An ideal gas goes through a cycle consisting of isothermal, polytropic, and adiabatic lines, with the isothermal process proceeding at the maximum temperature of the whole cycle. Find the efficiency of such a cycle if the absolute temperature varies `n-fold` within the cycle.

Explanation

The section from `(p_1, V_1, T_1)` to `(p_2, V_2, T_0//n)` is a polytropic process of index `alpha`. We shall assume that the corresponding specific heat `C` is `+ ve`.
Here, `dQ = CdT = C_V dT + pdV`
Now `pV^alpha = constant` or `TV^(alpha - 1) = constant`
so `pdV = (RT)/(V) dV = -( R)/(alpha - 1) dT`
Then `C = C_V -(R)/(alpha -1) =R ((1)/(gamma -1) - (1)/(alpha -1))`
We have `p_1 V_1 = RT_0 = p_2 V_2 = (RT_0)/(n) =(p_1 V_1)/(n)`
`p_0 V_0 = p_1 V_1 = b p_2 V_2, p_0 V_0^gamma = p_2 V_2^gamma`
`p_1 V_1^alpha = p_2 V_2^alpha` or `V_0^(gamma - 1) = (1)/(n) V_2^(gamma -1)` or `V_2 = V_0 n^((1)/(gamma -1))`
`V_1^(alpha -1) = (1)/(n) V_2^(alpha -1)` or `V_1 = n^(-(1)/(alpha -1)) V_2 = n^((1)/(gamma -1)-(1)/(alpha -1))V_0`
Now `Q'_2 = CT_0 (1 - (1)/(n)), Q_1 = RT_0 1n (V_1)/(V_0) = RT_0 ((1)/(gamma - 1)- (1)/(alpha -1))1n n = CT_0 1n n`
Thus `eta = 1 -(n -1)/(n 1n n)`.
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