Reading the content in the ideal gas law


You have undoubtedly seen the ideal gas law in classes before this one: either in the form common in chemistry

$pV = nRT$  (chemist's version)

or in the form typically given in a high school physics class

$pV = Nk_BT$   (physicist's version).

But despite the fact that this equation is given early and often in your science classes, it is one of the subtler and more difficult equations to understand the conceptual content of. This is not just because it has two versions, but for a number of other reasons:

  • It relates 4 (four!) independent variables, many of which may change together.
  • Two of the variables ($p$ and $T$) are emergent variables — connecting a microscopic and a macroscopic view of the world.
  • It's a static law that describes a uniform system but is often used for dynamic systems that are non-uniform.
  • It has very limited applicability, only holding for very dilute gases.

Any one of these would be worth taking some extra time with. All four inspire me to add a "Dangerous Bend" icon to our "see the dog in the equation" (sense-making) icon. Unpacking everything in this equation is going to be a bit of a challenge. Don't treat this equation as trivial or obvious!

1. = (Four variables) — In this class we have often made the point that equations in science are not simply mathematical ways to calculate something: they stand for relationships among physical quantities (often measurements). This equation shows this in spades. It relates four physical quantities that can change: $p$, $V$, $N$ (or $n$), and $T$. In a specific situation, two of these might be fixed, one of the remaining chosen as independent variable, and the last as dependent variable. These lead to special case laws:

When used in these special cases, the ideal gas law is often a "given the initial values and one of the final, calculate something."

But do not be misled! In many situations multiple variables are changing at the same time. In that case, it's not really right to think of it as a chain of paired changes ("changing the temperature changes the pressure, changing the pressure results in a change in the volume"). You really have to think of everything as changing together. The trap arises from the large number of variables. We so often have "clean" problems in which one thing is changed (the independent variable) and one thing changes as a result (the dependent variable) that we have a tendency to ignore other things that might be changing at the same time. You have to pay close attention to everything that is changing in an Ideal Gas Law problem.

2. $p$ — The variable $p$ stands for the pressure — a tricky concept. It has the units force/area and is defined by the equation "$p=F/A$", but inside a gas there is no $A$. And since there is no vector on the $p$ it indicates we are talking about a scalar and not a vector — pressure has no direction.

Pressure is a kind of 2-D analog to tension. If I pull a string in both directions with a given force, the net force on any piece of the string is 0, but there is a tension. Every bit of the string is pulled equally in opposite directions. Since there is no net pull, there is no net direction, but how hard it's being pulled — the tension — is important. The pressure is saying that the molecules of the gas are moving in all directions and would push on any surface that one inserted into it, but equally in both directions. The net result is zero — so no direction — but the magnitude of how hard the push would be if there were a surface there is important.

3. $V$ — This is straightforward: simply the volume of the gas that we are considering. 

4. $pV$ While we can simply think of $pV$ as the product of our two variables, thinking of them together and interpreting their meaning gives additional insight into what the ideal gas law is telling us. Since pressure looks like force/area and volume looks like area times distance, pressure times volume looks like force times distance or work.

Consider a gas inside a balloon that has a given pressure, volume, and surface area. If we now make a small change in either the volume (perpendicular to the surface of the balloon), the change in pV is the work associated with that change. 

5. $T$ — This is also moderately straightforward: simply the temperature of the gas. But we have to be careful. Since $pV$ can never be negative (for a dilute gas), our $T$  must also always be positive. The $T$ used in the ideal gas law must always be an absolute temperature — preferably degrees Kelvin.  

6. $N$ or $n$ and $R$ or $k_B$ — It's a bit confusing that we sometimes write the ideal gas law in terms of the number of moles or in terms of the number of molecules. It's clear that if you are concerned about chemical reactions, especially in a macroscopic experiment, that the number of moles, $n$, is entirely appropriate. But if you are interested in interpreting a meaning at the molecular level, it might be more appropriate to think about the number of molecules, $N$.

