Non-ideal Gas

1. Feb 11, 2006

Nusc

For a van der Waals gas experiencing an adiabatic process derive the following expression:

T(V-nb)^(R/Cv) = Constant

I tried using PV^gamma = Constant with gamma = Cp/Cv
and Cp - Cv = nR with PV = nRT but could not get it.

Any hints?

I would have to use Boyle's law to account for the factor of b but I'm not sure of its relavance to the problem.

Last edited: Feb 11, 2006
2. Feb 12, 2006

Staff: Mentor

http://theory.ph.man.ac.uk/~judith/stat_therm/node96.html [Broken]

Last edited by a moderator: May 2, 2017
3. Feb 12, 2006

Nusc

This problem is specifically non-ideal but the equations only apply to the ideal case.

4. Feb 12, 2006

Staff: Mentor

Well perhaps one can start with P (V-nb) = nRT, which ignores the +an2/V2 term, which correct pressure, i.e. (P + an2/V2).

5. Feb 12, 2006

Nusc

Okay so, it's non-ideal therefore V = V - nb
It's adiabatic so the other ideal equations still hold.
And this is a van der Waals gas.

If I do start with P (V-nb) = nRT then I would have to eventually end up with T(V-nb)^R/Cv = Constant. But how would I ever get that power R/Cv?

6. Feb 12, 2006

Nusc

I prove nothing:

P(V-nb)^R/Cv = nRT
Cp-Cv = nR
Cp = nR + Cv

gamma = Cp/Cv = (nR+Cv)/Cv = 1+nR/Cv

T(V-nb)^(gamma -1) = Constant
TP^(1/gamma -1) = Constant

T(V-nb)^(gamma -1) = TP^(1/gamma -1)

gamma root((V-nb)) / (V-nb) = gamma root(P)/P

P/(V-nb) = gamma root(P/(V-nb))

(P/(V-nb))^gamma = P/(V-nb)

(P/(V-nb))^(1+nR/Cv) = nRT

This gets me nothing, hints?

7. Feb 12, 2006

Gokul43201

Staff Emeritus
1. Replace V by V-nb
2. Write the new equation of state
3. Write the new adiabatic equation
4. Substitute and complete

That will give you the result of post#1

PS : If you have trouble, perform the above steps and we'll help from wherever you are stuck...

8. Feb 14, 2006

Nusc

So we know V = V-nb
PV^gamma = cst

gamma = Cp/Cv
Cp-Cv = nR
Cp = nR + Cv

P(V-nb) ^ gamma = cst
P(V-nb) ^ Cp/Cv = cst
P(V-nb) ^ (nR+Cv/Cv) = cst
P(V-nb) ^ (nR/Cv +1) = cst

Adiatbatic process so Q = 0 and dU = -W = CvMdT
Where does T supposed to come from in the T(V-nb)^R/Cv = cst?
How do I cancel n in the power?

Last edited: Feb 14, 2006
9. Feb 14, 2006

Nusc

Fine, let me try again.

Non-ideal Gas

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For a van der Waals gas experiencing an adiabatic process derive the following expression:

PV ^ gamma = cst.
P=nRT/V

nRTV^(gamma -1) = cst.
TV^(gamma -1 ) = cst.

(P + n(a/v)^2)(V-nb)^gamma = cst.
(P + n(a/v)^2)(V-nb) = nRT

Dividing those two we get

(V-nb)^(Cp/Cv - 1) = cst/nRT

T(V-nb)^(Cp/Cv - Cv/Cv) = cst/nR

But cst/nR is a cst. so

T(V-nb)^(nR/Cv) = cst

There I'm close but I still have that n term in the power.

What do I do eliminate it?

10. Feb 15, 2006

Nusc

Okay i got it. but what is the justification for not useing P:P + an2/V2)

Please tell me before 8 hours from this post

11. Feb 15, 2006

Gokul43201

Staff Emeritus
Guess this is too late...but for what it's worth, Cp and Cv are the molar specific heats (heat capacity per mole of gas).

This gives Cp - Cv = R