The Physics
Hypertextbook
Opus in profectus

# Gas Laws

## Problems

### practice

1. The graphs to the right show the pressure and temperature inside the cabin of a commercial jet airliner on a two hour flight. As can be seen in the pressure graph, the plane spent half an hour on the ground waiting to take off, fifteen minutes ascending, an hour and a quarter cruising, fifteen minutes descending, and fifteen minutes on the ground approaching the terminal. The interesting segment from a gas laws perspective occurred when the plane was cruising.

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 ℃, 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 ℃. (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?

2. There is a truly excellent book on food science written by Harold McGee called On Food and Cooking The Science and Lore of the Kitchen. Mr. McGee's book is vast in scope and interesting on every page. There is one peculiar essay in the chapter on legumes called "The Problems of Legumes and Flatulence" that lends itself particularly well to the gas laws.

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

Verify Mr. McGee's claim.

3. Determine…
1. the volume of one mole of an ideal gas at standard temperature and pressure
2. the dimensions of a cube that could hold one mole of an ideal gas at STP
3. the density of air at standard temperature and pressure (air has an average molecular mass of 28.871 u)
4. the density of air at room temperature (25 ℃) and one atmosphere of pressure
4. According to the current interpretation of the big bang theory, the universe began some 13.8 billion years ago when space, time, matter, and energy arose spontaneously in an infinitesimally small region of space called a singularity. Luckily for us, this tiny speck inflated, starting a journey of cosmic expansion that continues to this day. For the first 380,000 years of its existence, the space, time, matter, and energy of the universe were so dense that everything was effectively opaque. Light and other electromagnetic waves were tightly bound to the matter of the universe, much like the electrons in a wire are tightly bound to the metallic network of the metal from which it was constructed. (Have you ever been shocked while passing an electrical outlet in your house? No, of course not. And why not? Because the electrons are bound to the atoms of the solid metal conductor quite tightly.) After expanding for roughly 380,000 years, temperatures reduced to a relatively cool 3000 K and the universe finally became thin and cool enough for light and matter to live independent lives. When we look out at the universe around us now, all the radiation we see is at least 380,000 years younger than the universe as a whole. Everything before this moment is lost in time. This is also the time when nearly every free electron joined up with a hydrogen or helium nucleus — the period of recombination. In the intervening 13,799,620,000 years since recombination the oldest radiation has been diluted to the point where it is no longer visible, but instead lies wholly within the microwave part of the spectrum. This cosmic microwave background radiation (CMB) has been chilled to a mere 2.725 K by the overall expansion of the universe. Determine the following quantities at the moment of recombination in comparison to their current value for the universe as a whole…
1. volume
3. density

### conceptual

1. When incandescent light bulbs are manufactured, they are filled with an inert gas to insulate the filament and prevent its sublimation. The pressure inside a freshly manufactured light bulb is something like 80% of normal atmospheric pressure. Why are light bulbs filled with low pressure gas? Why not fill them with gas at normal atmospheric pressure? The gases normally used are argon and a little bit of nitrogen, both of which are relatively inexpensive. Cost is not therefore a determining factor.
2. Sketch a picture of a person holding a helium-filled balloon on the surface of the Earth under typical outdoor, environmental conditions. Sketch a picture of an astronaut holding that very same balloon on the surface of the moon. What has changed about the balloon between these two sketches and why?
3. A typical car tire has a volume of about 40 liters and should be inflated to roughly two atmospheres. How much air has to be added to a completely deflated tire to inflate it to the proper pressure?

