# Difference between Clausius-Clapeyron and Van't Hoff equation

I was wondering what is the difference between the Clausius-Clapeyron equation and the Van't Hoff equation. They appear to have the exact same physical meaning and are often used interchangeably.

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They're two forms of the same equation, but Clausius-Clapeyron uses vapor pressure ($p^*$) where Van 't Hoff uses the reaction equilibrium constant ($K$).

Why does this work out? Well, think of vaporization as a chemical reaction:

$$\text{X} (l) \longrightarrow \text{X} (\text{g})$$

The equilibrium constant is defined in terms of activity (a):

$$K=\left.\frac{a_\text{X(g)}}{a_{\text{X}(l)}}\right|_\text{equil}$$

For an ideal liquid solution at modest pressure, a is just the mole fraction x. And for an ideal gas, a is just its partial pressure in bar:

$$K=\left.\frac px\,\right|_\text{equil}$$

One equilibrium condition is $x=1\$ and $p=p^*,\$ so...

$$K=p^*$$

Or $p^*$ is the equilibrium coefficient for vaporization. Now since K is characteristic of a reaction at a given temperature, this would imply that if we change x, then the new partial pressure at equilibrium would change according to

$$K=p^*=\frac{p}{x} \qquad\Rightarrow\qquad p=xp^*$$

And that's exactly what happens. Neat, huh?

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