Specific Heats, Mayer Relation, and Gamma
For an ideal gas with fixed f, C_V=fR/2, C_P=C_V+R, and gamma=C_P/C_V=1+2/f; these are molar heat capacities.
Why this shows up in the exam
Finding gamma for molecular models · Comparing constant-P and constant-V heating · Converting between C_P, C_V, and f
Learn the idea
Ideal-gas heat capacities follow from degrees of freedom, while constant-pressure heating also supplies expansion work. At fixed volume all supplied heat changes internal energy; at fixed pressure the gas must additionally push back its surroundings.
🧠 Memory hook: Constant pressure pays one extra R for expansion.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- C_V = fR/2 — molar constant-volume heat capacity with f active degrees
- C_P - C_V = R — Mayer relation for an ideal gas
- gamma = 1 + 2/f — heat-capacity ratio for fixed active degrees
How to approach it
- 1Identify the heat-capacity convention
- 2Find f or one known capacity
- 3Use Mayer's relation and verify C_P>C_V
Common slip-ups that cost marks
- •Mixing molar and specific heat capacities
- •Using gamma formulas for a mixture without first finding effective heat capacities
- •Ignoring temperature-dependent mode activation
🌟 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.