Hydrogen is used in a Carnot cycle as a working substance. Find the efficiency of the cycle, if as a result of an adiabatic expansion
(a) the gas volume increases `n = 2.0` times ,
(b) the pressure decreases `n = 2.0` times.

Explanation

For `H_2, gamma = (7)/(5)`
`p_1 V_1 = p_2 V_2, p_3 V_3, p_4 V_4`
`p_2 V_2^gamma = p_3 V_3^gamma, p_1 V_1^gamma = p_4 V_4^gamma`
Define `n` by `V_3 = n V_2`
Then `p_3 = p_3 n^- gamma` so
`p_4 V_4 = p_3 V_3 = p_2 V_2 n^(1 - gamma) = p_1 V_1 n^(1 - gamma)`
`p_4 V_4^gamma =p_1 V_1^gamma` so `V_4^(1 - gamma) = V_1^(1 - gamma) n^(1- gamma)` or `V_4 = n V_1`
Also `Q_1 = p_2 V_2 1n (V_2)/(V_1), Q'_2 = p_3 V_3 1n (V_3)/(V_4) n^(1 - gamma) = p_2 1n (V_3)/(v_4)`
Finally `eta = 1 -(Q_2)/(Q_1) = 1 - n^(1 - gamma) = 0.242`
(b) Define `n` by `p_3 = (p_2)/(n)`
`p_2 V_3^gamma = (p_2)/(n) V_3^gamma` or `V_3 = n^(1//gamma) V_2`
So we get the formulae here by `n rarr n^(1//gamma)` in the previous case.
`eta = 1 - n^((1//gamma)-1) =1 - n^(-(2)/(7)) ~~ 0.18`.
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