Give Newton's formula for the speed of sound in a gas, show why it fails for air, and describe the correction made by Laplace. How do temperature, pressure and humidity affect the speed of sound in air?

1. 1 · State Newton's formula. Newton took the changes as isothermal, so the elasticity is P and v = √(P/ρ).
2. 2 · Show that it fails. For air at STP: v = √(1.01 × 10⁵ ÷ 1.29) ≈ 280 m s⁻¹, about 15 percent below the measured 331 m s⁻¹.
3. 3 · Give Laplace's correction.
   • Air is a poor conductor of heat
   • Compressions and rarefactions are very rapid
   • So the changes are adiabatic and the elasticity is γP
   • v = √(γP/ρ) = √(1.4 × 1.01 × 10⁵ ÷ 1.29) ≈ 331 m s⁻¹
4. 4 · Effect of temperature. v = √(γRT/M), so v ∝ √T; about 0.61 m s⁻¹ faster for each 1 °C rise.
5. 5 · Effect of pressure. At fixed temperature P/ρ is constant, so pressure has no effect.
6. 6 · Effect of humidity. Moist air is less dense than dry air, so sound travels faster on a humid day.
7. 7 · Real life. Thunder reaches you a little sooner on a hot, humid monsoon day than on a cold, dry one.
8. 8 · Close in one line. So, Laplace's formula v = √(γP/ρ) gives the correct speed of sound in a gas.

Scoring tip: Write both formulas, then the two reasons for an adiabatic change. Each reason usually earns a mark.