Computing the second virial coefficient

[JD_PM]

Homework Statement

$b_2$ is the second virial coefficient

Homework Equations

Virial expansion:

$$P = nkT(1 + b_2 (T)n + b_3 (T)n^2...)$$

$$b_2 = -\frac{1}{2} \int dr f(r) $$

r is the distance vector.

$$f(r) = e^{-\beta \phi(r)} - 1$$

The Attempt at a Solution

$$b_2(T) = 2 \pi r^2 \int_0^{\sigma} (e^{-\beta \phi(r)} - 1 )dr = 2\pi \int_0^{\sigma} r^2 dr - 2\pi \int_0^{\sigma} r^2 e^{-\beta \phi(r)} dr$$

I solved these integrals but did not get the stated comparison with VdW model. Am I going OK? I can post more algebraic steps.[/B]

 

[Charles Link]

In step 3, the limuts of your integral need to be from $\sigma$ to $+\infty$, with $\phi(r)=-\frac{\epsilon}{r^6}$. For the integral from $0$ to $\sigma$, the $- 1$ term will give $-\sigma$. $\\$ It's not completely clear what you are doing though. The $b_2$ as you have it written must be integrated over $dv=4 \pi r^2 \, dr$ if you are getting a $4 \pi r^2$ from it. $\\$ Edit: Yes, I googled it, and it is $dv$. $\\$ See equations 47-53 of this "link" that I googled: http://www.nyu.edu/classes/tuckerman/pchem/lectures/lecture_4.pdf $\\$ This one is too difficult to expect a student to derive on their own, so I am providing you with the solution here.

 

[Charles Link]

See the additions in post 2 above.

 

[JD_PM]

Thank you, this document is really helpful! I got the expressions for a and b!