Deriving the Adiabatic Expression for a Van der Waals Gas

[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.

 

[Astronuc]

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

 

[Nusc]

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

 

[Astronuc]

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

 

[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?

 

[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?

 

[Gokul43201]

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...

 

[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?

 

[Nusc]

Fine, let me try again.

Non-ideal Gas

--------------------------------------------------------------------------------

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?

 

[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

 

[Gokul43201]

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

What do I do eliminate it?

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