Heat Capacities and Gamma of Gas Mixtures
For nonreacting ideal gases at common temperature, C_V,mix = sum(n_i C_Vi)/sum n_i and C_P,mix = C_V,mix+R; gamma_mix=C_P,mix/C_V,mix.
Why this shows up in the exam
Monoatomic-diatomic mixtures · Inferring composition from effective gamma · Mixture sound-speed and adiabatic questions
Learn the idea
A mixture's heat capacity is the mole-weighted sum of component heat capacities, not an average of gamma values. Each component stores its own share of energy for the same temperature rise, so add capacities first and take their ratio afterward.
🧠 Memory hook: Add nC first; never average gamma directly.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- C_V,mix = sum(n_i C_Vi)/sum n_i — molar mixture heat capacity for ideal components
- gamma_mix = 1 + R/C_V,mix — mixture gamma after forming the effective molar C_V
- U_mix = sum_i n_i C_Vi T — total internal energy when zero references are consistent
How to approach it
- 1Convert each component to C_V
- 2Form mole-weighted C_V,mix
- 3Obtain C_P,mix and gamma_mix last
Common slip-ups that cost marks
- •Taking an arithmetic mean of component gammas
- •Weighting by mass when mole weighting is required
- •Forgetting different components can have different f
🌟 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
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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.