Proving a triple point (Thermodynamics)

[ebunny91]

Homework Statement

Prove that a on-component system must have a triple point. You may assume that ΔHfusion>0, if needed.

Homework Equations

C (components) = #of distinct substances - # of distinct chemical reactions

Gibbs Phase Rule: degrees of freedom= components - phases + 2 or (F=C-P+2)

The Attempt at a Solution

In a one-component system, c=1.

I know that when only one phase is present, F= 1 component - 1 phase + 2 = 2. So 2 variables (temperature or pressure) can be varied without changing the phase of the substance. This is the sold, liquid or gas regions in the phase diagram.

When two phases are present, F= 1 component - 2 phases + 2 = 1. So both variables (temperature and pressure) must be varied together along a certain curve to not change the phase of the substance. This is the melting point curve, sublimation curve or boiling point curve in phase diagrams.

Now, when three phases are present, F= 1 component - 3 phases + 2 = 0. So there are no degrees of freedom present, which means that the 3 phases can only be present at once at only one temperature and pressure.

The question is asking me to prove that one component systems must have a triple point and assume that ΔHfusion>0 (if needed). How do I go about doing that with the given equations, or any other ones that I may have missed?

 

[Borek]

Can't you just use Gibbs' phase rule? Or is the idea to show Gibbs' phase rule is true?

 

[ebunny91]

Yea I think I have to prove the Gibbs Phase Rule (somehow using something about ΔHfusion)