Microscopic Origin of Gas Pressure
For an isotropic dilute ideal gas, elastic wall collisions give P = (1/3)rho v_rms^2 = (2/3)(translational kinetic-energy density).
Why this shows up in the exam
Estimating pressure from collision rates · Relating pressure to molecular speed · Finding translational energy from PV
Learn the idea
Gas pressure is the rate at which molecular momentum is transferred to container walls. Every elastic wall hit reverses the normal velocity component; enormous numbers of such impulses create a steady macroscopic force.
🧠 Memory hook: Pressure is momentum delivered per area per time.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- P = (1/3) rho v_rms² — kinetic-pressure equation for isotropic molecular velocities and elastic wall collisions
- P = (2/3)(K_trans/V) — pressure in terms of total translational kinetic energy density
- impulse = 2m v_perp — normal momentum change in one elastic wall collision
How to approach it
- 1Choose a wall normal
- 2Compute impulse times collision rate
- 3Divide by area and verify pressure units
Common slip-ups that cost marks
- •Using total speed instead of the normal component
- •Forgetting the factor two for elastic reversal
- •Including rotational energy in the pressure relation
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A gas has rms molecular speed 300 m/s at 300 K. What is its rms speed at 1200 K, assuming ideal behavior?
More from Kinetic Theory of Gases
Ideal gas law and gas laws
The ideal gas law and related gas laws describe the relationships between pressure, volume, temperature, and number of moles for ideal gases.
Degrees of freedom and thermal properties
Degrees of freedom determine the distribution of energy among molecules, affecting internal energy, specific heats, and the ratio of specific heats (γ).
Kinetic theory and molecular motion
The kinetic theory explains the behavior of gases in terms of the motion and collisions of their molecules, relating properties like pressure, temperature, and kinetic energy.
RMS speed and temperature dependence
The root mean square (rms) speed of gas molecules depends on temperature and molar mass, and is a key measure of molecular motion in gases.
Mean free path and collisions
Mean free path is the average distance a molecule travels between collisions, and depends on molecular size and number density.
Ideal-Gas Equation and Molecular Form
For a dilute ideal gas in thermal equilibrium, the state variables satisfy PV = nRT = Nk_B T, where intermolecular potential energy and molecular volume are neglected.