Kinetic theory of gases

Pressure is a drum roll, not a push

• A balloon feels like it is being pushed steadily outwards. Nothing is pushing steadily. Molecules are striking the rubber at random, each blow lasting a fraction of a nanosecond.
• There are so many blows, so close together, that the average is perfectly smooth. Pressure is a drum roll heard from far enough away that the individual beats disappear.
• Taking that picture seriously, and applying nothing but Newton's laws to it, produces the gas laws from scratch and reveals what temperature actually is.
• This lesson is the assumptions of kinetic theory 分子动理论, the pressure derivation, r.m.s. speed and the meaning of temperature.

The assumptions

• A large number of identical molecules in continuous random motion 无规则运动.
• The molecules' own volume is negligible compared with the volume of the gas.
• There are no forces between molecules except during collisions, so they travel in straight lines between them.
• Collisions are perfectly elastic 弹性, losing no kinetic energy, and the time of a collision is negligible compared with the time between collisions. Newton's laws apply throughout.
• These fail at very high pressure, where molecular volume matters, and at very low temperature, where intermolecular forces matter.

Deriving the pressure

• Take a cube of side $L$ holding $N$ molecules of mass $m$, and follow one moving along $x$ with velocity $u_1$.
• One collision with the right wall reverses its velocity, so its momentum 动量 changes by $-2mu_1$ and, by Newton's third law, the wall receives $+2mu_1$.
• Between hits on that wall it travels $2L$, so $\Delta t = 2L/u_1$, giving an average force $F_1 = \Delta p/\Delta t = mu_1^2/L$.
• Summing over all molecules and dividing by the wall area $L^2$ gives $p = Nm\langle u_x^2\rangle / V$. In three dimensions, symmetry means $\langle u_x^2\rangle = \tfrac13\langle c^2\rangle$, so:

One molecule, one wall, then multiply up

Worked example: explain the pressure in words

The density form

• $Nm$ is the total mass of the gas, so $Nm/V$ is its density $\rho$. The same result reads:

• Use this version whenever a question gives a density instead of $N$ and $m$, which it often does.

Root-mean-square speed

• $\langle c^2 \rangle$ is the mean-square speed 均方速率: square each molecule's speed, then average. Its square root is the root-mean-square 均方根 speed:

• It is a single useful measure of molecular speed, slightly larger than the mean speed because squaring weights the fast molecules more heavily.
• The molecules do not all move at this speed. They have a wide spread, and raising the temperature shifts the whole spread to higher speeds.

A distribution, of which the r.m.s. speed is one summary

Worked example: r.m.s. speed from a density

What temperature actually is

• Two expressions for the same $pV$: $pV = NkT$ from the equation of state, and $pV = \tfrac13 Nm\langle c^2\rangle$ from kinetic theory. Set them equal, cancel $N$ and multiply by $\tfrac32$:

• The left side is the average translational kinetic energy 平动动能 of one molecule. So temperature is a measure of average molecular kinetic energy, and nothing else.
• Two consequences the exam asks for. Double the absolute temperature and the average kinetic energy doubles, so $c_{\text{r.m.s.}}$ grows by $\sqrt{2}$. And at the same temperature, lighter molecules move faster, since $\tfrac12 m\langle c^2\rangle$ is the same for both.

Internal energy of an ideal gas

• An ideal gas has no forces between molecules, so there is no molecular potential energy: its internal energy is entirely kinetic.
• Multiplying the average kinetic energy by the number of molecules:

• So the internal energy of an ideal gas depends only on its temperature, not on its pressure or volume. That single fact does most of the work in the next lesson.

Marks that slip away

You've got it