Speed of Sound in an Ideal Gas
For small adiabatic disturbances in an ideal gas, c = sqrt(gamma P/rho) = sqrt(gamma RT/M), with equilibrium properties evaluated locally.
Why this shows up in the exam
Comparing sound in helium and nitrogen · Gas-mixture sound speed · Linking sound speed with molecular rms speed
Learn the idea
Sound speed depends on adiabatic stiffness divided by density, giving a square-root dependence on gamma T/M. A pressure disturbance moves faster when the gas is harder to compress and slower when each mole carries more mass.
🧠 Memory hook: Sound follows square root of gamma T over M.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- c = sqrt(gamma P/rho) — sound speed from adiabatic bulk response
- c = sqrt(gamma RT/M) — ideal-gas form using SI molar mass
- c2/c1 = sqrt((gamma2 T2 M1)/(gamma1 T1 M2)) — comparison between ideal gases
How to approach it
- 1Find gamma and effective molar mass
- 2Use absolute temperature
- 3Form a ratio to cancel common constants
Common slip-ups that cost marks
- •Using isothermal compressibility
- •Dropping gamma
- •Using an unweighted molar mass for a mixture
🌟 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.