### numerical

1. An incandescent light bulb is filled with argon and a little bit of nitrogen at room temperature (say 20 ℃) and 0.8 atmospheres of pressure. When operating, the gas heats up to between 200 and 400 ℃. Determine the pressure range of the gas in such a light bulb when it is on.
2. A typical halogen lamp is filled with bromine or iodine (both of which are corrosive) at five atmospheres of pressure when manufactured. Halogen lamps operate at much higher temperatures than ordinary incandescent light bulbs. The glass envelope surrounding the filament can reach temperatures as high as 1200 ℃. (This combination of physical and chemical properties means that halogen lamps must be treated with extra respect.) Determine the pressure of the gas inside a halogen lamp after it has reached operating temperature.
3. A scuba diver in Lake Huron fills her lungs to their full capacity of 5 liters when 10 m below the surface.
1. What volume would her lungs acquire if she quickly rose to the surface?
2. What one word of advice should the diver heed as she rises?
4. The correct inflation of a tire at 20 ℃ is 200 kPa. After driving several hours, the driver checks the tires. If the temperature of the tires is now 40 ℃, what will the pressure gauge read?
5. A car tire has an outer diameter of 64.8 cm, an inner diameter of 43.1 cm, is 23.1 cm wide, and is inflated to 248 kPa. A mechanic removes the wheel and lays it horizontally on a work table. Approximately how much air escapes when the valve is opened and the pressure is allowed to equalize with the environment?
6. A hot air balloon has a mass of 300 kg when deflated and a volume of 2000 m3 when inflated. The balloon is to be launched on a day when the temperature is 27 ℃ and the air has a density of 1.61 kg/m3. The air inside the envelope is at 87 ℃ as the balloon floats horizontally.
1. How much mass is the balloon's gondola carrying?
2. How many adults are riding this balloon?
7. Read the following passage from The Science of Champagne, a lecture presented by Gerard Liger-Belair of the Faculté des Sciences de Reims at the New York Academy of Science 22 April 2008.

Just keep in mind, that after the final corking, when the bottles are ready to drink, the pressure inside the bottle due to the presence of carbon dioxide molecules rises up to 5 bars; that's to say, 5 times the atmospheric pressure. This is a huge pressure. You would have to be under about 40 meters of water to feel the same pressure. And the champagne contains approximately 10 grams of CO2 (of carbon dioxide molecule) per bottle. This is also a huge quantity, because this is also equivalent to about 5 liters of gaseous carbon dioxide molecules. So, six times the whole volume of the bottle. So this is a huge quantity of dissolved carbon dioxide molecules contained into [sic] the champagne.

Gerard Liger-Belair, 2008

Verify the claims of Mr. Liger-Belair through the following chain of calculations.

1. What is the absolute pressure under 40 m of water?
2. What is the molecular weight of carbon dioxide?
3. How many moles is 10 g of CO2?
4. What volume does this much CO2 occupy…
1. in the bottle at serving temperature (500 kPa gauge and 10 ℃)?
2. in the atmosphere at room temperature (100 kPa absolute and 20 ℃)?
5. How does the volume computed in part d. ii. compare to that in part d. i.?
6. Do Mr. Liger-Belair's numbers agree with the results of your calculations?
8. Determine the mass of air inside a football that satisfies the requirements of the International Football Association Board (IFAB) — the guardians of the rules of association football (a.k.a. soccer). Assume a reasonable temperature for the inside of the ball.

The ball is:

• spherical
• made of leather or other suitable material
• of a circumference of not more than 70 cm (28 ins) and not less than 68 cm (27 ins)
• not more than 450 g (16 oz) and not less than 410 g (14 oz) in weight at the start of the match
• of a pressure equal to 0.6–1.1 atmosphere (600–1,100 g/cm2) at sea level (8.5–15.6 lbs/sq in)

### statistical

1. constant-temperature.txt
These pressure-volume data sets were adapted from The Works of the Honorable Robert Boyle (1699). They show the volume of a column of air as it responded to first increasing and then decreasing absolute pressure. Use this data to verify the relationship between pressure and volume at constant temperature.
2. constant-volume.txt
Use these pressure-temperature data from two experiments done on the gas inside a rigid container to determine the value of absolute zero in degrees celsius. Be sure to convert the pressure values from gauge to absolute before proceeding (that is, add one atmosphere to each of them).