Sound travels through a gas as a series of alternating compressions and expansions. As a compression wave passes through the gas, a small region of the gas is compressed, increasing its pressure. As the wave moves onward, that region expands again and returns to approximately its original state.
Because sound propagation involves repeated compressions and expansions, the speed of sound must depend on how easily the gas can be compressed.
This behavior is described by a compressibility:
\[ \kappa = -\frac{1}{V} \left( \frac{\partial V} {\partial p} \right) \]
The question is: which compressibility should be used?
Isaac Newton assumed that the compressions and expansions associated with sound waves occurred slowly enough that the gas remained at constant temperature.
Under this assumption, the appropriate compressibility is the isothermal compressibility:
\[ \kappa_T = -\frac{1}{V} \left( \frac{\partial V} {\partial p} \right)_T \]
Newton's model predicted a speed of sound that was approximately 15% smaller than the experimentally measured value.
He attributed the discrepancy to imperfections in the behavior of real gases.
Pierre-Simon Laplace later realized that sound waves propagate much too quickly for significant heat transfer to occur during each compression and expansion.
Instead of being isothermal, the compressions and expansions are approximately adiabatic.
During compression, the gas becomes slightly warmer. During expansion, it becomes slightly cooler. Although these temperature changes are small, they have an important effect on the compressibility of the gas.
The appropriate compressibility is therefore the adiabatic compressibility:
\[ \kappa_S = -\frac{1}{V} \left( \frac{\partial V} {\partial p} \right)_S \]
Since adiabatic processes are also isentropic,
\[ dS = 0 \]
the entropy remains constant as the sound wave passes through the gas.
In Chapter 4, we showed that
\[ \kappa_S = \kappa_T \frac{C_V}{C_P} \]
Since
\[ C_P > C_V \]
it follows that
\[ \boxed{ \kappa_S < \kappa_T } \]
A gas is therefore less compressible during an adiabatic compression than during an isothermal compression.
Physically, this occurs because the temperature rises during compression, causing the pressure to increase more rapidly than it would under isothermal conditions.
The gas effectively behaves as a stiffer spring, allowing sound waves to travel more quickly.
The speed of sound in a gas is given by
\[ \boxed{ v = \sqrt{ \frac{1} {\rho\kappa_S} } } \]
where \(\rho\) is the density of the gas.
Using the adiabatic compressibility rather than the isothermal compressibility brings the theoretical prediction into agreement with experiment, resolving the discrepancy in Newton's original calculation.
Big picture: Sound waves involve rapid compressions and expansions of a gas. Because these changes occur too quickly for significant heat transfer, they are adiabatic rather than isothermal. Laplace's recognition of this fact corrected Newton's calculation and showed that the speed of sound depends on the adiabatic compressibility of the gas.
Calculate the speed of sound in nitrogen gas at \(25^\circ\mathrm{C}\) and \(1.00\ \mathrm{atm}\), assuming ideal-gas behavior.
For an ideal gas, the speed of sound is given by
\[ v = \sqrt{ \frac{\gamma RT} {M} } \]
where
For nitrogen gas,
\[ \gamma \approx 1.40 \]
and
\[ M = 28.0\ \mathrm{g\,mol^{-1}} = 0.0280\ \mathrm{kg\,mol^{-1}} \]
The temperature must be expressed in Kelvin:
\[ T = 25 + 273.15 = 298.15\ \mathrm{K} \]
Substituting into the equation:
\[ v = \sqrt{ \frac{ (1.40) (8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}) (298.15\ \mathrm{K}) } { 0.0280\ \mathrm{kg\,mol^{-1}} } } \]
\[ v = \sqrt{ 1.24\times10^5 \ \mathrm{m^2\,s^{-2}} } \]
\[ v = 352\ \mathrm{m\,s^{-1}} \]
\[ \boxed{ v = 352\ \mathrm{m\,s^{-1}} } \]
Test of reasonableness: The commonly quoted speed of sound in air at room temperature is approximately \(343\ \mathrm{m\,s^{-1}}\). Since nitrogen is the major component of air (about 78%), it is reasonable that the calculated value is very close to the speed of sound measured in air.
The small difference arises because air is a mixture of gases rather than pure nitrogen.
Big picture: Sound waves propagate through gases as adiabatic compression and expansion waves rather than isothermal ones. Laplace's correction to Newton's original model revealed that the speed of sound is governed by the adiabatic compressibility of the gas and depends on both the thermodynamic properties of the gas and its molecular mass.