Q1.Prove that PV^Y=C for reversible adiabatic process.

Solution:

δQ = dU + PdV

Reversible Adiabatic:

δQ=0 (:: There is no heat interaction between system and surroundings = adiabatic

dU + dV = 0

dU = -PdV

m(Cv)dT = -PdV ----------------------------------------------(1)

Now we want 'Y' , so,

We know H = U + PV

dH = dU + PdV + VdP

dH = δQ + VdP

(:: There is no heat interaction δQ =0)

dH = VdP

m(Cp)dT = VdP ------------------------------------------(2)

Dividing (2) by (1)

Y = -V/P(dP/dV) (:: Y= Gama=Cp/Cv)

Y = -V/dV(dP/P)

Integral both side

∫YdV/V + ∫dP/P =∫0

Y(lnV) + lnP = lnC

ln PV^Y = lnC

PV^Y = C ------------Since, this has been found for δQ =0 so, only apply it for reversible adiabatic

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