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What's the degree of freedom of a critical point of a single component system?

In the critical point the difference of density between the liquid phase and the vapour phase is so small the are indistinguishable, hence considered as single phase. So according the gibbs phase rule (. F=C - P +2) the degree of freedom should be 2.

But critical point is a point. And the degree of freedom of a point in the phase diagram should be zero( for example, the triple point).

So what's the correct degree of freedom of the critical point?

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There are 3 phases at the critical point. Sub-critical vapour, Sub-critical liquid and critical fluid. 3 phases -> DoF = 0

That is one way to see it - and for convenience that is how I see it. It might be a bit wobbly to regard that as rigorous analyse though.

You can also say, more rigorously, that there is 1 phase and 2 constraints: $$\left(\frac{\partial\rho}{\partial V}\right)_T = 0 $$ $$\left(\frac{\partial^2\rho}{\partial V^2}\right)_T = 0 $$ Subtract the number of constraints from the DoF and you get 0 again.

This defines a single point on the P/T curve.

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