(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have come across a rather interesting conundrum. Given the configurational potential energy partition function for a non-ideal gas:

[tex]

Z = Z_{internal}\frac{1}{N!}\left(\frac{2\pi m}{h^{2}\beta}\right)^{\frac{3N}{2}}(V^{N} - B_{2}(T)N^{2}V^{N-1})

[/tex]

where [itex] B_{2}(T) [/itex] is the second virial coefficient.

I'm supposed to solve for the pressure.

2. Relevant equations

[tex] p = \frac{1}{\beta Z}\frac{\partial Z}{\partial V} [/tex]

or

[tex] p = \frac{1}{\beta}\frac{\partial ln(Z)}{\partial V} [/tex]

3. The attempt at a solution

If we calculate it using the first expression for p we arrive at

[tex]

p = \frac{1}{\beta}\left(\frac{N}{V} - B_{2}(T)N^{2}(N-1)V^{-2} \right)

[/tex]

However if you do it using the second form for p you can arrive at two different answers depending on how you 'prepare' the 3rd term in Z

Ignoring terms that don't involve V, if you start with it as written and evaluate

[tex]

\frac{\partial}{\partial V} ln(V^{N} - B_{2}(T)N^{2}V^{N-1})

[/tex]

[tex]

= \frac{NV^{N-1} - B_{2}(T)N^{2}(N-1)V^{N-2}}{(V^{N} - B_{2}(T)N^{2}V^{N-1})}

[/tex]

in the denominator (for each case) we assume [itex] B_{2}(T) [/itex] is small, so we can write the denominator just as [itex] V^{N} [/itex]

So we arrive at

[tex]

\frac{N}{V} - B_{2}(T)N^{2}(N-1)V^{-2}

[/tex]

OK, great that's what it should be, but if you start out with:

[tex]

\frac{\partial}{\partial V}ln(V^{N}(1-B_{2}(T)N^{2}V^{-1}))

[/tex]

[tex]

= \frac{\partial}{\partial V}\left(ln(V^{N}) + ln(1-B_{2}(T)N^{2}V^{-1})\right)

[/tex]

[tex]

= \frac{N}{V}+ \frac{B_{2}(T)N^{2}V^{-2}}{(1-B_{2}(T)N^{2}V^{-1})}

[/tex]

Again, [itex] B_{2}(T)[/itex] is small, so:

[tex]

= \frac{N}{V} + B_{2}(T)N^{2}V^{-2}

[/tex]

Now we have a different answer...

Why is this happening??

In my prof's notes he uses the second derivation, where we get a different answer from every other calculation.

I suspect

[tex]

\frac{N}{V} - B_{2}(T)N^{2}(N-1)V^{-2}

[/tex]

is the correct form but I can't see what's wrong with the other way.

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# Homework Help: [Stat Mech] Different answers when using log space

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