Gas Laws
Practice
practice problem 1
Jet aircraft of the type from which this data was collected typically fly at altitudes greater than 10,000 m; well above the vertical limit of human survivability. Pressure and temperature outside the cabin on this flight are about 26 kPa (one-quarter atmosphere) and −60 °C, respectively. That's low enough that most humans would suffocate in under thirty seconds and freeze solid in a few hours.
Although the environmental conditions outside are harsh, life inside a jet isn't all that bad. While passengers would find it most comfortable if the cabin was kept at one atmosphere there are engineering and economic reasons to maintain the pressure at a slightly lower value. During this flight the cabin pressure was kept at a compromise value of 81 kPa — much higher than the pressure outside, but still lower than what most people live under. (80 kPa is the average atmospheric pressure at an elevation of 2000 m.) Temperature hardly varied at all, staying nearly constant at 23 °C. (People are far more sensitive to temperature changes than to pressure changes.) When air is drawn into the cabin from outside, to what temperature does it rise after it has been compressed?
solution
Despite the lengthy introduction, this is a comparatively simple question. Temperature and pressure are directly proportional (if we assume that volume remains constant).
| P1 = | 26 kPa |
| T1 = | −60 °C = 213 K |
| P2 = | 81 kPa |
| T2 = | ? |
| P1 | = | P2 | |
| T1 | T2 | ||
| 26 kPa | = | 81 kPa | |
| 213 K | T2 | ||
| T2 = 663 K = 390 °C | |||
This temperature is about the same as one would find inside a pizza oven. Since the volume of the air does not remain constant but is somewhat reduced during pressurization, the actual temperature of the air drawn into the cabin is even higher than we've calculated. To prevent roasting the passengers to golden brown crispness this air must be refrigerated — a comparatively expensive procedure given the size and weight limitations imposed by flight. Thus, a significant portion of the air breathed in a typical commercial airliner is recirculated. That is, the air exhaled by the passengers is stirred up by the plane's ventilation system with a small amount of fresh, refrigerated air is continuously added to the mix. This recirculation is what makes airplane air so particularly nasty.
practice problem 2
We are indebted to high-altitude aircraft flight and the space program for the recent spate of interest in flatulence. After World War II, it appeared that intestinal gas might prove a serious problem for test pilots. The volume of a given amount of gas increases as the pressure surrounding it decreases. This means that a pilot's intestinal gas will expand as he flies higher into the atmosphere in an unpressurized cockpit. At 35,000 feet, for example, the volume will be 5.4 times what it would be at sea level. The resulting distention could cause substantial pain…. So the word went out across the land: study flatulence.
Harold McGee, 1984 (paid link)
Verify Mr. McGee's claim.
solution
For those of you who are still a bit unclear, legumes are the third largest family of flowering plants (which includes beans, peas, and peanuts) and flatulence is the medical term for intestinal wind (which is the polite term for farts — yes, the gas laws apply even to "gas"). The typical atmospheric pressure at sea level is 101 kPa. According to the standard atmospheric tables, at 35,000 feet (11,000 m) typical atmospheric pressure is more like 22.7 kPa. If we assume that a person's body temperature doesn't change much while flying, its volume would be inversely proportional to the external pressure acting on it.
| P1V1 = P2V2 | |
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At 35,000 feet the volume would be 4.4 times what it would be at sea level, not 5.4 times as Mr. McGee claims.
practice problem 3
- the volume of one mole of an ideal gas at standard temperature and pressure
- the dimensions of a cube that could hold one mole of an ideal gas at STP
- the density of air at standard temperature and pressure (air has an average molecular mass of 28.871 u)
- the density of air at room temperature (25 °C) and one atmosphere of pressure
solution
Use the complete ideal gas law to determine this somewhat famous number.
PV = nRT V = nRT P V = (1 mol)(8.314462618 J/K mol)(273.15 K) 101,325 Pa The volume of a cube is the cube of one side. Conversely the side of a cube is the cube root of its volume.
V = s3 s = ∛V s = ∛(0.022414 m3) For those familiar with the English system of units this is about 11 inches.
Start with the definition of density and substitute the value just computed for volume. Air is a mixture of gases, so its molecular weight is the weighted average of its constituent molecules. Watch the units. Molecular weights are almost always given in grams per mole, but the SI unit of mass is the kilogram.
ρ = m V ρ = 0.28871 kg/mol 0.022414 m3/mol ρ = 1.29 kg/m3 This is a problem of proportionality. Density and volume are inversely proportional for a constant mass of gas while volume and temperature are directly proportional at constant pressure. Thus density and temperature are inversely proportional when mass and pressure are constant. Be sure to use absolute temperatures.
ρ ∝ 1 (m constant) ⎫
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⎭⇒ V ρ ∝ 1 (m & P constant) V ∝ T (P constant) T ρ2 ρ1 = T1 T2 ρ2 1.288 kg/m3 = 273 K 298 K