An adiabatically isolated cylinder with a mobile piston contains N = 1 mol of an

Question

An adiabatically isolated cylinder with a mobile piston contains N = 1 mol
of an ideal gas. The piston slowly moves and the volume of the gas increases
from the initial value V0 to the final value V1. The temperature T and volume
V of the gas during this adiabatic expansion process are related by equation
TV^2/5 = T_0V_0^2/5, where T_0 is the initial temperature.

Knowing that the gas satisfies the ideal-gas equation of state
pV = NRT

Show that the pressure P and volume V of the gas during the adiabatic expansion
described above are related by equation of the form
pV^k = p_0V_0^k

Explanation / Answer

TV^2/5 = T_0V_0^2/5

From ideal gas equation, T = pV/NR

Putting it in above equation, (pV/NR)V^2/5 = (p_0V_0/NR)V_0^2/5
Simplifying it we get, pV^3/5 = p_0V_0^3/5

This is of the form pV^k = p_0V_0^k

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