---
title: "Maxwell-Boltzmann Distribution — AP Physics 2 Guide"
description: "The Maxwell-Boltzmann distribution graphs the spread of molecular speeds in a gas at one temperature. Learn how it shifts with T and links to rms speed on AP Physics 2."
canonical: "https://fiveable.me/ap-physics-2-revised/key-terms/maxwell-boltzmann-distribution"
type: "key-term"
subject: "AP Physics 2"
unit: "Unit 9"
---

# Maxwell-Boltzmann Distribution — AP Physics 2 Guide

## Definition

The Maxwell-Boltzmann distribution is a graph showing how the speeds (and kinetic energies) of atoms in an ideal gas are spread out at a given temperature. Higher temperature shifts the peak to faster speeds and flattens the curve, since temperature measures average kinetic energy (AP Physics 2, Topic 9.1).

## What It Is

The Maxwell-Boltzmann distribution is the answer to a question the kinetic theory forces you to ask. If [temperature](/ap-physics-2-revised/unit-9/1-kinetic-theory-of-temperature-and-pressure/study-guide/wWjb2NGJDLNmMhB3 "fv-autolink") only tells you the *average* [kinetic energy](/ap-physics-2-revised/key-terms/kinetic-energy "fv-autolink") of gas atoms, what are the individual atoms actually doing? The answer is that they're doing a little bit of everything. Some atoms crawl, some scream along at enormous speeds, and most cluster somewhere in the middle. The Maxwell-Boltzmann distribution is the graph of that spread, with molecular speed on the x-axis and the number of molecules at each speed on the y-axis.

The shape of the curve depends on temperature and on the mass of the gas particles. Heat the gas up and the peak slides right (faster typical speeds) while the curve gets shorter and wider, because the atoms now span a bigger range of speeds. Cool the gas and the opposite happens. The peak shifts left and the curve gets taller and narrower. One thing never changes when you heat or cool a sealed container, though. The total area under the curve stays the same, because it represents the total number of molecules, and no atoms appear or vanish when the temperature changes.

## Why It Matters

This term lives in **Topic 9.1, Kinetic Theory of Temperature and Pressure**, in [Unit 9](/ap-physics-2-revised/unit-9 "fv-autolink") (Thermodynamics). It directly supports learning objective **9.1.B**, which asks you to describe temperature in terms of atomic motion. The CED's essential knowledge names the Maxwell-Boltzmann distribution explicitly as the graphical representation of the energies and speeds of atoms at a given temperature.

The distribution is what makes the whole kinetic picture honest. Equations like $K_{avg} = \frac{3}{2}k_B T = \frac{1}{2}mv_{rms}^2$ deal only in averages, but real gases are messy collections of trillions of particles. The Maxwell-Boltzmann curve is how you reason qualitatively about that messiness, and qualitative graph reasoning is exactly the kind of thinking [AP Physics 2](/ap-physics-2-revised "fv-autolink") rewards. Head to the [Topic 9.1 study guide](#) for the full kinetic theory picture.

## Connections

### [Root-mean-square speed (Unit 9)](/ap-physics-2-revised/key-terms/root-mean-square-speed)

The rms speed is one specific point pulled from the distribution. It's the speed that corresponds to the average kinetic energy, via $K_{avg} = \frac{1}{2}mv_{rms}^2$. Think of the Maxwell-Boltzmann curve as the whole population and v_rms as one representative statistic from it.

### Kinetic theory of pressure (Unit 9)

Pressure comes from atoms colliding with the container walls (LO 9.1.A), and the distribution tells you those atoms hit at a whole range of speeds. Raise the temperature, the curve shifts right, collisions get harder and more frequent, and pressure goes up. The graph and the pressure equation $P = F_{\perp}/A$ are two views of the same atomic chaos.

### [Degrees of freedom (Unit 9)](/ap-physics-2-revised/key-terms/degrees-of-freedom)

The $\frac{3}{2}k_B T$ in the average kinetic energy formula comes from the three translational [degrees of freedom](/ap-physics-2-revised/key-terms/degrees-of-freedom "fv-autolink") a gas atom has (motion in x, y, and z). The distribution describes how that translational energy gets shared unevenly across the atoms.

