9.1 Kinetic Theory of Temperature and Pressure

AP Physics 2 was revised for the 2024–25 school year. Its units are now numbered 9 to 15, picking up where AP Physics 1's Units 1–8 leave off, so thermodynamics is Unit 9 and this topic is 9.1. Older books and reviews number the course's units 1 to 7, with fluids as Unit 1 and thermodynamics as Unit 2.

Fluids (pressure in liquids, buoyancy and Bernoulli's equation) moved to AP Physics 1, so if an older Physics 2 review opens with fluids, you can skip that part. Going the other way, electric charge, simple circuits, and waves and sound left AP Physics 1, so this course now teaches them from the start (Units 10, 11 and 14).

The free-response questions changed in May 2025 to four set types: mathematical routines, translation between representations, experimental design and analysis, and qualitative/quantitative translation. Physics 2 free-response questions from 2024 and earlier are built differently.

Starting in May 2027, the multiple-choice section has 42 questions in 85 minutes (up from 40 in 80), and the four free-response questions take 95 minutes (down from 100). You can use a calculator and the equation sheet on both sections.

A gas is a huge number of atoms flying around in random directions. When an atom hits a wall and bounces back, its momentum changes, so the wall must have pushed on it. By Newton's third law, the atom pushed back on the wall just as hard. One collision is a tiny tap, but trillions of them every second add up to a steady force.

Pressure is that force spread over an area: $P = \dfrac{F_\perp}{A}$. Only the part of each push that is perpendicular to the surface counts. The unit is the pascal (1 Pa = 1 N/m²), and normal air pressure is about 1.0 × 10⁵ Pa, or 1 atmosphere.

Pressure isn't only at the walls. It exists everywhere inside the gas. If you placed a tiny surface anywhere in the middle of the container, atoms would hit both sides of it.

Atoms in an ideal gas collide elastically, with each other and with the walls. Momentum is conserved in atom–atom collisions, in one or two dimensions, just like the carts and pucks you studied in Physics 1.

For an atom of mass m that hits a wall head-on at speed v and bounces straight back, the change in momentum is 2mv (from +mv to −mv). A glancing hit changes only the perpendicular component of velocity, so it delivers less push.

Pressure goes up when collisions happen more often (more atoms, or a smaller container) or hit harder (faster atoms). Heating a gas does both at once.

The temperature of a gas tells you the average translational kinetic energy of its atoms:

$$K_{\text{avg}} = \frac{3}{2}k_BT$$

Here $k_B = 1.38 \times 10^{-23}$ J/K is Boltzmann's constant and T must be in kelvins (K = °C + 273). Two big consequences follow. First, average kinetic energy is proportional to absolute temperature: double T in kelvins and you double $K_{\text{avg}}$. Second, it doesn't depend on which gas it is. Helium and argon at the same temperature have the same average kinetic energy per atom.

Setting $\frac{1}{2}mv_{\text{rms}}^2 = \frac{3}{2}k_BT$ gives the root-mean-square (rms) speed, $v_{\text{rms}} = \sqrt{\dfrac{3k_BT}{m}}$. The rms speed is the square root of the average of the squared speeds. It's a typical speed, not the speed of every atom. Because m is in the denominator, lighter atoms move faster at the same temperature.

Atoms in a gas don't all share one speed. Collisions constantly speed some up and slow others down. The Maxwell–Boltzmann distribution is a graph of how many atoms have each speed. It starts at zero (almost no atoms are nearly still), rises to a peak at the most probable speed, then falls off with a long tail toward high speeds. The curve is lopsided, with the tail stretching out to the right.

You need to know how its shape changes, not its formula.

• Hotter gas: the peak moves to a higher speed and gets lower, and the curve spreads out. More atoms are very fast.
• Same gas, same number of atoms: the area under the curve stays the same, because the area stands for the total number of atoms.
• Heavier atoms at the same temperature: the peak sits at a lower speed and the curve is taller and narrower.