The Physics
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Opus in profectus

Gas Laws

Discussion

introduction

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 are a mixture of sugar, air, and gelatin. Sugar makes them sweet, air makes them fluffy, and gelatin makes them elastic. 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 is made from wheat flour, water, yeast, and a bit of sugar. Yeast are tiny microorganisms. They are quite possibly the 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 Charles (1746–1823) in 1787 and much, much later by Joseph 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…

• Doubling the absolute temperature of the air in an engine cylinder will double its volume.
• Halving the absolute temperature of the air in a bag of potato chips will cause it to shrink to one-half its original volume.
• The absolute temperature of a bread oven is one and a half times that of room temperature. Therefore, the loaf of baked bread that comes out of an oven has 50% more volume than the ball of dough that went into it.

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 (V constant)

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 1,000 m of altitude.

 P1 = P2 = constant T1 T2

absolute temperature

In 1703, Amontons stated… ?

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 21st century as they were in the 19th 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 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

PV = nRT

where…

 P = absolute pressure T = absolute temperature V = volume

and

 n = number of moles R = gas constant = 8.314 J/mol K

statistical thermodynamics

PV = NkT

where…

 P = absolute pressure T = absolute temperature V = volume

and

 N = number of particles k = Boltzmann constant = 1.381 × 10−23J/K