Gas Laws

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Discussion

The gas laws are a set of intuitively obvious statements to most everyone in the Western world today. It's hard to believe that there was ever a time when they weren't understood. And yet someone had to notice these relationships and write them down. For this reason, many students are taught the three most important gas laws by the names of their discoverers. However, since the laws are known by different names in different countries and (more importantly) since I can never remember who gets credit for which law without referring to notes, I will not follow this convention.

pressure-volume (constant temperature)

What happens to the volume of a gas as the pressure on it changes. Let's try the following experiment using equipment that might be found in your kitchen.

Marshmallows in a kitchen vacuum pump. Do try this experiment at home. Do it!
The volume of a marshmallow increases as the pressure on it decreases … and vice versa.

Marshmallows are a mixture of sugar, air, and gelatin. The sugar makes them sweet, the air makes them fluffy, and the gelatin holds everything together. Marshmallows are a frozen foam and are mostly air by volume. When placed in a vacuum pump, they expand as the pressure decreases. Break the seal on their container and they shrink during the return to normal atmospheric pressure. Since the vacuum pump pulls on the marshmallows hard enough to burst some of the air bubbles, they are actually a bit smaller and more shriveled at the end of this experiment. This illustrates a fundamental, yet important, property of gases. The pressure of a gas is inversely proportional to its volume when temperature is constant. Symbolically …

P ∝  1  (T constant)
V

or

P1V1 = P2V2 = constant

This correlation was discovered independently by Robert Boyle (1627-1691) of Ireland in 1662 and Edme Mariotte (1620-1684) of France in 1676. In Great Britain, America, Australia, the West Indies and other remnants of the British Empire it is called Boyle's law, while in Continental Europe and other places it is called Mariotte's law.

Mariotte added the important provision that temperature remain constant. Boyle neglected to mention it, but the data he used to derive his law were most likely collected during a period in which the temperature did not experience any significant change. Since the gas needs to be in thermal equilibrium with its environment (or some other heat reservoir) to maintain an even temperature, the pressure-volume relationship normally applies only to "slow" processes. The marshmallow-vacuum experiment shown above is an example of a "slow" process. The pressure is reduced at a rate slow enough that heat from the environment is able to keep the jar and its contents at nearly room temperature. Such a transformation that takes place without a change in temperature is said to be isothermal.

Pumping a bicycle tire with a hand pump is an example of a "fast" process. The work done pushing the piston transforms into an increase in the internal energy (and thus an increase in the temperature) of the air molecules within the pump. People familiar with hand bicycle pumps will attest to the fact that they get hot after use. Likewise, when a gas is allowed to expanded into a region of reduced pressure it does work on its surroundings. The energy to do this work comes from the internal energy of the gas and so the temperature of the gas drops. You can experience this yourself without the aid of any apparatus other than your mouth. Purse your lips so that your mouth has only a tiny opening to the outside and blow hard. The air rushing from your mouth will be quite cool despite coming from the core of your body, which is normally quite hot (around 37 °C). During a "fast" process like the ones just described, pressure and volume are changing so rapidly that heat doesn't have enough time to get into or out of the gas to keep the temperature constant. Such a transformation that takes place without any flow of heat is said to be adiabatic.

volume-temperature (constant pressure)

What happens to the volume of a gas when its temperature changes? Let's try another kitchen experiment.

Bread dough before and after baking. Do try this experiment at home. Do it!
[dough] [bread]
Increasing the temperature of bread dough increases its volume.

Bread is made from wheat flour, water, yeast, and a bit of sugar. Yeast are tiny microorganisms. They are quite possibly the very first domesticated animals and, much like dogs and horses, yeast have been bred for different purposes. Just as we have guard dogs, lap dogs, and hunting dog; draft horses, race horses, and war horses; we also have brewer's yeast, champagne yeast, and bread yeast. Bread yeast have been selectively bred to eat sugar and burp carbon dioxide (CO2). When wheat flour and water are mixed together and kneaded, the protein molecules are mashed and stretched until they line up neatly to form a substance called gluten that, like chewing gum, is both elastic and plastic. Let this special matrix sit and the the CO2 vented from the yeast get trapped in thousands of tiny resilient, stretchy pockets. As this process continues these tiny pockets expand, which causes the volume of the dough to expand or rise in a process called proofing. We now have a fluffy gummy blob ready for the oven.

