An ideal gas is heated from temperature T₁ to temperature T₂ by keeping its volume constant. The gas is expanded back to its initial temperature according to the law PVⁿ = constant. If the entropy changes in the two processes are equal, find the value of n in terms of the adiabatic index γ

Change in entropy during constant volume process is given as:
= m c_v ln (T₂ / T₁)
Change in entropy during polytropic process i.e. (PVⁿ = constant)

= m c_v [(γ – n)/ (n – 1)] ln (T₂ / T₁)

For the same entropy, equating both the above equations, we will get:
[(γ – n)/ (n – 1)] = 1
(γ – n) = (n – 1)
2n = γ +1
n = γ +1 / 2

Hence, the value of n in terms of the adiabatic index γ is equal to γ +1 / 2