--- title: "Kinetic Theory: Pressure from Molecular Motion" description: "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 en" canonical: https://lightmysky.com/learn/science/kinetic-theory-pressure-from-molecular-motion-mt_2ARUheICqu source: https://lightmysky.com/learn/science/kinetic-theory-pressure-from-molecular-motion-mt_2ARUheICqu.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Matter & Materials · Ages 16 to 18 Page: https://lightmysky.com/learn/science/kinetic-theory-pressure-from-molecular-motion-mt_2ARUheICqu ## Ready when they can - 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 ## Lesson: Pressure from bouncing molecules 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. *(drawing: For a fixed amount of gas in a rigid container, faster particles transfer more momentum to the walls each second.)* 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. 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. **Example.** 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. 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. **Example.** 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. **Recap.** 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.* ## Practice 13 questions on this page, each with its working shown. ## Needs first - [The Gas Laws and the Ideal Gas Equation](https://lightmysky.com/learn/science/the-gas-laws-and-the-ideal-gas-equation-mt_G7DOTVl4KM) ## Optional connections - [Drawing Particle Diagrams](https://lightmysky.com/learn/science/drawing-particle-diagrams-mt_htAYR-iCFF) - [Elastic and Inelastic Collisions](https://lightmysky.com/learn/science/elastic-and-inelastic-collisions-mt_JMwUuh1YUq) ## Opens up - [The Maxwell-Boltzmann Distribution of Molecular Speeds](https://lightmysky.com/learn/science/the-maxwell-boltzmann-distribution-of-molecular-speeds-mt_njr5r5YtuT)