Problem No. 5 (15 points)

With the same amount of monatomic ideal gas, two cyclic processes $1-2-3-1$ and $1-3-4-1$ are performed. Find the ratio of their efficiencies.

![](https://cdn.mathpix.com/cropped/2024_05_06_7ac9c551b647ccda756fg-1.jpg?height=740&width=939&top_left_y=1035&top_left_x=587)

Answer: $\frac{13}{12} \approx 1.08$

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# Solution and Evaluation Criteria:

For the cycle $1-2-3-1$: work done by the gas over the cycle: $A=\frac{1}{2} p_{0} V_{0}$

Heat received from the heater:

$$
Q=Q_{12}+Q_{23}=\Delta U_{123}+A_{123}=\frac{3}{2}\left(4 p_{0} V_{0}-p_{0} V_{0}\right)+2 p_{0} V_{0}=6.5 p_{0} V_{0}
$$

Efficiency of this cycle: $\eta_{123}=\frac{A}{Q}=\frac{1}{13}$

For the cycle 1-3-4-1: work done by the gas over the cycle: $A=\frac{1}{2} p_{0} V_{0}$

Heat received from the heater:

$Q=Q_{13}=\Delta U_{13}+A_{13}=\frac{3}{2}\left(4 p_{0} V_{0}-p_{0} V_{0}\right)+1.5 p_{0} V_{0}=6 p_{0} V_{0}$

Efficiency: $\eta_{134}=\frac{A}{Q}=\frac{1}{12}$

Ratio of the efficiencies of these cycles: $\frac{\eta_{134}}{\eta_{123}}=\frac{13}{12} \approx 1.08$