Adiabatic expansion of gas

Question

If a real gas is adiabatically expanded against constant pressure, then which of the following will definitely increase?

Compressibility factor (Z) or Entropy (S)

Correct answer: Entropy

My answer: Compressibility

My Reasoning :

$Z=\frac{PV}{nRT}$

In adiabatic $TV^{γ-1}=\text{constant}$

Therefore, $T$ ∝ $\frac{1}{V^{γ-1}}$

$γ-1$ is greater than zero always. So we can say is volume increases, temp decreases. Therefore Z should increase as its numerator increases and denominator decreases. Entropy for isobaric adiabatic has $\ln\frac{T_2}{T_1}$. Since $T_2$ is less than $T_1$, entropy decreases. What am I doing wrong ?

Answer 1

First note that the process is an irreversible adiabatic expansion. If it were a reversible one, the gas has to expand against a varying pressure because pressure of gas decreases during expansion and this varying gas pressure must be matched with external pressure in all infinitesimal steps of expansion. "The gas expanded against constant pressure" ⇒ the gas must have a pressure which is always higher than the constant external pressure during the expansion ⇒ irreversible.

Will entropy definitely increase?

ΔS_sur = 0 as q = 0. Irreversible ⇒ ΔS_tot > 0. ΔS_tot = ΔS_sys + ΔS_sur = 0 + ΔS_sys = ΔS_sys. So ΔS_sys > 0, so entropy of system will definitely increase.

Will compressibility factor definitely increase?

In adiabatic irreversible expansion all p, V, T changes. If Z is to increase, the increment in pV must be greater than the increment in T, or the magnitude of decrement in T must be greater than that in pV, which cannot be sure will happen as all p, V, T change.

What did you do wrong?

First of all, TV equation is for perfect gas doing reversible adiabatic expansion only. Second of all, a gas expands against constant external pressure does not mean the pressure of gas does not change. All p, V, T will change. You said as volume increases, temperature decreases, but pressure of gas will decrease too.