# Open Mind

## Gases

#### June 29th, 2007 · 2 Comments

This is the first in a series of posts designed to share some of the nitty-gritty science behind global warming. The first part of my study is to brush up on thermodynamics. This is extremely important for climate science! We’re especially interested in the behavior of earth’s atmosphere, and thermodynamics deals with (among other things) the behavior of gases. So, that’s where we’ll start. I’ll start at the very beginning, so this post will be very elementary for those who are already familiar with basic physics, but very essential for those who are not.

We are (for the moment) mainly concerned with the behavior of a gas. Gas is a phase of matter in which the molecules of a substance move about freely, not bound to each other. They can, in fact, be quite far apart. The molecules tend to fly hither and yon, at different speeds and directions. But there are certain properties of the gas that we can measure, which apply to the bulk material of the gas.

One of the important properties of a gas is the number of molecules it contains. We’ll call this number $N$. Generally, $N$ is very large. For example, one liter of ordinary air (at room temperature and sea-level pressure) has about $2.5 \times 10^{22}$ molecules.

One of the most important properties of a gas is its volume, which we’ll denote as $V$. Gas can freely expand and contract, so its volume is highly variable (in fact, this extreme compressibility/expandibility is what makes it a gas).

Another important property of a gas is its mass. The mass of a liter of ordinary air is about 1.2 grams, or 0.0012 kg (kilograms). We can compute the mass of a quantity of gas, if we know the number of molecules and the average mass of each molecule. For air, the average mass of each molecule is about 29 amu, or atomic mass units. Since 1 amu is about $1.66 \times 10^{-27}$ kg, the average mass of a molecule of air is about $4.8 \times 10^{-26}$ kg. For the $2.5 \times 10^{22}$ molecules in a liter of ordinary air, this gives 0.0012 kg (1.2 grams).

Yet another important property of a gas is its pressure, which we’ll denote $P$. Gas molecules, as they fly about, tend to bounce of everything the gas touches. This exerts a force on every surface which the gas touches. The average force per unit area is called the pressure of the gas. Under ordinary conditions (at sea level), the pressure of air is 101325 Newtons per square meter (101325 Pascals). This is the same as 14.7 pounds of force per square inch (psi). The unit called the bar is 100000 Pascals, so ordinary atmospheric pressure is 1.01325 bars, or 1013.25 mb (millibars). Sometimes pressure is expressed in units called “millimeters of Mercury” (mm Hg), because one of the most common ways to measure pressure is to see how high it can raise liquid mercury in a narrow tube. At standard pressure, the mercury will rise 760 mm. Finally, pressure is sometimes measured in atm, or atmospheres, which is just the pressure divided by the standard sea-level pressure of earth’s atmosphere. So, standard pressure can be expressed as $P$ = 1 atm = 101325 Pascal = 1013.25 mb = 14.7 psi = 760 mm Hg (there certainly are a lot of different units for measuring pressure).

We’re not yet done with the important properties of a gas! Another is the temperature, which we’ll call $T$. Temperature is a rather complex property, but we can think of it as the average energy per mode of the molecules of the gas. “Modes” are the ways that kinetic energy (energy of motion) can exist in the molecules. Temperature is often thought of as the average kinetic energy of the gas molecules.

Temperature is usually measured in degrees Centigrade, although in the U.S. it’s usually measured in degrees Fahrenheit. But a more physically meaningful unit is degrees Kelvin. This is because when the average energy per mode is zero, the temperature on the Kelvin scale is zero. But on the Centigrade or Fahrenheit scale, the temperature would be negative (on the Centigrade scale, -273.15 degrees). It is more sensible to have zero energy correspond to zero temperature, and the Kelvin scale accomplishes this. This temperature is called absolute zero, and it is 0 K = -273.15 deg.C = -459.67 deg.F.

Then there’s the is energy (which we’ll call $E$). There are many forms of energy besides kinetic energy, for example the potential energy between the molecules of a substance. $E$ will denote the total energy of the gas — not just the kinetic energy, and not the energy per molecule but the total energy.