While there the dough expands again, but his time it's not due to the action of microorganisms (they all die around the boiling point of water). This time it's the heat, or rather the temperature. The temperature inside a bread oven is roughly 50% greater (in absolute terms) than the temperature outside. And similarly, the baked bread that comes out of a bread oven is also roughly 50% greater than the room temperature dough that goes in. This domestic example illustrates quite nicely a fundamental property of gases. The volume of a gas is directly proportional to its temperature when pressure is constant. Symbolically …

V ∝ T (P constant)

While no doubt known and understood informally by billions of bakers since the dawn of civilization, the precise mathematical relationship was first discovered by the French physicist Guillaume Amontons (1663-1705) in 1699. The experiment was repeated much later by Jacques Alexander Cèsare Charles (1746-1823) in 1787 and much, much later by Joseph Louis Gay-Lussac (1778-1850) in 1802. Charles did not publish his findings, but Gay-Lussac did. It is most frequently called Charles' law in the British sphere of influence and Gay-Lussac's law in the French, but never Amonton's law.

An isobaric process is one that takes place without any change in pressure.

Let's recall what it means when two quantities are directly proportional like volume and temperature. Heat up a gas and it's volume will expand. Cool it down and it's volume will contract. The two quantities change in the same direction. More specifically, an increase in one results in a proportional increase in the other and a decrease in one results in a proportional decrease in the other. For example …

There's a symmetry at work here somewhere. A symmetry is a change in one quantity that leaves another, more fundamental quantity unchanged. It's something like multiplying both the numerator and denominator of a fraction by the same thing.

a  ·  ⎛
⎝		 x ⎞
⎠		  =  a · x  =  a
b x b · x b

No wait, it's exactly like that. The only way two quantities can change in direct proportion is if their ratio remains constant. Thus …

V1  =  V2  = constant
T1 T2

pressure-temperature (constant volume)

Fix this, too.

The pressure of a gas is directly proportional to its temperature when volume is constant. Symbolically …

P ∝ T

An isochoric process is one that takes place without any change in volume.

This relationship doesn't really have a name, but I have heard it called the "pressure law" or (mistakenly) "Gay-Lussac's law".

Temperatures drop 6 °C for every 1000 m of altitude.

P1  =  P2  = constant
T1 T2

absolute temperature

In 1703, Amontons stated … ?


[magnify].

Double room temperature, 293 K = 20 °C, and you get 586 K = 313 °C not 40 °C.

a complete ideal gas law

Proportionality statements aren't as popular today in the Twenty first Century as they were in the Nineteenth Century and earlier. We live in an era where it's all about the equation. There's good and bad in this focus. Equations convey a lot of information in a very few symbols, which is why they're so popular, but they're also a crutch; a device used to support a weak understanding and make it seem strong. Equations can be used by a student with no understanding to fake competency.

"I put the numbers into the equation and I got the right answer. Since I have the right answer, I am smart."

Skilled? Certainly. Smart? Not necessarily.

Still, it would be nice to have an equation around for those times when all you want to do is just get the job done with a minimum of hassle.

Combine the three together.

P1V1  =  P2V2  = constant
T1 T2

There are two ways to write the complete statement of the ideal gas law as an equation …

functional thermodynamics statistical thermodynamics
PV = nRT PV = NkT
where …              
    P =  absolute pressure    
    T =  absolute temperature    
    V =  volume    
and …       or …      
n =  number of moles N =  number of particles
R =  gas constant = 8.315 J/mol·K k =  Boltzmann's constant = 1.382 J/K

Summary

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.
     
    [magnify]
     
    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.
     
    [magnify]
     
    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   P1  =  P2  
    T1  =  −60 °C = 213 K T1 T2
    P2  =  81 kPa 26 kPa  =  81 kPa
    T2  =  ?? 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.
  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.

    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 ⇒ V2  =  P1  =  101 kPa  = 4.4
    V1 P2 22.7 kPa
     
    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.
  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), and
    4. the density of air at room temperature (25 °C) and one atmosphere of pressure.
    Solutions …
    1. Use the complete ideal gas law to determine this somewhat famous number.
               
