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### Boiling Point of Water

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Date: 01/29/97 at 09:39:15
Subject: boiling point of water under pressure

Is there an equation that predicts the boiling point of water as the
pressure is increased? The research I am doing is in reference to the
boiling point of a 50-50 mix of antifreeze and water that is under the
pressure of a cooling system of a small gasoline powered engine. I can
not find much information on >1 atm. There are copious amounts of

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Date: 01/29/97 at 16:06:46
From: Doctor Mitteldorf
Subject: Re: boiling point of water under pressure

Dear Bruce,

Chemistry is messy, and there are few exact formulas. But this is
an area where there's an approximate formula from thermodynamics that
should help. It works whether the pressure is more or less than 1 atm.

The thing I always like to remember is the general rule that the
probability of a molecule being in a state of energy E is exp(-E/kT)
times a "phase space factor". If you think quantum mechanically, then
the space factor tells how many different ways there are for the
molecule to be in that state. If you think classically, the phase
space factor can be thought of as how much room you have in physical
space (x,y,z) times how much room in velocity space (vx,vy,vz).

Now, amazingly, this core bit of general knowledge is enough to get
us an approximate formula. If the molecules require some energy E to
be torn from their cozy home in the liquid and set on their own in the
air, then the probability of them being out in the air will be
proportional to exp(-E/kT). To the extent that the water vapor
behaves like an ideal gas, the pressure is proportional to T times
this probability.  So:

p ~ T exp(-E/kT)

(This formula can't be inverted explicitly to give T as a function of
p. You have to solve it numerically.)

This is a useful formula because E should depend on temperature only
weakly.  You can treat it approximately as a constant.  In theory,
E is the heat of vaporization; in practice, you should treat it as an
experimental parameter. There's one more free parameter in the system
which is the proportionality constant in the expression above. So you
should use experimental data to adjust these two parameters for the
best fit to the pressure/temperature curve.

(All this assumes that the water evaporates, but that the antifreeze
component makes a negligible contribution to the pressure.)

-Doctor Mitteldorf,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math
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Associated Topics:
High School Physics/Chemistry

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