Entropy Difference of an Unknown Gas (not an ideal gas)

[albertov123]

Homework Statement

Temperature, pressure and volume measurements performed on 1 kg of a simple compressible substance in three stable equilibrium states yield the following results.

State 1 (T₁=400 C , V₁= 0,10 m³, P₁=3 MPa)
State 2 (T₁=400 C , V₁= 0,08 m³, P₁=3,5 MPa)
State 3 (T₁=500 C , V₁= 0,10 m³, P₁=3,5 MPa)

Estimate the difference in entropy S₂-S₁

Homework Equations

We don't know the gas. So I can't assume this is an ideal gas and I can't go to thermodynamics tables. I don't know the relevant equation.

The Attempt at a Solution

First, I didn't get why the question identifies state 3. I think we can completely ignore state 3 because the question is entropy difference between state 2 and state 1.

This is clearly a compression process, (volume decreases, pressure increases) but temperature stays still. But when a gas is compressed, its pressure and temperature rises. So there must be heat transfer going on.

Entropy change = S_gen + Q/T
But we don't know the entropy generation so we can't go from here.

I assume I need a relation related with volume and pressure that yields entropy but in constant temperature. Is there a relation like that?

 

[Chestermiller]

Yes. There is a relation for the partial derivative of entropy with respect to pressure at constant temperature in terms of the P-V-T properties of a gas.
Have you studied that yet? Are you learning about the Maxwell equations in your course yet? The derivation starts off with dG=-SdT+VdP.

 

[albertov123]

Yes we are learning maxwell equations in fact. That derivation must be dG=-SdT+VdP+∑μ_idN_i from my notebook.

The relation you have mentioned is probably -(∂S/∂P)_T=(∂V/∂T)_P and with that equation my solution would include state 3. But how could i go with this equation I'm not clear. Although I'm very doubtful, can I think the above equation as -(ΔS/ΔP)_{between state 1-2}=(ΔV/ΔT)_{between state 2-3}? Otherwise I don't know what to do.

 

[Chestermiller]

Yes we are learning maxwell equations in fact. That derivation must be dG=-SdT+VdP+∑μ_idN_i from my notebook.

The relation you have mentioned is probably -(∂S/∂P)_T=(∂V/∂T)_P and with that equation my solution would include state 3. But how could i go with this equation I'm not clear. Although I'm very doubtful, can I think the above equation as -(ΔS/ΔP)_{between state 1-2}=(ΔV/ΔT)_{between state 2-3}? Otherwise I don't know what to do.

Yes, that's what you do. They said "estimate" the entropy change, so using finite differences is OK.