Since the left hand side of the law has units of work (or energy), so must the right hand side. Since both $n$ and $N$ are pure numbers (at least by dimensional analysis — we have to keep track of "mole" as a unit), we need something to convert $T$ — in degrees Kelvin — to an energy unit. For the chemists' form, this is $R$, the gas constant = 8.314 Joules/mole-oK. The physicists' constant is called Boltzmann's constant, kB. Since we need to have the forms have the same value, we must have $nR = Nk_B$. Since the number of molecules in a mole is Avogadro's number, NA = 6.022 x 1023 molecules/mole, $N = nN_A$ so $k_B = R/N_A$ = 1.381 x 10-23 Joules/oK.

7. $k_BT$ — Since $pV$ has units of energy, $k_BT$  is a kind of an energy, and the ideal gas law looks in structure like a work-energy theorem (especially when you look at changes). From our derivation of the ideal gas law by averaging Newton's laws for many molecules hitting a wall (see Kinetic theory: the ideal gas law), kBis essentially (up to a factor of 2/3) the average kinetic energy of a single molecule. (The reason for the 2/3 factor is somewhat technical and won't be discussed here.)

When does this equation hold?  While every equation we discuss in our "Reading the content in ..." series has limitations as to how you can apply them, this is one where the limitations are often ignored and lead to wrong or even nonsensical results. We've therefore added a second "Dangerous Bend" icon to warn you against these common and easy to fall using the equation where it does not apply.

The first and most important limitation is that this is a GAS law. This means that it only holds for gases. It does NOT hold for liquids and it does not hold for solids. It doesn't even hold for gases dissolved in a liquid. (Though the way partial pressure measurements are defined in liquid can be very misleading about this fact.) Since it is an IDEAL gas law, it only holds for dilute gases.

As we saw in the discussion above, the Ideal Gas Law is saying that the work done by the pressure in expanding the gas comes from the kinetic energy of the molecules. Since potential energies are ignored, this means that the law ONLY holds when the molecules are very rarely close enough that they have significant potential energy of interaction. This is a bit tricky since if they didn't interact at all, they would never share energy and each molecule would keep the same energy it started with. This is NOT the assumption in the Ideal Gas Law: rather, it is assumed that the molecules are continually sharing energy through collisions and on the average each molecule will then have the same average energy (per degree of freedom). What this means is that the molecules has to collide often enough to share energy -- pretty quickly on a macro level, but rarely on a micro level so that they are interacting only a very small fraction of the time and are mostly moving freely without interaction. The Ideal Gas Law also assumes that the sizes of the molecules in the gas are very small and can be neglected compared to the total volume in which they are confined.

This is a particularly simple model of a gas — interactions are neglected in looking at energies and volume of the molecules are neglected. When gases are compressed both these factors start to be important. A more sophisticated model equation that takes into account the effect of both of these factors is the Van der Waals equation of state. This equation even provides a decent model for describing the condensation of a gas into a liquid!

A technical point: A second limitation on the Ideal Gas Law arises from the fact that it is an emergent law. Two of its variables, p and T are explicitly macroscopic quantities. This can become confusing when we try to map them onto simple mathematics of functions. We often define a pressure or temperature field — a function $p(x,y,z,t)$ or $T(x,y,z,t)$ that is supposed to represent the pressure or temperature at a particular point in space and instant in time. You'll see this on weather maps, but you may also see "heat maps" showing the variation in the temperature of the skin of a patient taken with an infrared camera.

This is fine as long as you are careful not to really think of the temperature "at a particular point in space." This is especially important for the ideal gas law which only works with dilute gases. At a particular point in space, there may or may not (most likely not) be a molecule there. If you tried to define pressure or temperature at a single point in space it would fluctuate like crazy as individual molecules pass through that point. What we mean when we say "temperature at this point in space" is really "the temperature in a small volume of space that is too small for me to care about at a macroscopic level but still contains lots and lots of molecules". 

As long as we don't think about going down to the molecular level, this technicality causes no problem, but it nicely illustrates the point that the mathematics we use is a model of a physical system and that model may work fine under some circumstances (when the "point" at which we are measuring our $p$ or $T$ is small but not a real "point") and be totally ridiculous under other circumstances (when we are looking at points on the molecular level).

Joe Redish 7/19/16

Article 475
Last Modified: May 27, 2019