How Do You Calculate the Critical Point in Thermodynamics?

[moonman]

How would you go about calculating the volume, pressure or temperature for the critical point in a phase plane? I know that there's a Clapeyron equation for finding the equation of the coexistance curve dP= L/TV dT, but can this be used to find the critical point? and if not, what will?
I've already worked out the fundamental relation for the gas. what do I do next?

 

[Dr.Brain]

The variation of P with T on the transition line in phase diagram is :

log (P/Po) = dH/R (1/To - 1/T)

You should know the three of these terms to calculate the fourth one. Just to give you an idea , critical point is the greatest volume at which liquid-gas equilibria exists at a particular pressure.

BJ

 

[Physics Monkey]

The critical point is determined by the conditions
$$\frac{\partial P}{\partial V} = 0$$
$$\frac{\partial^2 P}{\partial V^2} = 0$$
plus the equation of state. The first condition tells you where your system is about to become thermodynamically unstable. For example, in the van der Waals equation of state, the first derivative of pressure with respect to volume naively becomes negative below a certain temperature, but this is impossible in a stable system. The second condition also has to do with stability. Quite generically, when the first derivative of pressure vanishes at some point, the second must also for the system to be stable. Together with the equation of state, you have three equations for three unknowns.