1 Show that for one mole of an ideal gas undergoing adiabati

1) Show that for one mole of an ideal gas undergoing adiabatic expansion: (It asks to derive the equation below using ideal gas law and knowledge about adiabatic process)

2. Using part 1, show that for an ideal gas undergoing an adiabatic expansion,

Solution

A. Since the process is adiabatic

n = 1

dQ = 0

first law

dU + PdV = 0

dU = n*Cv*dT = Cv*dT

Cv*dT + P*dV = 0

[PV = nRT, P = RT/V]

Cv*dT + RT*dV/V = 0

dividing by Cv*T

dT/T + R*dV/Cv*V = 0

2.

dT/T + R*dV/Cv*V = 0

dT = -P*dV/Cv (1)

as you know

dU = n*Cv*dT = Cv*dT

also dU = -P*dV

R = Cp - Cv

Cp/Cv = y

PV = RT

P*dV + V*dP = R*dT

using eq. 1

P*dV + V*dP = R*( -P*dV/Cv)

dV/V + dP/P = [-(Cp - Cv)/Cv]*(dV/V)

dP/P = (1 - Cp/Cv)*(dV/V) - dV/V

dP/P = y*dV/V

ln P = y*ln V + C

PV^y = C

3. Cp - Cv = R

Cp/Cv = y

(Cv + R)/Cv = y

y = 1 + R/Cv