The last property of a gas which I’ll mention is the entropy, which we’ll denote by $S$. Entropy is a difficult quantity to define (except mathematically), but we can think of it as a measure of the degree of “randomness” of the gas. Essentially, it’s a measure of how many different ways the gas molecules can exist at the microscopic level, while still having the same essential properties at the macroscopic level. If this doesn’t make much intuitive sense, don’t worry. Even physicists have difficulty conceptualizing exactly what entropy “means.”

That’s certainly a lot of properties! But not all of them are independent. It was discovered centuries ago that if you take a certain quantity of gas, and add more gas (increase the number of molecules) while keeping the total volume and temperature unchanged, then the pressure will increase. Or, if you reduce the volume while keeping the number of molecules and temperature unchanged, the pressure will likewise increase. Finally, if you keep the number of molecules and the volume constant, but increase the temperature (heat the gas), the pressure will increase. These variables — pressure, volume, number of molecules, and temperature — are related to each other. The relationship between them is called the equation of state of the gas.

How many of the variables are independent? It turns out that for a given gas, three of the variables are needed to define its state. We might, for example, choose $N,~V,~T$ (the number of molecules, volume, and temperature) as our “defining variables.” Given these three, if we know what the gas is made of (its chemical composition), we can compute the other variables. We could, alternatively, choose to give $N,~V,~S$ (the number, volume, and entropy), and again given these three we can compute the others. It’s worth noting that some of the variables are extrinsic and others are intrinsic. Extrinsic variables scale with the size of the system, while intrinsic variables do not. If we take two identical boxes of gas (same number, type of molecules, same temperature, volume, etc.) and join them together into one larger box, then we’ll have twice as many molecules (double $N$), twice as much volume ($V$), twice the energy ($E$), twice the entropy ($S$), so $N,~V,~E,~S$ are extrinsic properties. However, the temperature and pressure will be unchanged, so $T,~P$ are intrinsic properties.

Perhaps the most important relation between the various variables is the equation of state. A typical gas obeys (almost exactly) what is called the ideal gas equation of state. This is a relationship between the pressure $P$, the volume $V$, the number of molecules $N$, and the temperature $T$, which involves a constant called the Boltzmann constant (denoted by $k$, which takes the value $1.38066 \times 10^{-23}$ Joules per Kelvin). The ideal gas equation of state is

$PV = NkT$.

This involves four gas properties ($P,~V,~N,~T$) and one constant ($k$). If we know any three of the gas properties, we can compute the fourth using this equation of state. If, for example, we know the volume, number of molecules, and temperature, we can compute the pressure as

$P = {NkT \over V}$.

Alternatively, if we know the number of molecules, volume, and pressure, we can compute the temperature as

$T = {PV \over Nk}$.

The ideal-gas equation of state is not quite exact, but it’s close enough to the behavior of real gases that it gives sufficiently accurate answers for almost all physical situations.

Another important quantity is the energy equation. For an ideal gas (and real gases behave almost exactly like ideal gases), the energy can be determined if we know the number of molecules and the temperature, as well as a constant (depending on the chemical constitution of the gas) known as the specific heat at constant volume (denoted $C_V$). In this case, the energy is

$E = N C_V T$.

There’s a lot of information to digest, just in this brief post. What do all these properties mean? How are they related? How can we use the relationships to determine the behavior of the gas? Those will be topics of future posts.

### 2 responses so far ↓

• Excellent post (as always), Tamino. I’m looking forward to seeing how far you’re willing to take this.

The usage of “centigrade” seems a bit quaint — it’s been half a century since most of the world switched over to “Celsius”.

[Response: I plan to go quite a distance. It’ll take time.

As for “centigrade” — I guess I’m just old-fashioned.]

• Down here, MetService uses hectoPascals on its weather maps…

[Response: The hectopascal is 100 Pascals, and is equal to the millibar (mb). That’s one *more* unit for pressure!]