# Van Der Waals Phase Transition

Diracobama2181
Homework Statement:
Show that $$\rho_{gas}-\rho_{liquid}\propto |T_C-T|^\frac{1}{2}$$.
Relevant Equations:
$$P=\frac{\rho RT}{1-\rho \beta}-\alpha \rho^2$$
Not sure where to actually start. Do I need to do a virial expansion? Any tips on on where to start would be greatly appreciated.

Mentor
Start by writing down alpha and beta in terms of TC and PC.

Abhishek11235
Rearrange the given equation for T. Now,at critical point density is infinite. So, use limiting process for density.

Diracobama2181
Rearrange the given equation for T. Now,at critical point density is infinite. So, use limiting process for density.
Would T and P be different for $$\rho_{gas}$$ and $$\rho_{liquid}$$?
Right now, after rearranging, I get
$$T=\frac{P-\rho \beta P+\alpha \rho^2-\alpha \beta \rho^3}{R\rho}$$
which gives
$$T=\frac{- \beta P+\alpha \rho-\alpha \beta \rho^2}{R}$$
when I let $$\rho$$ go to $$\infty$$

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Abhishek11235
which gives
$$T=\frac{- \beta P+\alpha \rho-\alpha \beta \rho^2}{R}$$
when I let $$\rho$$ go to $$\infty$$

As it should(There is very exciting physical phenomenon related to this). Now you want to find relation between phase change and density. For this,you have to approach one temperature(The critical temperature)(Why?). Next,the pressure should be same(This should become clear if you P-T graph of phase change relationship)

Fred Wright
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• Raihan amin
WidomFisherLine
Homework Statement:: Show that $$\rho_{gas}-\rho_{liquid}\propto |T_C-T|^\frac{1}{2}$$.
Homework Equations:: $$P=\frac{\rho RT}{1-\rho \beta}-\alpha \rho^2$$

Not sure where to actually start. Do I need to do a virial expansion? Any tips on on where to start would be greatly appreciated.
Use the Widom Insertion Method