Helium gas goes through a cycle $A B C D A$ (consisting of two isochoric and two isobaric lines) as shown in figure. Calculate the approximate efficiency of the cycle. (Assume the gas to be close to ideal gas).

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Solution

We know that the efficiency of a process is the ratio of work done to energy supplied. Here,

$η = \frac{Δ W}{Δ Q} = \frac{A r e a u n d e r p - V d i a g r a m}{Δ Q_{A B} + Δ Q_{B C}} = \frac{p_{0} V_{0}}{n c_{v} Δ T_{1} + n c_{p} Δ T_{2}}$

Where $c_{p}$ and $c_{v}$ are molar heat capacities at constant pressure and volume, respectively

Thus,

$η = \frac{p_{0} V_{0}}{\frac{3}{2} n R (T_{B} - T_{A}) + \frac{5}{2} n R (T_{C} - T_{D})}$

$η = \frac{p_{0} V_{0}}{\frac{3}{2} (2 p_{0} V - p_{0} V) + \frac{5}{2} (4 p_{0} V - 2 p_{0} V)}$

$η = \frac{p_{0} V_{0}}{\frac{3}{2} p_{0} V_{0} + 5 p_{0} V}$

$η = \frac{1}{6.5} = 15.4 %$

Hence, this is the required efficiency.

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