For a given amount of nitrogen gas the initial, minimum temperature is $T_{0}$ , while the maximum temperature is $4 T_{0}$ . The gas is first heated at constant volume, then it is allowed to expand at constant pressure. Then it is cooled at constant volume and finally it is compressed at constant pressure. This way the gas returns to its initial state. Find the maximum possible efficiency of the cyclic process. If it is n, find 10 n _____ (n is a fraction not percentage) (nearest integer)

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Detailed Solution

$\frac{p_{0} V_{0}}{T_{0}} = \frac{p_{1} x V_{0}}{4 T_{0}}$

from which $p_{1} = \frac{4 p_{0}}{x}$
With this, the area enclosed by the graph is $W_{u s e f u l} = (\frac{4 p_{0}}{x} - p_{0}) (x V_{0} - V_{0}) =$

$= p_{0} V_{0} (\frac{4}{x} - 1) (x - 1) = \frac{p_{0} V_{0}}{x} (4 - x) (x - 1) η (x) = \frac{W_{u s e f u l}}{Q_{i n}} = \frac{\frac{p_{0} V_{0}}{x} (4 - x) (x - 1)}{\frac{p_{0} V_{0}}{x} (11.5 x - 4)} = \frac{(4 - x) (x - 1)}{11.5 x - 4} = \frac{x^{2} - 5 x + 4}{4 - 11.5 x} = η_{\max} = 0.1059$