Laplace Correction and the Speed of Sound
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Direct answer
Newton treated sound compressions as isothermal and got v = √(P/ρ) ≈ 280 m/s for air at standard conditions — about 15 percent below the measured value. Laplace's correction recognises that compressions and rarefactions alternate too fast for heat to escape, so the process is adiabatic and the relevant elastic modulus is γP, giving v = √(γP/ρ) = √(γRT/M) ≈ 331 m/s for air (γ = 7/5, M = 28.8 g per mole), matching experiment. The formula carries three consequences JEE tests constantly: speed is independent of pressure at fixed temperature (P and ρ scale together), it grows as the square root of absolute temperature (about 0.61 m/s per degree Celsius near room temperature), and it increases with humidity because water vapour lowers the mean molar mass of the air mixture.
What you must remember
- Laplace's result: v = √(γP/ρ); the adiabatic bulk modulus of a gas is γP, not P — that single factor of √1.4 = 1.18 lifts Newton's 280 m/s to 331 m/s.
- Molecular form: v = √(γRT/M) with M in kg per mole; speed rises with lighter molecules and higher absolute temperature, never with pressure alone.
- Temperature rule: v ∝ √T in kelvin; in practice v ≈ 332 + 0.61 t m/s with t in degrees Celsius, so sound travels about 347 m/s on a 27-degree-Celsius day (300 K).
- Pressure independence: at fixed temperature, doubling pressure doubles density, leaving v unchanged — the classic assertion–reason pair.
- Humidity: moist air is lighter (water vapour M = 18 replaces dry air's M ≈ 29), so v increases; sound travels faster on a humid day than a dry one at the same temperature.
- Benchmark speeds: air 331 m/s at 0 degrees Celsius, hydrogen about 1284 m/s (γ = 7/5, M = 2), helium about 965 m/s (γ = 5/3, M = 4) — the reason inhaling helium raises voice pitch.
- Exact numeric anchor: for air, γ = 1.4, P = 1.013 × 10⁵ Pa and ρ = 1.29 kg/m³ give v = √(1.4 × 1.013 × 10⁵/1.29) ≈ 331.3 m/s; the arithmetic itself has appeared as a JEE Main question.
Why Newton was wrong by 51 m/s
Newton's logic was flawless for the model he assumed: sound is a pressure wave, its speed is √(elastic modulus/density), and for an isothermal gas the bulk modulus equals the pressure. The model failed because the physics of the compressions is thermal. In one second a 1000 Hz wave compresses and rarefies each layer of air a thousand times; the hot compressed zones and cold rarefied zones are separated by fractions of a millimetre, and heat simply cannot diffuse that fast. The oscillation is therefore adiabatic, and an adiabatic gas is stiffer: compressing it raises its temperature, so the pressure rises more for the same volume change. Quantitatively the bulk modulus becomes γP instead of P, and the speed picks up the factor √γ = √1.4 ≈ 1.183: 280 × 1.183 ≈ 331 m/s, within experimental error of measurement. This correction, published by Laplace in 1816, is a standing lesson that thermodynamic assumptions inside a wave equation decide the answer.
The companion calculations are quick. At 27 degrees Celsius, v = 331 × √(300/273) ≈ 347 m/s — the same number the 0.61-per-degree rule gives. In hydrogen, v = √(1.4 × 8.31 × 300/0.002) ≈ 1320 m/s, roughly four times air's speed at the same temperature, which is why a hydrogen-filled organ pipe of the same length sounds about two octaves higher.
How the exam frames it
JEE Main asks the formula book: compute v at a stated temperature, compare speeds in two gases, or convert a Celsius reading to kelvin before touching a square root — that last conversion is where a third of the class loses the question. JEE Advanced prefers the reasoning: justify pressure-independence, explain the humidity effect through mean molar mass, or place the Laplace correction in an assertion–reason format where "sound propagation is adiabatic because compressions are too rapid for heat exchange" must be judged for both halves. The recurring traps: using Celsius in √T; forgetting M in kg per mole (28.8 × 10⁻³, not 28.8); and assuming higher pressure means faster sound at the same temperature. One more: v = √(γP/ρ) applies to the medium, not the source — a louder whistle travels no faster, only with bigger amplitude.
Frequently asked questions
What exactly was Laplace's correction to Newton's formula?
He replaced the isothermal bulk modulus P with the adiabatic value γP, converting v = √(P/ρ) ≈ 280 m/s into v = √(γP/ρ) ≈ 331 m/s for air.
Why is sound propagation adiabatic rather than isothermal?
Compressions and rarefactions alternate thousands of times per second across sub-millimetre distances, far too fast for heat to flow between them and equalise temperature.
Does raising the pressure of a gas raise the speed of sound in it?
No — at constant temperature density rises in the same proportion as pressure, so √(γP/ρ) is unchanged; only temperature or molar mass matters.
Why does sound travel faster on a humid day?
Water vapour (M = 18) displaces heavier dry air (M ≈ 29), lowering the mixture's mean molar mass, and v = √(γRT/M) rises accordingly.
How does the speed vary between hydrogen and air at the same temperature?
Hydrogen, with M = 2 against air's 28.8, carries sound about √(28.8/2) ≈ 3.8 times faster — roughly 1300 m/s at room temperature.