An ideal gas with the adiabatic exponent `gamma` goes through a cycle (Fig. 2.3) within which the absolute temperature varies `tau-fold`. Find the efficiency of this cycle.
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Explanation

Because of the linearity of the section
`B C` whose equation is
`(p)/(p_0) = (v V)/(V_0) ( = p = alpha V)`
We have `(tau)/(v) = v` or `v = sqrt(tau)`
Here `Q''_2 = C_V T_0 (sqrt(tau) -1)`,
`Q'''_2 = C_p T_0(1 - (1)/(sqrt(tau))) = C_p (T_0)/(sqrt(tau)) (sqrt(tau) -1)`
Thus `Q'_2 = Q''_2 + Q'''_2 = (RT_0)/(gamma -1) (sqrt(tau) -1) (1 + (gamma)/(sqrt(tau)))`
Along `BC`, the specific heat `C` is given by
`CdT = C_V dT + pdV = C_V dT + d((1)/(2) alpha V^2) = (C_V + (1)/(2))dT`
Thus `Q_1 = (1)/(2) R T_0 (gamma + 1)/(gamma -1) (tau -1)/(sqrt(tau))`
Finally `eta = 1-(Q'_2)/(Q_1) = 1-2 (sqrt(tau) + gamma)/(sqrt(tau) + 1) (1)/(gamma + 1) = ((gamma - 1)(sqrt(tau) -1))/((gamma + 1)(sqrt(tau) + 1))`.
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