# Vapor Pressure

I was thinking about the cause of vapor pressure and thought of how it might work. But I’m not sure if these ideas are on the right track or not.

For example water at 373K

The fraction of molecules with energy above the heat of vaporization (6.8E-20 J/molecule or about 0.4eV) is about 8E-6.
(From numerical integration of the energy distribution function up to very many kTs – it surprised me that, at the boiling point, so few molecules were more energetic than the heat of vaporization).

Molecular surface density is ~1E19 / sq meter if the molecular spacing is about 3E-10m.

So number of surface molecules with enough energy is ~9E13 per sq meter.
Maybe a little more if some molecules from a little under the surface can find a clear path out of the liquid.
Or maybe a little less since some molecules with enough energy will be traveling sideways and just dissipating the energy among the others.

The molecules are moving up and down and back and forth for distances constrained by the presence of the others. Maybe each trip is something like 5E-11m.

Then the number of round trips per second (frequency that a molecule has a chance to get out) is the speed divided by twice that distance. Using the average speed of 661m/s there are 6E12 per second.
Maybe it should be a little more if on the average the faster ones get more frequent motion, except each molecule becomes fast and slow very often. I have to think more about that.
Also this seems like a very large frequency of motion. But did you see that high speed pulsed lasers measured the rate of switching the hydrogen bonding partner for a water molecule is 160 billion times per second? http://www.physorg.com/news193636695.html

So the exit rate is the number of fast molecules times the frequency of motion. The exit rate is 6E26 molecules per sec per sq meter.

Supposing there is only water vapor above the surface, at the same temperature, the pressure required so that the arrival rate balances the exit rate can be found from the ideal gas law and from kinetic theory where arrival rate is the density times the average speed divided by 4.
My calculated pressure is about 18000 Pa.
Maybe a little more if some of the molecules that arrive don’t stick.
The real vapor pressure is about 5 or 6 times higher.

I repeated it with 300K and the relationship between my (incorrect) vapor pressures and temperature followed the pattern of the Clausius-Clapeyron relation.

Do you think this reasoning is on the right track? Do you see anything major left out?

Also, I’m assuming that although there are all kinds of molecular distortion going on, that there is a translational speed (of the center of mass perhaps?) that for an equilibrium liquid (or solid) is distributed according to the Maxwell distribution and is related to the temperature just like in gasses. Does this seem right?