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Next: Van der Waals Equation Up: Imperfect Gases and Liquids. Previous: Interaction Energy


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 $\exp(-U/kT)\approx1-U/kT$
Since $u(0)=\infty$, this never works!
Let us discuss Mayer function instead:  
 f(r) = \exp\bigl(-u(r)/kT\bigr) - 1
 \end{displaymath} (3)
When $u(r)\approx0$ Mayer function is small. When $u(r)\to\infty$ we have $f(r)\to-1$.
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)\end{displaymath}

Express through Mayer function:

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

 \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 \end{displaymath} (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 ZN, integrate (4) over $d\mathbf{r}_1\,d\mathbf{r}_2\,\dots$

First term:

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

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: $N(N-1)/2\approx N^2/2$
Integration over $\mathbf{r}_1$--factor V
Integration over $\mathbf{r}_2$--factor $\int f(\mathbf{r})\,d\mathbf{r}$
Integration over $d\mathbf{r}_3\,d\mathbf{r}_4\dots$--factor VN-2


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

B(T) is second virial coefficient

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

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

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

Second Virial Coefficient

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

Since f(r) is dimensionless, B has dimension of volume.
Hard core plus attraction, equation (2).
High Temperatures:
If $T\to\infty$, Mayer function is

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


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


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

 \end{displaymath} (6)
with positive a (at $r\gt\sigma$ potential u is attractive!)

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

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):

Higher Order Terms


If we include the terms fikflm in expansion ZN, we will obtain:  
 P= \frac{NkT}{V}\left(1+\frac{NB(T)}{V} + \frac{N^2C(T)}{V^2}+ \dots
 \right) \end{displaymath} (7)
C(T) is the third virial coefficient. Similarly fourth, fifth....

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


Will these series converge? Yes, if

\frac{NB(T)}{V}\ll 1\end{displaymath}

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.

next up previous
Next: Van der Waals Equation Up: Imperfect Gases and Liquids. Previous: Interaction Energy

© 1997 Boris Veytsman and Michael Kotelyanskii
Thu Sep 18 22:50:29 EDT 1997