[plu]

I am trying to calculate the entropy change involved in the expansion of 1 mole of O₂ gas at an initial temperature of 120 K and pressure of 4 atm to a final temperature of 90 K and pressure of 1 atm.  The process is stated to be adiabatic, so dS_surr = 0.  I used the following equation to calculate dS_sys:

dS_sys = nRln(V₂/V₁) + nC_pln(T₂/T₁)

C_p of O₂ is given to be 28.2 J/K.mol .  Then, using the ideal gas law, V₁ and V₂ can be found to be 2.46 L and 7.39 L respectively.  My final answer is thus 1.03 J/K.mol .  However, the solution that my instructor gives has the answer as 3.41 J/K.mol and uses this equation:

dS_sys = nRln(P₁/P₂) + nC_pln(T₂/T₁)

This suggests that there is an inverse relation between P and V during the expansion, but this cannot be true because T is not constant.  Which solution is correct?

[Donaldson Tan]

Since entropy is a state variable, then we can divide the pathway into 2 consercutive path.

The first path involves constant temperature, ie. 120K, 4atm to 120K, 1atm

The second path involves constant pressure, ie. 120K, 1atm to 90K, 1atm.

dS_sys = nRln(P₁/P₂) + nC_pln(T₂/T₁)

The expression in bold represents the first path.

The underlined expression represents the second path.

Hope this clears things up for you.