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

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 *Z*_{N}, 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
*V*^{N-2}

Result:

Taking log:

Since , we obtained:

(5) |

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) *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
*T*_{b}it is zero*B*(*T*_{b})=0. This is*Boyle point*of gas.

- At high

If we include the terms *f*_{ik}*f*_{lm} in expansion *Z*_{N}, we will
obtain:

(7) |

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

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*.

© 1997

Thu Sep 18 22:50:29 EDT 1997