# Equation to calculate the triple point

Is there any equation that can calculate the triple point of a substance? Because a method for calculating the critical temperature and the Boyle temperature of a substance that I posted in another question I recently found that it also seems to be able to predict the triple point for some substances.

I searched the Internet and couldn't find any other such equation, in fact I came across a site that said there was none. Is this true? My equation predicts it from the Lennard-Jones parameters. Source:

Here is an excerpt from a document that I'm putting together. It also predicts the reduced triple point temperature to be 0.5

• Well, there is no general relation between these temperatures. – Mithoron Jun 6 '16 at 18:56
• Oh, you copied those L-J constants... Well as you can see this constant isn't equal to triple point temp. – Mithoron Jun 6 '16 at 21:00
• @Mithoron What constant? – KingChem Jun 7 '16 at 2:23
• I agree. We can't predict the triple point from just liquid and vapor characteristics alone. The properties of the solid phase must enter in. In the case of water, you have many phases each of which would potentially make its own triple point with liquid and vapor, unless you know which solid phase to pick. – Oscar Lanzi Jun 7 '16 at 9:57

In principle, you can use the integrated Clapeyron and Clausius–Clapeyron equations to calculate $P$ vs. $T$ co-existence lines and find the crossing point which is at the triple point. You need the $\Delta H$ for fusion and evaporation and the densities of the solid and liquid phases. The equations are $$p_\text{liq-vap}=\exp\left(\frac{\Delta H_\text{vap}}R\left(1/T_1-1/T\right)\right)$$ and for the solid vapour line use $\Delta H_\text{fus} +\Delta H_\text{vap}$

For the solid-liquid line use $$p_\text{sol-liq} = P_1 + \frac{\Delta H_{\text{fus}}}{\delta V}\left(\ln\left(T\right)-\ln\left(T_1\right)\right)$$

where $\delta V = M\left(1/d_\mathrm l-1/d_\mathrm s\right)$ where $M$ is the molecular weight and $d$ the densities of liquid and solid. $T_1$ and $P_1$ are temperature and pressure at the triple point. Thus you have to iterate $T_1$ and $P_1$ by fitting (say by least squares) to the calculated curves to whatever data you have to find the triple point. Tricky!

Naively, I am surprised that using Lennard-Jones parameters would work as they describe interaction potential between two atoms and not in a solid or liquid.

• But does my method work?I mean it appears to to me at least but I dunno what a trained chemist would say. If it does it appears to be simpler than the method you posted, just a simple proportionality. – KingChem Jul 8 '16 at 19:44
• From the list you show, the comparison between calculation and experiment is sometimes good, but sometimes it is poor. You do need to understand the reason for this before you can proceed further. – porphyrin Jul 9 '16 at 8:49
• I have no idea. I would like a qualified individual to help me develop the theory further but so far nobody has been interested. – KingChem Jul 11 '16 at 18:16
• Note that the integrated equations you gave are oversimplifications - for example, you have assumed that the solid and liquid have negligible volume relative to the vapor. They're pretty good approximations, but will not give exact values for triple point even if solved as described. – Andrew Nov 9 '19 at 13:00
• @Andrew You are free to answer with the exact equations if you wish to :) – porphyrin Nov 10 '19 at 8:47