Equipartition and Internal Energy
For an ideal gas with f active quadratic degrees of freedom, U = (f/2)nRT and Delta U = nC_V Delta T, assuming f remains unchanged across the temperature interval.
Why this shows up in the exam
Internal energy of monoatomic and diatomic gases · Thermalization after a moving container stops · Energy of gas mixtures
Learn the idea
Each active quadratic degree contributes equal mean energy, making ideal-gas internal energy a temperature function. A molecule's thermal energy is shared equally among all active quadratic storage modes; multiplying per-particle energy by particle count gives the sample energy.
🧠 Memory hook: Half kT per quadratic degree; then multiply by particles.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- U = (f/2) nRT — internal energy of an ideal gas with fixed active degree count f
- Delta U = n C_V Delta T — internal-energy change when molar C_V is valid over the interval
- U = N(f/2)k_B T — particle-count form of equipartition energy
How to approach it
- 1Determine active f
- 2Choose per molecule or per mole
- 3Apply energy conservation and check signs
Common slip-ups that cost marks
- •Using total mass instead of moles
- •Adding an arbitrary constant when only changes matter
- •Keeping f fixed across a mode-activation transition without justification
🌟 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.