Subsections

# Virial Expansion

## Mayer Function

For ideal gas U=0. Let us discuss slightly non-ideal gas, and expand when .

Simple idea:
Expand Problem:
Since , this never works!
Solution:
Let us discuss Mayer function instead: (3)
When Mayer function is small. When we have .
Assumption:
For slightly non-ideal gas Mayer function is small on average: ## Partition Function and Free Energy

Gibbs factor: Express through Mayer function: Expand: (4)

When is f(r) is non-zero? For hard spheres--when . 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 1.
First term: 2.
Second term: a collection of equal contributions like Number of terms: (a)
Integration over --factor V
(b)
Integration over --factor (c)
Integration over --factor VN-2

Result: B(T) is second virial coefficient

Taking log: Since , we obtained: Pressure: (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 , Mayer function is and 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 and from to . First integral gives b, second is Expansion: and (6)
with positive a (at potential u is attractive!) 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 Tb it is zero B(Tb)=0. This is Boyle point of gas.

## Higher Order Terms

### Expansion

If we include the terms fikflm in expansion ZN, we will obtain: (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 It means that distances between molecules their sizes!

This is true for gases, but not for liquids!

Virial expansion does not work for liquids.   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