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(C) Given, temperatures are $T_1 = 500 \ K$ and $T_2 = 300 \ K$.The maximum theoretical efficiency (Carnot efficiency) of an engine operating between

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$C$

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(C) Given, temperatures are $T_1 = 500 \ K$ and $T_2 = 300 \ K$.The maximum theoretical efficiency (Carnot efficiency) of an engine operating between these temperatures is given by $\eta_{max} = 1 - \frac{T_2}{T_1}$.$\eta_{max} = 1 - \frac{300}{500} = 1 - 0.6 = 0.4$ or $40\%$.Now, calculate the efficiency $(\eta = \frac{W}{Q_{in}})$ for each design:For design $A$: $\eta_A = \frac{150}{1000} = 0.15$ $(15\%)$.For design $B$: $\eta_B = \frac{450}{1000} = 0.45$ $(45\%)$.For design $C$: $\eta_C = \frac{300}{1000} = 0.30$ $(30\%)$.According to the Second Law of Thermodynamics, no engine can have an efficiency greater than the Carnot efficiency. Since $\eta_B = 0.45 > 0.4$, design $B$ violates the Second Law of Thermodynamics and is impossible. Design $C$ has an efficiency of $0.3$, which is less than $0.4$, making it physically possible. Therefore, design $C$ is the most suitable choice.

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