How to solve this?

Question

When an ideal gas at pressure P, temperature T and volume V is isothermally compressed to V/n, its pressure becomes Pi. If the gas is compressed adiabatically to V/n, its pressure becomes Pa. The ratio Pi/Pa is?

Answer 1

Isothermal compression:

 

PV = constant

 

PV = P_i(V/n) ⇒

 

P_i = nP

 

Adiabatic compression:

 

PV^γ = constant

 

γ = specific heat ratio of the gas

 

PV^γ = P_a(V/n)^γ ⇒

 

P_a = n^γP

 

P_i/P_a = nP / n^γP = n^{1 - γ}