Kinetic Theory: Pressure from Molecular Motion
For identical particles in a classical ideal gas, wall collisions produce pressure and give pV = N m mean(v^2)/3. Comparing this with pV = nRT links kelvin temperature to mean translational kinetic energy per particle. Root mean square speed describes particle motion even when there is no bulk flow.
What a learner can do afterwards
- States the ideal-gas assumptions, explains the one-third factor using equal mean squared velocity components, and identifies conditions where real gases depart from the model
- Calculates root mean square speed and explains zero mean velocity for an equilibrium gas in its rest frame
- Uses mean translational kinetic energy = 3kT/2 to compare temperatures, gases and total translational energies
1 · Read
You used pV = nRT to relate gas measurements. Kinetic theory explains those measurements using particles. A particle carries momentum, mass times velocity. When it hits a wall, its momentum changes and it exerts a force on the wall. Pressure is the average force perpendicular to a wall divided by its area. At fixed volume and temperature, adding gas particles raises pressure.
Heating a fixed amount of ideal gas in a rigid container raises typical particle speeds. Collisions with each wall become more frequent and transfer more momentum per collision on average. More momentum reaches the wall each second, so its average force and the pressure rise. A sealed container keeps gas inside; rigid walls keep its volume fixed. Do not heat sealed containers.
The model treats particle size as negligible compared with particle spacing. Particles move in straight lines with no interparticle forces between collisions. Collisions are elastic: total kinetic energy is conserved, although two colliding particles can exchange energy. We consider a classical gas in equilibrium with no bulk flow and no preferred direction of motion.
For N identical particles of mass m, pV = N m mean(v^2)/3. Here p is pressure, V is volume, and mean(v^2) is the average of squared speeds. Use Pa, m^3, kg and m/s. Squared speed is v_x^2 + v_y^2 + v_z^2. No preferred direction means the three component averages are equal; each is mean(v^2)/3.
Each component average is 90000 m^2/s^2. Their sum gives mean(v^2) = 270000 m^2/s^2.
In pV = nRT, n counts moles. With N_A particles per mole, N = n N_A. Boltzmann's constant is k = R/N_A, giving pV = NkT. Comparing this with the collision equation gives mean translational KE = m mean(v^2)/2 = 3kT/2. Translational energy describes motion through space, excluding rotation and vibration. T is in kelvin. At equal temperatures, ideal gases have equal mean translational energy per particle.
Use k = 1.38e-23 J/K. At 200 K, mean translational KE = 1.5 x 1.38e-23 x 200 = 4.14e-21 J per particle. Here 1e-21 means 10^-21. For a temperature change, multiply mean energy by final T / initial T. Total translational energy = N x mean energy per particle. At fixed temperature, doubling N doubles total energy but leaves the mean unchanged.
Bar values are mean translational energy per particle in units of 1e-21 J. Doubling kelvin temperature doubles this mean.
Velocity includes direction; speed does not. For an equilibrium gas with no bulk flow, opposite directions balance, giving zero mean velocity in its rest frame. A finite sample can fluctuate around zero. Root mean square speed is v_rms = sqrt(mean(v^2)). Square each speed, average the squares, then take the square root. At positive temperature, rms speed is positive.
For sample speeds of 300 and 900 m/s, the squares are 90000 and 810000 m^2/s^2. Their mean is 450000 m^2/s^2. Its square root gives an rms speed of about 670.8 m/s. The arithmetic mean speed is 600 m/s, so these two averages differ.
Mean translational KE = m v_rms^2/2. At the same temperature, this mean is equal for light and heavy gas particles. A smaller mass therefore needs a larger v_rms^2, and hence a larger rms speed. This compares averages. Individual particles have different speeds and energies. Samples with different particle counts can still have different total energies.
Real particles occupy space and exert forces on one another. At high density, particle size and interactions can affect pressure, contrary to the ideal assumptions. At low temperature, attractions matter more compared with translational energy. Real gases can then depart from ideal-gas predictions. Elastic collisions do not mean there are no interactions during a collision.
Wall collisions produce pressure. Kelvin temperature sets mean translational energy per particle. Rms speed uses squared speeds; zero mean velocity does not mean particles are still.
Lesson, questions and flashcards written for this dataset. OpenStax College Physics and Chemistry were consulted for scientific facts; their prose and worked examples were not reproduced.
Readings, questions and flashcards written for this dataset, with OpenStax consulted for scientific facts. Video references: Khan Academy, The kinetic molecular theory of gases and Kinetic molecular theory and the gas laws. External media terms remain unverified.
More to read
Pressure and momentum transfer
A container wall feels many brief forces as gas particles strike it. Each collision changes the particle's momentum perpendicular to the wall. The wall receives momentum too. Pressure is the average perpendicular force divided by wall area.
Two changes can raise this average force. Particles can strike more often, or each collision can transfer more momentum. Heating a fixed amount of ideal gas in a rigid container does both on average. Its equilibrium pressure rises.
Sealing and rigidity describe different constraints. Sealing prevents gas from entering or leaving. Rigidity prevents the walls from expanding. A flexible sealed container can change volume, so sealing alone does not give the same pressure prediction. Do not heat sealed containers.
For identical particles moving with no preferred direction, the collision model gives pV = N m mean(v^2)/3. It links measurable pressure and volume with particle count, mass and squared speed.
Using the ideal-gas assumptions
The kinetic model leaves out the volume occupied by particles and the forces between them during travel. Between collisions, particles follow straight paths. Elastic collisions preserve total kinetic energy, but they can redistribute it between particles.
The pressure equation also uses motion with no preferred direction. The mean squared velocity component along the x-axis then equals those along the y-axis and z-axis. Adding the three component averages gives mean squared speed. This is why each contributes one third of the total.
A real gas can depart from these assumptions. At high density, finite particle size and interparticle forces can affect pressure. At low temperature, attractions become more significant compared with translational energy. The ideal equation may then give a different pressure from the measured value.
Averages of energy and motion
Temperature describes an average, not the energy of every particle. In a classical ideal gas at equilibrium, mean translational kinetic energy per particle is 3kT/2. Translational energy is energy of motion through space. Rotation and vibration are excluded from this formula.
Particle count does not enter that mean. It does enter the sample's total translational energy, which is N times the mean. Two samples at equal temperature can therefore have equal mean energies but different totals.
Velocity and speed also need different averages. Opposite directions balance in a still equilibrium gas, giving zero mean velocity in its rest frame. A finite sample can fluctuate around zero. Rms speed is calculated by squaring speeds, averaging the squares and taking the square root.
Mean translational energy also equals m v_rms^2/2. Equal temperatures give equal mean energies, so particles with less mass have a higher rms speed. Individual particle speeds still vary.
Explore more
2 · Watch
Take it off screen
Where it sits
Learn first
Optional connections
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
13 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.