      PV =  nRT      
         
      V =  nRT  =  (1 mol)(8.31450 J/K mol)(273.15 K)  
      P (101,325 Pa)  
      V =  0.022414 m3 = 22.414 liter  
       
               
    2. 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 = ∛(0.022414 m3)
      s =  0.28195 … m = 28.2 cm
         
      For those familiar with the English system of units, this is roughly eleven inches.
    3. 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  =  0.28871 kg/mol  = 1.29 kg/m3
      V 0.022414 m3/mol
               
    4. 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) ⇒      
      V ρ ∝  1  (m & P constant)
      V  ∝  T  (P constant) T
           
                     
      ρ2  =  T1 ⇒ ρ2  =  273 K ⇒ ρ2 = 1.18 kg/m3
      ρ1 T2 1.2881 kg/m3 298 K
                       
  4. According to the current interpretation of the big bang theory, the universe began some 13.7 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 little speck of something in a vast sea of nothing quickly blew up (or inflated in more appropriate terms) and started itself on 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 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 380,000 years, when temperatures had reduced to a relatively cool 3000 K, the universe finally became thin enough for light to separate from matter and live an independent life. When we look out at the universe around us now 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 moment when the energy density of the universe dropped low enough to allow nearly every free electron to join up with a hydrogen or helium nuclei. Since these entities were all created at the same moment, this event is known as the period of recombination. In the intervening 13,699,620,000 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,
    2. radius, and
    3. density.
    Solutions …
    If your life in physics is entirely determined by your ability to solve problems, then you must surely regard the previous paragraph as 10% useful and 90% wasteful information. If you appreciate physics as an opportunity to be exposed to and to learn new ways of thinking, then you must surely view the previous paragraph as wholly incomplete and lacking in interesting detail. If you believe that all knowledge prior to 6000 years before present is nonsense for some imagined religious reason, then you must surely believe that reading every word is a waste of your precious time. Since one of the fundamental principles of writing is write to your audience and since my audience has multiple reasons for using this page, I am certain that no one will be entirely satisfied with the information presented or the nature of the questions asked. Nonetheless, let's answer them …
    1. The basic principle behind this problem, and the underlying assumption given that it appears in this section, is that the universe is some kind of ideal gas and that it obeys one of the basic gas laws. My guess would be that temperature and volume are directly proportional when pressure is constant. (A more realistic appraisal would be that the transition is adiabatic -- with no heat transfer between the universe and whatever's outside it; which would be nothing.)
                           
      V2  =  V1 ⇒ V1  =  T2  =  2.75 K  =  1
      T2 T1 V2 T1 3000 K 1100
                           
      The universe was roughly one thousandth its current volume at the time of recombination.
    2. Knowing the way the volume changed, we need only us the fact that volume is proportional to the cube of radius to determine the change in radius.
                                             
      V1  =  ⎛
      ⎜
      ⎝      		 r1 ⎞
      ⎟
      ⎠      		 3 ⇒ r1  =  ⎛
      ⎜
      ⎝      		 T2 ⎞
      ⎟
      ⎠      		 ⅓  =  ⎛
      ⎜
      ⎝      		 1 ⎞
      ⎟
      ⎠      		 ⅓  =  1
      V2 r2   r2 T1   1100   10.3
                                             
      The universe was roughly one tenth its current diameter at the moment of recombination.
    3. Density is inversely proportional to volume, so this last part is quite easy to calculate.
                         
      ρ =  m ⇒ ρ1  =  V2  =  1  =  1100
      V ρ2 V1 1100 1
                         
      The density was roughly one thousand times greater at the moment of recombination than it is now.

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. A typical incandescent light bulb is filled with argon and a little bit of nitrogen at room temperature (say 20 °C) and 0.8 atmospheres of pressure. When operating, the gas heats up to between 200 and 400 °C. 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 °C. (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 °C is 200 kPa. After driving several hours, the driver checks the tires. If the temperature of the tires is now 40 °C, 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 °C and the air has a density of 1.61 kg/m3. The air inside the envelope is at 87 °C as the balloon floats horizontally.
    1. How much mass is the balloon's gondola carrying?
    2. How many adults are riding this balloon?

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).

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