Wavelength of Sound Waves in Hydrogen Gas

Formula: v = √(γRT/M)

Where: • γ = adiabatic index (1.4 for diatomic gases including H₂) • R = 8.314 J/(mol·K) • T = temperature in Kelvin • M = molar mass of gas (H₂ = 0.002 kg/mol)

At T = 273 K (0°C): v = √(1.4 × 8.314 × 273 / 0.002) = √(1,589,000) ≈ 1260 m/s

Standard accepted value: ≈ 1270 m/s at 0°C

Comparison with air (M = 0.029 kg/mol, same γ): v_air = √(1.4 × 8.314 × 273 / 0.029) ≈ 331 m/s

Ratio: v_H₂/v_air = √(M_air/M_H₂) = √(29/2) ≈ 3.8

Sound travels ~3.8 times faster in hydrogen than in air.

Relationship: v = f × λ → λ = v/f

For sound in hydrogen (v ≈ 1270 m/s):

Frequency → Wavelength in H₂ → Wavelength in Air 100 Hz → 12.7 m → 3.4 m 500 Hz → 2.54 m → 0.68 m 1000 Hz → 1.27 m → 0.34 m 5000 Hz → 0.254 m → 0.068 m

Note: Same frequency sound has a longer wavelength in hydrogen than in air, because the wave travels faster.

The pitch (frequency) depends on the source, not the medium. The speed and wavelength change with medium; frequency does not.

Speed of sound in a gas depends on:

1. Elasticity (bulk modulus) of the gas
2. Density of the gas

v = √(γP/ρ) = √(γRT/M)

Hydrogen is the lightest gas (M = 2 g/mol). Lower molar mass → lower density → sound travels faster.

Comparison at 0°C: Gas → Molar mass → Speed of sound Hydrogen (H₂) → 2 g/mol → ~1270 m/s Helium (He) → 4 g/mol → ~970 m/s Neon (Ne) → 20 g/mol → ~433 m/s Nitrogen (N₂) → 28 g/mol → ~337 m/s Air → 29 g/mol → ~331 m/s Oxygen (O₂) → 32 g/mol → ~316 m/s CO₂ → 44 g/mol → ~259 m/s

Lighter gas = faster sound = longer wavelength at same frequency.