Example

Question:

Estimate the speed of sound in air at standard temperature and pressure. The mass of 1 mole of air is \(29.0 \times 10^{-3}\,\mathrm{kg}\).

Solution:

Step 1: Calculate density of air at STP.
1 mole of any gas occupies \(22.4\,\mathrm{L}\) at STP.
\[ \rho_0 = \frac{\text{mass of 1 mole}}{\text{volume of 1 mole at STP}} = \frac{29.0 \times 10^{-3}\,\mathrm{kg}}{22.4 \times 10^{-3}\,\mathrm{m^3}} = 1.29\,\mathrm{kg\,m^{-3}} \] Step 2: Using Newton's formula for sound speed:
\[ v = \sqrt{\frac{P}{\rho}} = \sqrt{\frac{1.01 \times 10^5\,\mathrm{N\,m^{-2}}}{1.29\,\mathrm{kg\,m^{-3}}}} = 280\,\mathrm{m\,s^{-1}} \] Step 3: Laplace correction.
Newton's result is ~15% lower than experimental value (\(331\,\mathrm{m\,s^{-1}}\)).
Laplace showed that sound propagation is adiabatic, not isothermal.
For adiabatic processes: \(PV^{\gamma} = \text{constant}\)
The correct formula becomes: \[ v = \sqrt{\frac{\gamma P}{\rho}} \] For air, \(\gamma = 1.4\), giving \(v = 331\,\mathrm{m\,s^{-1}}\), which matches experiment.