# Homework Help: Help with Compressibility of liquid and relation to pressure

1. May 6, 2012

### bearit88

1. The problem statement, all variables and given/known data
Assume that the distance across a microscopic cell is larger than the correlation length
of the liquid, so whatever is happening in one cell is statistically uncorrelated with what
is happening in an adjacent cell. Further, assume that each cell has two distinct possible
behaviors: The number of molecules in the ith cell is either ni = 0 or ni = ρ(0) Δv. The
probability of the former (an empty cell) is 1 - x; the probability of the latter (a full
cell) is x. The total number of molecules in the system is N =Ʃni, where the sum
is over all M cells.

The statistical weight for a particular microstate can be expressed as a product over
M factors, where the ith factor depends upon ni and the model parameters x, ρ(0) and
Δv. With that statistical weight, you can compute the average of N, the average of
(N-<N>)^2, and so forth. The compressibility of the liquid can be thus determined. The
model applies at conditions of high density, i.e., from a lowest density <N>/V = ρ(0)/2,
where the pressure is p = p(0), to the highest density <N>/V approaches ρ (0), where p approaches infinity.

3. The attempt at a solution
We have found <N> to be Mxρ(0)Δv
and the compressibility (dρ/dβp) as ρ(0)Δv(1-X).

This is just a set up to get us thinking about our upcoming final. We must infer the questions.
How can we find pressure or entropy changes? I assume that is what will be done with the density part?