# Speed in ideal gases and in air

For a gas, K (the bulk modulus in equations above, equivalent to C, the coefficient of stiffness in solids) is approximately given by thus Where: is the adiabatic index also known as the isentropic expansion factor. It is the ratio of specific heats of a gas at a constant-pressure to a gas at a constant-volume(), and arises because a classical sound wave induces an adiabatic compression, in which the heat of the compression does not have enough time to escape the pressure pulse, and thus contributes to the pressure induced by the compression. p is the pressure. is the density Using the ideal gas law to replace with nRT/V, and replacing ? with nM/V, the equation for an ideal gas becomes: where is the speed of sound in an ideal gas. (approximately 8.3145 J·mol?1·K?1) is the molar gas constant. is the Boltzmann constant (gamma) is the adiabatic index (sometimes assumed 7/5 = 1.400 for diatomic molecules from kinetic theory, assuming from quantum theory a temperature range at which thermal energy is fully partitioned into rotation (rotations are fully excited), but none into vibrational modes. Gamma is actually experimentally measured over a range from 1.3991 to 1.403 at 0 degrees Celsius, for air. Gamma is assumed from kinetic theory to be exactly 5/3 = 1.6667 for monoatomic molecules such as noble gases). is the absolute temperature in kelvin. is the molar mass in kilograms per mole. The mean molar mass for dry air is about 0.0289645 kg/mol. is the mass of a single molecule in kilograms. This equation applies only when the sound wave is a small perturbation on the ambient condition, and the certain other noted conditions are fulfilled, as noted below. Calculated values for have been found to vary slightly from experimentally determined values. Newton famously considered the speed of s und before most of the development of thermodynamics and so incorrectly used isothermal calculations instead of adiabatic. His result was missing the factor of but was otherwise correct. Numerical substitution of the above values gives the ideal gas approximation of sound velocity for gases, which is accurate at relatively low gas pressures and densities (for air, this includes standard Earth sea-level conditions). Also, for diatomic gases the use of requires that the gas exist in a temperature range high enough that rotational heat capacity is fully excited (i.e., molecular rotation is fully used as a heat energy "partition" or reservoir); but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity (i.e., insignificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode, have energies too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See the section on gases in specific heat capacity for a more complete discussion of this phenomenon. For air, we use a simplified symbol . Additionally, if temperatures in degrees Celsius(°C) are to be used to calculate air speed in the region near 273 kelvin, then Celsius temperature may be used. Then: For dry air, where (theta) is the temperature in degrees Celsius(°C). Making the following numerical substitutions: is the molar gas constant in J/mole/Kelvin; is the mean molar mass of air, in kg; and using the ideal diatomic gas value of Then: Using the first two terms of the Taylor expansion: The derivation includes the first two equations given in the Practical formula for dry air section above.