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Next: Quiz Up: Imperfect Gases and Liquids. Previous: Virial Expansion

Van der Waals Equation

We know that liquids are (almost) incompressible. So A(N,V,T) should diverge at $V\sim Nb$. How can we arrange this?

Start from virial equation:
A=A_{\text{ideal}} + \frac{N^2kTB(T)}{V} =\  -kTN\ln\frac{eV}{N} + Nf(T) +
 \frac{N^2kTb}{V} - \frac{N^2a}{V}\end{multline*}
Rearrange the terms:  
 A = Nf(T)-NkT\ln\frac{e}{N} -NkT\left(\ln V-\frac{N b}{V}\right) -
 \frac{N^2a}{V} \end{displaymath} (8)

We want an equation, that:

Idea: substitute $\ln V - Nb/V$ by $\ln(V-Nb)$.

Result: van der Waals formula  
 A = Nf(T)-NkT\ln\frac{e}{N} -NkT\ln \left(V-N b\right) -
 \frac{N^2a}{V}\end{displaymath} (9)

P= -\left(\frac{\partial A}{\partial V}\right)_{N,T} =
 \frac{NkT}{V-Nb} - \frac{N^2a}{V^2}\end{displaymath}

At $V\to Nb$ it diverges.

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