11 Jul 2010 22:12
While my part of the world was embroiled in a heat wave last week, brutal midwinter cold was gripping Vostok Station high atop the mountains and ice of central Antarctica. At times, it was more than two hundred degrees colder in Vostok than it was here in the U.S. Northeast.
A few days earlier, by coincidence, I’d installed the last of my three SodaStream carbonators and ordered two refills. On the 7th, in the middle of the heat wave, the local distributor exchanged my empties, leaving me with four pounds of freshly compressed carbon dioxide and two questions.
- Is four pounds of CO2 a lot compared to the amount already inside my condo, not counting the basement?
- When the temperature drops below -109.3 °F, the sublimation point of carbon dioxide, does it snow dry ice?
I’m happy to share with you the answers.
1. Yes, quite a lot. The volume of my condo, not counting the basement, is about 7,000 cubic feet, or about 200,000 liters, and while I have far too much junk, my place is still mostly full of air. And like air most places, the air in my condo is mostly nitrogen, oxygen, and argon, and about 400 parts per million of it (by volume) is carbon dioxide. My condo contains a fifth of a million liters of air, I figured, so I’ve got a fifth of 400 liters of CO2, or about 80 liters.
At room temperature, CO2 is effectively an ideal gas, so I don’t need Google to convert 80 liters to moles, which is going to help with the calculation. A mole of an ideal gas at standard temperature and pressure¹ occupies 22.4 liters, so I’m living with about 4 moles of CO2. (And maybe a few voles, too, but that’s not relevant here.) A mole of any chemical compound means 600 sextillion molecules of the stuff. Fortunately, we don’t have to do any sextillionary arithmetic. What makes a mole a mole is that it’s the number of Daltons (neutron or proton+electron masses, more or less) in a gram, so the mass of a mole of a chemical compound is simply its molecular weight in grams. The molecular weight of CO2 is² 12+2×16, or 44, so a mole of CO2 has a mass of about 44 grams or an ounce and a half.
Four moles of CO2 in my condo — that’s about six ounces, or less than a tenth of what’s squeezed into my two new Sodastream carbonators.
2. No. The temperature -109.3 °F is the sublimation point of CO2 at one atmosphere. That’s the temperature at which dry ice in your house or cooler chest (which is pure CO2 solid under an atmosphere of air pressure) sublimates; it’s not the temperature at which CO2 in plain air would precipitate into “snow.” Carbon dioxide won’t precipitate out of the air unless it’s cooled to the sublimation point for its partial pressure in air. That’s (see above) about 400 millionths of an atmosphere at sea level, and roughly two-thirds of that at Vostok Station, elevation 11,000 feet. Smart guy David R. Cook at Argonne National Laboratory has generously used some tax dollars to spread the word. He proposes that the temperature would have to be much colder than –110 — in fact about 220 degrees below zero on the Fahrenheit scale (and no, that’s not “twice as cold” as -110) for CO2 to precipitate at Vostok Station.
Typical partial pressures of CO2 on earth are unfortunately off the scale of the phase diagrams I could find, so I can’t provide more details or confirm what David says. You can, however, see the trend in the phase diagram below. The boundary between the periwinkle and salmon regions is where CO2 sublimates or “snows.” The lower the pressure, the lower the temperature.
Public domain image from Wikimedia Commons.
The phase diagram also answers a question I didn’t think to ask.
- Are full SodaStream carbonators filled with liquid or gas? Liquid.
They’re about a liter big, which (using the calculations above) translates to an internal pressure of several hundred atmospheres, which is well within the green region. How cool is that? Liquid CO2 right here in my house.
Call it chemistry or call it physics, this is all great fun.
¹ Standard temperature and pressure is 20 °C and one (sea level) atmosphere of pressure. That’s close enough to the conditions in my condo. Around room temperature, gas expands at a rate of about 1% per 4 °C rise in temperature. It also expands as you climb above sea level, by about 1% for each 300 feet. The second derivatives are small, and the contributions from the partial derivative of volume with respect to temperature and altitude add, so yadda yadda yadda, we’re within a few percent. It was worth a quick thinking through, though.
² There are a some simplifications here, but again, they don’t affect the overall calculation by more than a percent or two. The value 44 assumes that every molecule of CO2 is made up of the predominant isotopes of carbon and oxygen, carbon 12 and oxygen 16, and that single atoms of these isotopes weigh exactly 12 and 16 Daltons, respectively. The “official” atomic weights as periodically reviewed by IUPAC would be more accurate, and they give a molecular weight of 44.01. The discrepancy reflects the distribution of isotopes on earth as well as other details that probably involve higher physics and chemistry well beyond my understanding. In addition, it would seem to me that this measurement ought to be called the molecular mass, but when measuring any property of such tiny things, you’re going to run up against quantum mechanics and other complications, so I suppose whoever understands all of this can call it what they want.