16.1: The Formation Of The Lightest Elements - Physics LibreTexts

Going Further 14.2: Expanding Gases Cool

In this chapter, and throughout much of the module, we have repeatedly stated that the Universe was denser and hotter when it was younger. Or equivalently, we have said that the Universe must have cooled as it expanded. Why must this be the case? If you have had a course in chemistry or physics you know the answer, but if not (or maybe even if you did) you might find these statements confusing. We will now explain why gases exhibit this property.

Consider a gas inside a closed container from which it cannot leak out. If you imagine that the container is a cylinder with one end free to move, then you could push or pull on the cylinder to change the volume occupied by the gas, and thus its density. A hand bicycle pump is an example of this sort of device. What effect would this have on the gas?

From experience you probably know that you must push on such a cylinder in order to compress it, and the more you manage to squeeze the gas, the harder it becomes to compress it further, sort of like trying to squeeze a tube of toothpaste with the cap on. The reason for the increasing difficulty is two-fold.

The first cause of your increasing effort is that you are expending energy to compress the gas. That energy is not lost: energy is always conserved. The only place for the energy to go in this case is into the gas. The energy you expend pushes on the gas particles and imparts a higher kinetic energy into them, so they move faster because you have sped them up. Since the particles are moving faster, they strike the walls of the container with a greater force. That increases the pressure on the walls and is the first reason you must push harder.

The second reason you have to exert more effort to compress a canister of gas is because the gas volume is getting smaller as you push inward. That means that the particles do not have to travel as far when they bounce between one wall and the opposite wall; they strike each wall more often. Because the particles are moving faster, they travel that shorter distance in a shorter time than would would be true if they moved at a lower speed. This also causes the particles to hit the walls more frequently.

The net result is that by compressing the gas you speed up the particles and cause them to hit the walls more often and with greater force. The fact that the gas particles are moving faster is generally conveyed by saying their temperature (T) has increased. That they are confined to a smaller space means that their volume (V) has decreased, and these changes have led the pressure (P) to increase. The total number of particles (N) in this case has remained the same because no gas has leaked out of the cylinder. We could also think about what happens in terms of the number density, N / V, which is increasing in this case.

The pressure, volume, number of particles, and temperature of a gas are related by the following equation.

\[PV=NkT \nonumber \]

The lowercase k is called Boltzmann’s constant, which we have already seen. The equation is known as the ideal gas law. You might be familiar with this equation from a chemistry course. It is often written as below.

\[PV=nRT \nonumber \]

Now, n is the number of moles and R is the ideal gas constant. Chemists like to use moles, physicists use the number of particles instead, but they are essentially the same. According to the ideal gas law, as the density or temperature of an ideal gas goes up, the pressure also tends to go up. Don't let the word "ideal" confuse you. Most gases follow this law quite closely. It fails to work when the particles interact strongly with one another, but at that point you are probably no longer dealing with a gas.

Conversely, if we let the volume of our cylinder expand, the gas inside would have to push against an outside pressure, expending energy to do so. The energy would have to come from the internal energy of the gas itself, thus lowering its temperature, density and pressure.

That is why gases cool as they expand and heat up when they are compressed. It is the basis of technologies like refrigerators and air conditioners. In the case of the Universe, it expands not against an external pressure, but against its own self-gravity. The result is the same: the gas within the Universe cools as a result of the expansion.