Question
Why does the speed of sound in air increase with temperature but stay (almost) independent of pressure? And why does sound travel faster in hydrogen than in oxygen, even though hydrogen is “lighter”? Students struggle with these because the textbook formula seems to suggest pressure matters and lighter gas should be slower (less inertia, but less restoring force?).
Solution — Step by Step
When you increase pressure at constant temperature (Boyle’s law), density increases proportionally: constant. So in , the ratio doesn’t change, and doesn’t change. Pressure alone is a red herring.
For an ideal gas, where is molar mass. Substituting:
Now appears explicitly, has cancelled. So . A jump from K to K raises sound speed by about .
For the same and same (both diatomic), . Molar mass: , . So . Sound is about 4 times faster in hydrogen.
The “lighter gas, slower sound” intuition is wrong because lower mass means molecules dart around faster on average, and sound is just collective molecular motion. Lower , higher .
Why This Works
Sound speed is fundamentally a molecular speed scaled by . The RMS speed of gas molecules is . Sound speed is , which is the same molecular RMS speed multiplied by . Sound cannot travel faster than the molecules carrying it.
This is why temperature (which sets molecular speed) matters and pressure (which only sets density at fixed ) doesn’t.
Quick formula for speed of sound in air at temperature : m/s. Memorize this — appears in NEET almost every alternate year.
Alternative Method
Dimensional argument: from , , , , the only combination with units of speed is . So sound speed must have this form, with as the dimensionless prefactor (Laplace correction). No need to derive from first principles.
Common Mistake
Students see “Newton’s formula ” and think Laplace’s correction adds because of friction or viscosity. It’s neither — Newton assumed isothermal compression in sound waves; Laplace pointed out the compressions are too fast for heat exchange, so they’re adiabatic, hence the factor.