Proving Internal Pressure of Ideal & VdW Gases

Jennifer Lyn
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Like in the other problem I posted- This is the other question that I missed and just can't find a solution for.

Homework Statement


Prove the internal pressure is 0 for an ideal gas and ((n^2)a)/(v^2) for a Van der Waals gas.

Homework Equations


1. VdQ Eqn: p= (nRT)/(v-b) - ((n^2)a)/(v^2)
2. (partial S/partial V) for constant T = (partial p/partial T) for contant V.
3. dU = TdS - pdV
4. pi sub t (internal pressure) = (partial U/partial V) for constant T


The Attempt at a Solution



a) Ideal Gas
0 = (partial U/partial V) for const T
int 0 dv = int du
0 = int (TdS - pdV)
int p dv = int T ds
int (nRT/v) dv = int (Pv/nR) dS
nRT x int(1/V) dv = pv/nR x int 1 dS
... and I get kind of lost here, though I know that what I've already done is wrong.. :(

b) VdW gas
I actual have to get going to school, but I'll come back and type up what I've done (incorrectly :( for this part afterwards).
 
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Hello Jennifer,

I suppose you are given the so called thermal equation of state p=p(T,V,n) for both

the ideal gas

p=\frac{nRT}{V}

and the Van der Waals gas

p=\frac{nRT}{V-nb}-\frac{n^2a}{V^2}

The inner pressure \left(\frac{\partial U}{\partial V}\right)_T can be calculated after finding the so called caloric equation of state U=U(T,V,n) for both cases.

Another straightforward method would be to use the following identity which shows that the caloric and thermal equations of state are not independent of each other:

\left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial p}{\partial T}\right)_V-p

Do you know how to derive this identity?
 
I think so..
\Pit = \partialu/\partialv for constant t
= ( 1/\partialv\times(Tds - pdv) )
= T \times (\partialp/\partialt) - p

I think that's right. I still don't know how to get from that Maxwell relation to the ideal gas and Van der Waals eqn, though.
 
Last edited:
Ok, I think I figured it out, from my previous post (sorry- I am still getting used to using the tools for math on this board) I replace the vanderwaals eqn into P in my partial p and then just solve from there.
 
Thanks everyone!
 
To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
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