v ∝ √T. Not affected by pressure. Effect of humidity.

Speed of Sound in Air — Factors Affecting It

Question

What factors affect the speed of sound in air? Why does pressure NOT affect the speed of sound, even though air is a compressible medium?

This is a classic NCERT Class 11 question that also shows up regularly in JEE Main as a concept-based MCQ.

Solution — Step by Step

Newton derived the speed of sound in a gas as:

v = \sqrt{\frac{B}{\rho}}

where $B$ is the bulk modulus (elasticity) and $\rho$ is the density. This is the master formula — everything else follows from it.

For an ideal gas, pressure $P$ and density $\rho$ are related by $PV = nRT$ , which gives $\rho = \frac{PM}{RT}$ (where $M$ is molar mass).

When pressure increases, $B$ increases by the same factor (since $B = \gamma P$ for adiabatic conditions), and so does $\rho$ . The ratio $B/\rho$ stays constant — pressure cancels out completely.

Substituting $B = \gamma P$ and $\rho = PM/RT$ :

v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M}}

Since $\gamma$ , $R$ , and $M$ are constants for a given gas, we get:

v \propto \sqrt{T}

Temperature $T$ here is in Kelvin — never Celsius.

Water vapour ( $M = 18$ g/mol) is lighter than dry air ( $M \approx 29$ g/mol). When humidity increases, the effective molar mass of the air mixture drops.

From $v = \sqrt{\gamma RT / M}$ , a smaller $M$ means a larger $v$ . So sound travels faster in humid air than in dry air.

Why This Works

The speed of a mechanical wave depends on the restoring force (elasticity) and the inertia (density) of the medium. Higher elasticity → faster wave. Higher density → slower wave.

When we increase pressure at constant temperature, yes, the air becomes “stiffer” (higher $B$ ). But it also packs more molecules into the same space (higher $\rho$ ). These two effects exactly cancel each other — a beautiful symmetry of ideal gases.

Temperature breaks this symmetry. Heating air at constant pressure increases $v$ because the molecules move faster and the medium transmits the disturbance more quickly, while the density change is less significant in the $\sqrt{T}$ dependence.

Alternative Method

You can arrive at $v \propto \sqrt{T}$ using dimensional reasoning combined with kinetic theory.

The RMS speed of gas molecules is $v_{rms} = \sqrt{3RT/M}$ . Since the speed of sound is essentially the speed at which a pressure disturbance propagates through molecular collisions, it must be proportional to $v_{rms}$ . Hence $v_{sound} \propto \sqrt{T}$ .

This isn’t a derivation, but it’s a great way to quickly recall the temperature dependence in an exam when you’ve forgotten the exact formula.

Effect of Temperature: Numerical Form

v_T = v_0 \sqrt{\frac{T}{273}}

where $v_0 = 332$ m/s at 0°C (273 K).

For small changes, there’s an approximation used in NCERT:

v_T \approx 332 + 0.6t \text{ m/s}

where $t$ is temperature in °C. At 20°C, $v \approx 344$ m/s.

Common Mistake

Using Celsius instead of Kelvin in the ratio.

Students write $v \propto \sqrt{T}$ and then plug in $T = 27°C$ directly instead of $T = 300$ K. If a question asks “by what factor does speed change when temperature doubles from 27°C to 54°C?”, the correct answer uses $T = 300$ K and $T = 327$ K — not 27 and 54. Doubling the Celsius value does NOT double the Kelvin temperature.

The ratio comes out to $\sqrt{327/300} \approx 1.04$ , a 4% increase — not $\sqrt{2}$ .

Quick recall for JEE MCQs: The four factors and their effects in one line — Temperature ↑ (speed ↑), Pressure (no effect), Humidity ↑ (speed ↑ slightly), Molar mass ↑ (speed ↓). Pressure is the classic trap option.