}$. Total heat lost in 60 s: $dQ \approx P_{avg} \times \Delta t = 4.9875 \times 60 = 299.25 \mathrm{~J}$.

Equating the two expressions for $dQ$: $0.1C = 299.25 \mathrm{~J}$ $C = 2992.5 \mathrm{~J} /{ }^{\circ} \mathrm{C} = 2.9925 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}$.