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Name: Nelson
Status: student
Grade: 9-12
Country: Brazil
Date:  Fall 2011  

Question:
What is the equation relating the boiling point of water with altitude on Earth? (without the variable "pressure"). At sea level, the boiling point of water (100% H2O) is 100 degrees Celsius. This everybody knows... :D I am saying in generic terms, say, altitude h, in meters (sea level: h = 0). Whenever possible, please justify where the "magic numbers" (physical constants) are. This way these constants will mean something.

Replies:
Nelson,

You will need to find an equation that relates altitude to atmospheric pressure. And then relate the atmospheric pressure to boiling point.

For the second task, there is an equation called the Clausius-Clapeyron equation which relates the boiling point of any liquid based on two variables, the external pressure and the enthalpy of vaporization. Since enthalpy is a function of the substance, it is essentially something that can be analyzed or looked up in a table. The external pressure becomes the only variable factor.

Greg (Roberto Gregorius) Canisius College


Nelson,

It is a good question. I have just worked something out by hand, but since I have not seen it elsewhere I wonder if I have done it right or not. Still here it is, working from two famous formulas (in my case I pulled them from Atkins and DePaula's Physical Chemistry text).

By combining the barometric formula with the Clausius -Clapeyron equation and assuming that the atmospheric pressure at sea level is 1 atm, I get the boiling point T_B at height h as being given approximately by the following reciprocal formula:

(1/T_B) = (1/T_B^*) + (ah/T_a)

where T_a is the ambient temperature, i.e, the temperature of the room (this determines the atmospheric pressure); T_B^* is the boiling point at 1 atm pressure; and a = (Mg/enth), where M is the average molar mass of air (about 29 g/mole), g is the gravitational contant, and "enth" is the enthalpy of vaporization of water (about 44 kJ/mole I believe). All units must be converted to SI in order for this formula to work at all.

Note that if h=0, the boiling point equals T_B^*, and also that if h > 0, the boiling point is less than T_B^*, as expected. Also, if the ambient temperature is cold that will lower the boiling point as well (because it decreases the ambient pressure).

This formula is highly approximate. The C-C equation is sometimes only good to two significant figures, etc etc. However, it does seem to capture the essentials.

Thanks for the great question! best, Dr. Topper


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