Subsections

• Mayer Function
• Partition Function and Free Energy
• Second Virial Coefficient
• Higher Order Terms
  • Expansion
  • Convergence

Virial Expansion

Mayer Function

For ideal gas U=0. Let us discuss slightly non-ideal gas, and expand $\int\exp(-U/kT)\,d\tilde r$ when $U\approx0$.

Simple idea:

Expand $\exp(-U/kT)\approx1-U/kT$

Problem:

Since $u(0)=\infty$, this never works!

Solution:

Let us discuss Mayer function instead:  

$$f(r) = \exp\bigl(-u(r)/kT\bigr) - 1$$ (3)

When $u(r)\approx0$ Mayer function is small. When $u(r)\to\infty$ we have $f(r)\to-1$.

Assumption:

For slightly non-ideal gas Mayer function is small on average:

$$\left\langle f(r)\right\rangle\ll1$$

Partition Function and Free Energy

Gibbs factor:

$$\exp(-U/kT) = \prod_{i\gt j} \exp\bigl(-u(r_{ij})/kT\bigr)$$

Express through Mayer function:

$$\exp(-U/kT) = \prod_{i\gt j} \bigl(1+f(r_{ij})\bigr)$$

Expand:  

$$\exp(-U/kT) = 1 + \sum_{i\gt j}f(r_{ij}) + \sum_{i\gt j,k\gt l}f(r_{ij})f(r_{kl})+\dots$$ (4)

When is f(r) is non-zero? For hard spheres--when $r\le\sigma$. This is collision. Low density gas--not many collisions! Take a volume V, where collisions are rare. It means neglecting products of f-functions.

To obtain Z_N, integrate (4) over $d\mathbf{r}_1\,d\mathbf{r}_2\,\dots$

1.

First term:

$$\int d\mathbf{r}_1\,d\mathbf{r}_2\,\dots = V^N$$

2.

Second term: a collection of equal contributions like

$$\int f(\mathbf{r}_{12})\,d\mathbf{r}_1\,d\mathbf{r}_2\,d\mathbf{r}_3\,\ldots\,d\mathbf{r}_N$$

Number of terms: $N(N-1)/2\approx N^2/2$

(a)

Integration over $\mathbf{r}_1$--factor V

(b)

Integration over $\mathbf{r}_2$--factor $\int f(\mathbf{r})\,d\mathbf{r}$

(c)

Integration over $d\mathbf{r}_3\,d\mathbf{r}_4\dots$--factor V^N-2

Result:

$$\begin{gathered} Z_N = V^N - B(T)N^2V^{N-1}+\dots\ B(T) = -\frac12 \int f(\mathbf{r})\,d\mathbf{r} \end{gathered}$$

B(T) is second virial coefficient

Taking log:
$$\begin{multline*} -kT\ln Z_N = -kT\ln\left(V^N - B(T)N^2V^{N-1}+\dots\right)=\ -kT\ln V^N -kT\ln\left(1-B(T)N^2/V+\dots\right)\end{multline*}$$
Since $\ln(1-x)\approx1-x$, we obtained:

$$A=A_{\text{ideal}} + \frac{N^2kTB(T)}{V}$$

Pressure:  

$$\begin{gathered} P= -\left(\frac{\partial A}{\partial V}\r... ...\ \frac{NkT}{V}\left(1+\frac{NB(T)}{V}\right) \end{gathered}$$ (5)

Second Virial Coefficient

If we know u(r), we can calculate B!

Dimension:

Since f(r) is dimensionless, B has dimension of volume.

Model:

Hard core plus attraction, equation (2).

High Temperatures:

If $T\to\infty$, Mayer function is

$$f(r) = \exp\bigl(-u(r)/kT\bigr) -1 = \begin{cases} -1, & r<\sigma\ 0, & r\gt\sigma \end{cases}$$

and

$$B(T)=-\frac12\int f(\mathbf{r})\,d\mathbf{r}= \frac12\int_0^{\sigma} 4\pi r^2\, dr= \frac{2\pi}{3}\sigma^3\equiv b$$

This is four times the volume of one molecule

At high temperatures B(T)=b describes the hard core volume of gas.

Lower Temperatures:

Integrate from to $\sigma$ and from $\sigma$ to $\infty$. First integral gives b, second is

$$\frac12\int_{\sigma}^{\infty}\Big[1-\exp\big(-u(r)/kT\big)\Big]4\pi r^2\,dr$$

Expansion:

$$1-\exp\big(-u(r)/kT\big)\approx \frac{u(r)}{kT}$$

and  

$$B=b-\frac{a}{kT}$$ (6)

with positive a (at $r\gt\sigma$ potential u is attractive!)

$$a = -2\pi\int_{\sigma}^{\infty}u(r) r^2\,dr\gt$$

Assumption:

We assume (6) is valid for all temperatures.

Meaning of the coefficients:

b describes steric repulsion, a describes long range attraction.

Behavior of B(T):

• At high T we have B(T)>0 (repulsion)
• At lower T we have B(T)<0 (attraction)
• At some temperature T_b it is zero B(T_b)=0. This is Boyle point of gas.

Higher Order Terms

Expansion

If we include the terms f_ikf_lm in expansion Z_N, we will obtain:  

$$P= \frac{NkT}{V}\left(1+\frac{NB(T)}{V} + \frac{N^2C(T)}{V^2}+ \dots \right)$$ (7)

C(T) is the third virial coefficient. Similarly fourth, fifth....

We can compute any virial coefficient if we know u(r)!

Convergence

Will these series converge? Yes, if

$$\frac{NB(T)}{V}\ll 1$$

It means that distances between molecules $\gg$ their sizes!

This is true for gases, but not for liquids!

Virial expansion does not work for liquids.