Worked example 1
Sketch and describe how the distribution for a sample of N₂ at 300 K differs from the same sample at 600 K.
Try it first: Decide first whether anything about the number of particles has changed.
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Gases, Solutions & SpectroscopyKinetic Molecular TheoryContent level: Core 20 min
Maxwell-Boltzmann Speed Distributions
What you'll be able to do: Read and sketch Maxwell-Boltzmann curves, and explain how temperature and molar mass change their shape.
Introduction
Not every particle in a gas moves at the average speed. The Maxwell-Boltzmann curve shows the whole spread, and reading it correctly is a skill the exam tests directly.
These are recommended, not required. You can start this lesson at any time.
Learning objectives
- Interpret the axes of a Maxwell-Boltzmann distribution
- Predict how raising temperature changes the curve
- Compare the curves of two gases of different molar mass
- Explain why the area under every curve for a fixed sample is the same
Lesson
What the curve shows
The horizontal axis is particle speed and the vertical axis is the relative number, or fraction, of particles at that speed. The curve rises from zero, passes through a peak at the most probable speed, and trails off in a long tail to the right because a few particles are always moving very fast.
The curve never touches zero on the right. There is always a small population of very fast particles, and that tail matters enormously for reaction rates later in the course.
Raising the temperature
Heating the sample gives every particle more energy on average. The peak moves right to a higher most probable speed and drops in height, so the curve becomes broader and flatter. The total area stays the same because you still have the same number of particles.
Changing the gas
At a fixed temperature, a heavier gas has the same average kinetic energy but a smaller average speed. Its curve peaks further left and is taller and narrower. Compare helium with xenon at 300 K and the helium curve is a wide, low sprawl while xenon is a tall, tight spike.
Reading questions on the exam
Most questions ask you to identify which curve belongs to which condition. Work through two checks in order: does the peak shift left or right, and does the curve get taller or shorter. A shift right with flattening is a temperature increase; a shift left with sharpening at the same temperature is a heavier gas.
Relate molar mass to speed at one temperature
At the same temperature, different gases have the same average translational kinetic energy but different speed distributions. Root-mean-square speed varies as 1/sqrt(molar mass). A lighter gas therefore has a distribution shifted toward higher speeds, while the total area under a normalized distribution stays the same.
Equal temperature means equal average kinetic energy, not equal particle speed.
Key ideas
Definition
Most probable speed
The speed at the peak of the curve, where the largest fraction of particles is found.
Key concept
Constant area
The area under the curve counts all particles, so it does not change when the sample is heated.
Rule
Temperature effect
Higher temperature moves the peak right and lowers it.
Rule
Mass effect
At fixed temperature, greater molar mass moves the peak left and raises it.
Equation
Root mean square speed
urms = sqrt(3RT/M)
- R = 8.314 J/(mol*K)
- T = temperature in kelvin
- M = molar mass in kg/mol
Worked examples
Worked example 2
Calculate the root mean square speed of O₂ at 298 K.
Try it first: Convert the molar mass into kilograms per mole before substituting.
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Worked example 3
At one temperature, how does the rms speed of H₂ compare with O₂?
Try it first: Use molar masses 2.0 and 32.0 g/mol in a speed ratio.
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Common mistakes
Believing all particles in a gas move at the same speed.
Why it's wrong: Collisions constantly redistribute energy, producing a wide spread of speeds.
Check instead: Quote the most probable speed and note the distribution around it.
Drawing a heated curve with a larger area.
Why it's wrong: Heating does not create particles, so the area is unchanged.
Check instead: Broaden and lower the curve instead of enlarging it.
Forgetting to convert molar mass to kg/mol in urms.
Why it's wrong: The joule is built on kilograms, so grams give an answer that is off by about thirty times.
Check instead: Check that your speed is in the hundreds of metres per second.
After the lesson · Question bank
Independent practice
Work through new problems on this topic on your own. The lesson's quick checks and worked examples stay above if you need to review a step.
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What you should now know
A Maxwell-Boltzmann distribution plots the fraction of particles against speed. Raising the temperature shifts the peak to higher speed and flattens the curve, while increasing molar mass shifts the peak to lower speed and sharpens it. The area under the curve is fixed because it represents all of the particles.
- The curve plots fraction of particles against speed
- Higher temperature shifts the peak right and flattens the curve
- Greater molar mass shifts the peak left and sharpens it
- Area under the curve is fixed by the number of particles
- urms = sqrt(3RT/M) with M in kg/mol
Sources and further reading
This lesson is original Chem Help content. No external sources were adapted.