Re: Is there an equation directly relating altitude to boiling point?

Date: Fri Jan 1 15:59:31 1999
Posted By: Matthew Buynoski, Senior Member Technical Staff,Advanced Micro Devices
Area of science: Earth Sciences
ID: 914618212.Es

Message:

Some of the answer you seek probably has to do with how accurate you want to be. I can not find a highly accurate empirical equation that will account for the actual pressure profile of the atmosphere.

I can, however, get you about half way there...

If we model the earth's atmosphere as an ideal gas, then the pressure at any height Y (expressed in kilometers) is

        P = P_o * e^(-AY)   
 
        where P_o is sea level atmospheric pressure
        A is 0.116 per km

                        (see Resnick and Halliday, "Physics" Part 1, pg. 360).

Now there *must* somewhere be an equation, probably a polynomial curve fit to experimental data, for the vapor pressure of water vs. temperature. There is extremely accurate data on this (see the Handbook of Chemistry and Physics). My ancient edition has a very complete table of this, and I'm reasonably sure that more recent editions do as well). But I can't find an equation fitted to this kind of data, even after looking into a book on atmospheric modelling.

If one could find such an equation, then it will probably have the form of

	VP(T) =  C + DT + ET² + FT³     

        where C, D, E, F are constants
        T is the temperature
        VP(T) is the vapor pressure

Now, boiling occurs when the vapor pressure equals atmospheric, so that the "final answer" is

    C + DT + ET² + FT³ = P_o * e^(-AY)

Now all we need are C, D, E, and F :-)

To complete your quest for a single equation, I can think of two ways to continue:

a. find a real expert in atmospheric modelling (try web pages for professors at departments of meteorology at some universities).
b. Fit the tabular data in the "Handbook of Chemistry and Physics" for the vapor pressure of water vs. temperature to a polynomial equation yourself. This can be done by linear least squares fitting of the unknown coefficients (C,D,E,F) to the data. Your physics or chemistry teachers should be able to help you do this. It basically means solving a set of algebraic equations, somewhat messy but nothing that should be beyond your skills once you are introduced to the least squares method. One advan- tage of this method is that you can concentrate on the range of altitudes that interest you. Compute the atmospheric pressures at those altitudes and then investigate the temperature range at which the vapor pressure of water will equal those pressures. You might also try graphing the vapor pressure vs. temperature data in that range. With any luck, it will be "close enough" to a straight line that you can get by with a linear fit. You can then read C and D off the graph as intercept and slope, and go from there.

Good luck! Again, apologies for not being able to come up with a final form equation.

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