Degrees of Freedom and Molecular Models
In the classical active-mode model, a monoatomic molecule has f=3, a rigid linear molecule usually f=5, and a rigid nonlinear molecule f=6; each active vibration adds two quadratic terms.
Why this shows up in the exam
Classifying monoatomic and polyatomic gases · Determining heat capacities from geometry · Recognizing rigid versus nonrigid diatomic models
Learn the idea
Degrees of freedom count independent quadratic ways in which a molecule can store thermal energy. Translation is always available, rotation depends on molecular geometry, and vibration contributes only when the mode is thermally active.
🧠 Memory hook: Translate three; rotate by shape; an active vibration counts twice.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- f = f_trans + f_rot + 2 f_vib — effective quadratic degree count when the listed vibrational modes are active
- f_trans = 3 — three spatial translations for any freely moving molecule
How to approach it
- 1Identify molecular geometry
- 2State whether vibration is active
- 3Count independent quadratic terms
Common slip-ups that cost marks
- •Giving a linear molecule three rotational modes
- •Counting a vibration only once
- •Assuming all modes are active at every temperature
🌟 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.