Thermo Inferno Challenge

A gas possessing isentropic expansion factor $\gamma$ is made to undergo a thermodynamic cycle consisting of $2$ isobaric and $2$ isothermal processes each.If during the cycle,the ratio of maximum to minimum volume $\dfrac{V_{max}}{V_{min}} = \alpha$ and ratio of maximum to minimum temperature $\dfrac{T_{max}}{T_{min}} = \beta$ .Then express the efficiency $\eta$ of the thermodynamic cycle in terms of $\alpha$ and $\beta$ .Also assume $\alpha > \beta > 1$ to get a positive efficiency.

If $\eta = \dfrac{(\beta - a) ln (\dfrac{\alpha}{\beta})^b}{\dfrac{(\beta-c)\gamma}{\gamma - 1}+ \beta ln (\dfrac{\alpha}{\beta})^d}$ find $a + b + c+ d$ .

Details and Assumptions :

1) The isentropic expansion factor, also known as the adiabatic index of a gas is the ratio of its molar specific heat capacities at constant pressure and constant volume ; $\gamma = \dfrac{C_{p}}{C_{v}}$ ; where $C_{p} - C_{v} = R =$ a constant.

2) The gas is assumed to be ideal.The ideal gas law $PV =\mu RT$ is always applicable.

3) $ln (x)$ refers to the natural log of $x$ ; $log_{e} (x)$ .

4) $a,b,c,d$ are positive integers in their simplest form.They may or may not be distinct.

5) Just as a sidenote,you may want to investigate the cases when $\alpha$ and $\beta$ approach $0$ or $\infty$ .

This problem is Original and made by me.Hope you liked it ! :D