A Carnot engine having a perfect gas as the working substance is driven backward and is used for freezing water already at $0^{\circ} C .$ If the engine is driven by a 500 W electric motor having a efficiency of 60%, how long (in seconds) will it take to freeze 15 kg of water. Take $15^{\circ} C and 0^{\circ} C$ as the working temperatures of the engine and assume there are no heat losses in the refrigerating system. Latent heat of ice $= 333 \times 10^{3} J / kg$

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Detailed Solution

$\begin{array}{l} G i v e n, T_{1} = 273 K \\ T_{2} = 15 + 273 = 280 K \\ Useful power = η P = 0.6 \times 500 = 300 J / s \\ We know that coefficient of performance \\ β = \frac{Q_{2}}{W} = (\frac{T_{2}}{T_{1} - T_{2}}) \\ \Rightarrow Q_{2} = 300 \times (\frac{273}{288 - 273}) = 5460 J / s \\ Heat needed to melt the ice Q = m L = 15 \times 333 \times 10^{3} J \\ Time taken in freezing water t = \frac{Q}{Q_{2}} = \frac{15 \times 333 \times 10^{3}}{5640} = 914.84 s \end{array}$