### [Conservation of momentum (Unit 4)](/ap-physics-2-revised/key-terms/conservation-of-momentum)

Atom-atom and atom-wall collisions in a gas are [momentum](/ap-physics-2-revised/key-terms/momentum "fv-autolink") problems. The same conservation principles you used for colliding carts explain how individual atoms constantly trade speed, which is why the gas settles into a distribution of speeds instead of every atom moving identically.

## On the AP Exam

Maxwell-Boltzmann questions are almost always qualitative graph-reasoning questions, often framed as claim-evaluation. A classic stem gives you a student's claim, such as "heating a gas makes the distribution narrower and taller," and asks you to judge it. That claim is backwards. Heating makes the curve wider and shorter with the peak shifted right. Another favorite setup cools a sealed gas (say, 400 K to 200 K) and asks what happens to the peak and width. Peak left, curve narrower and taller, area unchanged.

The sneakiest version compares two gases at the same temperature, like U-235 and U-238 in a centrifuge. Same temperature means same average kinetic energy, full stop. The lighter gas does NOT have more energy; it has a higher rms speed, so its distribution peaks at a faster speed. You should be able to sketch or compare two curves and explain shifts in terms of $K_{avg} = \frac{3}{2}k_B T$. No released FRQ has used the term verbatim, but it's named in the CED's essential knowledge for 9.1.B, so it's fair game for multiple-choice and for justifying answers in paragraph-length responses.

## Maxwell-Boltzmann distribution vs Root-mean-square speed

The Maxwell-Boltzmann distribution is the whole graph, the full spread of speeds in the gas. The rms speed is a single number derived from that spread, the speed matching the average kinetic energy. If an MCQ asks which quantity corresponds to the average kinetic energy of helium atoms at 300 K, the answer is v_rms, not the distribution itself. The distribution shows variety; v_rms summarizes it.

## Key Takeaways

- The Maxwell-Boltzmann distribution graphs how many gas atoms are moving at each speed at a given temperature, showing that atoms in a gas have a wide range of speeds, not one shared speed.
- Increasing temperature shifts the peak to higher speeds and makes the curve wider and flatter; decreasing temperature does the opposite.
- The area under the curve equals the total number of molecules, so it stays constant when you heat or cool a sealed container.
- Two different gases at the same temperature have the same average kinetic energy, but the lighter gas has a higher rms speed and a distribution peaked at faster speeds.
- The rms speed is one point connected to the distribution through the equation K_avg = (3/2)k_B T = (1/2)mv_rms², linking the graph to temperature quantitatively.

## FAQs

### What is the Maxwell-Boltzmann distribution in AP Physics 2?

It's a graph showing the spread of speeds (or kinetic energies) of atoms in an ideal gas at a given temperature. It appears in Topic 9.1 under learning objective 9.1.B, which connects temperature to atomic motion.

### Does heating a gas make the Maxwell-Boltzmann curve taller and narrower?

No, that's backwards and it's a claim AP questions love to test. [Heating](/ap-physics-2-revised/unit-9/3-thermal-energy-transfer-and-equilibrium/study-guide/B2UC1jOK2bqVTMMH "fv-autolink") shifts the peak to higher speeds and makes the curve wider and shorter, because atoms spread over a bigger range of speeds. Cooling is what makes the curve taller and narrower.

### Do all atoms in a gas move at the same speed?

No. Temperature only fixes the average kinetic energy. Individual atoms constantly trade speed through collisions, so at any instant some are slow, some are very fast, and most sit near the peak of the distribution.

### How is the Maxwell-Boltzmann distribution different from rms speed?

The distribution is the entire graph of speeds; rms speed is one value pulled from it, the speed corresponding to the average kinetic energy via K_avg = (1/2)mv_rms². Exam questions sometimes ask which single quantity matches the average kinetic energy, and that answer is v_rms.

### Do lighter gas atoms have more kinetic energy at the same temperature?

No. At the same temperature, all ideal gases have the same average kinetic energy, since K_avg = (3/2)k_B T depends only on T. Lighter atoms like U-235 just move faster than heavier ones like U-238, so their distribution peaks at a higher speed.

## Related Study Guides

- [9.1 Kinetic Theory of Temperature and Pressure](/ap-physics-2-revised/unit-9/1-kinetic-theory-of-temperature-and-pressure/study-guide/wWjb2NGJDLNmMhB3)

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