Question

An adiabatic process has thermodynamic relationships between two thermodynamic variables and the coefficient of adiabatic expansion. Starting from the differential mass balance in the closed system for an ideal gas, deduce the equation that relates P and T for this process.

Answer 1

From the first law of thermodynamics for a closed system:

dU = dQ + dW

For an adiabatic process:

dQ = 0

Therefore;

dU = dW

For and ideal gas;

dU = n Cv dT

dW = - P dV

Substituting these in the expression:

n C_v dT = - P dV

From the Ideal gas equation of state:

P V = n R T

then ; T = P V / nR

Substituting this in the expression:

n C_v d (PV / nR) = - P dV

But n and R is constant and therefore comes out of the differential.

(n C_v / n R) d(PV) = - P dV

(C_v / R) { V dP + P dV } = - P dV

- (C_v / R) V dP = ( (C_v / R) + 1) P dV

- (C_v / R) V dP = ( (C_v + R) / R ) P dV

Multiplying both the sides by R

- C_v V dP = ( C_v + R ) P dV

But for an ideal gas;

C_v + R = C_p

Therefore;

- C_v V dP = C_p P dV

Rearranging the above expression we get;

dP / P = - (C_p / C_v) dV / V

Let us denote Cp / Cv = r

Then;

dP / P = - r dV / V

Integrating both the sides;

ln (P) = - r ln (V) + constant

ln (P) = - ln (V^r) + constant

ln (P) + ln (V^r) = constant

ln (P V^r) = constant

or

P V^r = constant

But for an ideal gas:

V = nRT / P

P (nRT / P)^r = constant

P (T / P)^r = constant

P^{(1 - r)} T^r = constant

where r is the coefficient of adiabatic expansion

This is the required relation between T and P for an ideal gas undergoing an adiabatic process.