Read a spread of particle speeds
You will be able to: Interpret speed-distribution shape, area and temperature without memorizing its equation.
What changes when a gas gets hotter?
A class can have the same average height without every student having that height. Similarly, gas particles at one temperature have a range of speeds. A speed-distribution graph shows how particles are spread across those speeds.
A useful starting point: Temperature measures an average, not a total →
Words and symbols before equations
- Speed axis
- Horizontal axis, in m/s; speed is nonnegative.
- Probability density
- Vertical axis, in (m/s)⁻¹; an interval’s area represents its fraction of particles.
- Peak or mode
- Most probable speed; not generally the mean or rms speed.
- Normalized distribution
- The full area under the probability-density curve is one.
What this picture assumes
Normalized Maxwell–Boltzmann speed densities for identical particles of mass 4.6×10⁻²⁶ kg; blue reference 300 K, orange selected T. Full area of each curve is 1; the plotted 0–3500 m/s window omits a small tail. No memorization of the distribution formula is required.
Connect the picture to the physics
The height at one exact speed is not the number of particles at that precise speed. Compare areas over speed intervals. For the same gas, increasing temperature spreads the distribution over a wider range and shifts its peak toward greater speeds.
When normalized curves have equal total area, broadening lowers the peak. A lower peak therefore does not mean fewer total particles. Temperature fixes average kinetic energy, which depends on squared speed; the peak is not itself v_rms.
The interactive curves use the ideal Maxwell–Boltzmann speed distribution for the same particle mass. Its equation is not needed for AP Physics 2; focus on relative width, peak location and the conserved total area. The displayed finite speed axis omits a very small far-right tail.
A worked example, step by step
Two normalized curves describe the same gas at 300 K and 1200 K. Compare the rms speed, characteristic speed scale and peak height.
- The temperature ratio is 1200/300=4.
- Speeds characteristic of this distribution scale as √T, so the horizontal speed scale doubles.
- The rms speed doubles; it remains different from the peak speed.
- To keep total probability one when the speed scale doubles, the peak height becomes half as large.
A taller peak does not by itself mean more particles. Read the axis definition and compare areas.
Why can the hotter curve have a lower peak?
Compare with an explanation
Its fixed total probability is spread across a wider speed range.
Predict. Change one thing. Explain.
Keep the 300 K reference curve fixed and raise the comparison temperature. Describe the peak shift and width before checking the rms readout. Both curves are normalized; no particles have been removed.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Selected T=600 K: peak speed 600 m/s; rms speed 734.8 m/s. Reference rms=519.6 m/s. Both full curve areas equal one. Peak and rms are distinct.
Normalized Maxwell–Boltzmann speed densities for identical particles of mass 4.6×10⁻²⁶ kg; blue reference 300 K, orange selected T. Full area of each curve is 1; the plotted 0–3500 m/s window omits a small tail. No memorization of the distribution formula is required.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant particle, temperature or energy relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionCompare normalized speed distributions of the same gas at 200 K and 800 K. (a) Predict the rms-speed ratio. (b) Predict the peak-position ratio. (c) Compare total areas. (d) Explain why the colder peak is taller.
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Compare with the answer and four-point rubric
- 1 point: v_rms,hot/v_rms,cold=√4=2.
- 1 point: The most probable speed also doubles for the same Maxwell–Boltzmann model.
- 1 point: Both full areas equal one.
- 1 point: The colder distribution is concentrated over a narrower speed range.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does an interval’s area mean?
The fraction of particles with speeds in that interval.
RECALL 2What happens to width as T rises?
It increases for the same gas.
RECALL 3Must you memorize the distribution formula?
No. Understand the graph’s features and their relation to temperature.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Read a spread of particle speeds
- Same gas: characteristic speeds scale with √T.
- Full probability-density area = 1.
- Peak speed, mean speed and rms speed are different averages or descriptors.
Remember: A taller peak does not by itself mean more particles. Read the axis definition and compare areas.
Conditions: Normalized Maxwell–Boltzmann speed densities for identical particles of mass 4.6×10⁻²⁶ kg; blue reference 300 K, orange selected T. Full area of each curve is 1; the plotted 0–3500 m/s window omits a small tail. No memorization of the distribution formula is required.
Refresh Kid · AP Physics 2 Unit 1 (official Unit 9) · Objectives 9.1.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 9.1, objectives 9.1.B. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Refresh Kid calls this the first AP Physics 2 unit; College Board numbers it Unit 9, continuing after AP Physics 1. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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