A piston divides a closed gas cylinder into two parts. Initially the piston is kept pressed such that one part has a pressure P and volume 5V and the other part has pressure 8P and volume V, the piston is now left free. Find the new pressure and volume for the isothermal and aidabatic process. `(gamma=1.5)`

Explanation

`P_0` : common pressure
Isothermal process
Chamber 1: `P.5V=P_0V_1`
Chamber 2, `8P.V=P_0V_2`
`(i)/(ii)implies(5)/(8)=(V_1)/(V_2)impliesV_2=(8)/(5)V_1`
`V_1+V_2=5V+V`
`V_1+(8)/(5)V_1=6VimpliesV_1=(30V)/(13)`
`V_2=(8)/(5)V_1=(48)/(13)V`
`P.5V=P_0V_1implies5PV=P_0(30)/(13)`
`P_0=(13)/(6)P`
Adiabatic process:
`P.(5V)^(gamma)=P_0V_1^(gamma)`
`8P.(V)^(gamma)=P_0V_2^(gamma)`
`(i)/(ii)implies(1)/(8)(5)^(gamma)=((V_1)/(V_2))^(gamma)implies(V_1)/(V_2)=5((1)/(8))^((1)/(gamma))`
`(V_1)/(V_2)=5((1)/(2^3))^((2)/(3))=(5)/(4)`
`V_2=(4)/(5)V_1`
`V_1+V_2=5V+VimpliesV_1+(4)/(5)V_1=6VimpliesV_1=(10)/(3)V`
`V_2=(4)/(5)V_1=(4)/(5).(10)/(3)V`
`V_2=(4)/(5)V_1=(4)/(5).(10)/(3)V=(8)/(3)V`
`P.(5V)^(gamma)=P_0V_1^(gamma)impliesP_0=P((5V)/(V_1))^(gamma)=P((5V)/((10)/(3)V))^(gamma)`
`P_0=((3)/(2))^(3/2)P=1.84P`