Ideal Gas Law

The ideal gas law can be written as

where

The alternate form comes from the statistical mechanics derivation in which is the number of gas molecules in the volume, and is Boltzmann's constant. In this formulation (the kinetic theory of ideal gases), the average kinetic energy of the gas molecules is given by . Thus, temperature is proportional to average kinetic energy of the gas molecules, where the kinetic energy of a molecule with translational speed is given by .

In an ideal gas, the molecules are like little rubber balls (or rubbery assemblies of rubber balls) in a weightless vacuum, colliding with each other and the walls elastically and losslessly (an ``ideal rubber''). Electromagnetic forces among the molecules are neglected, other than the electron-orbital repulsion producing the elastic collisions; in other words, the molecules are treated as electrically neutral far away. (Gases of ionized molecules are called plasmas.)

The mass of the gas in volume is given by , where is the molar mass of the gass (about 29 g per mole for air). The air density is thus so that we can write

That is, pressure is proportional to density at constant temperature (with being a constant).

We normally do not need to consider the (nonlinear) ideal gas law in audio acoustics because it is usually linearized about some ambient pressure . The physical pressure is then , where is the usual acoustic pressure-wave variable. That is, we are only concerned with small pressure perturbations in typical audio acoustics situations, so that, for example, variations in volume and density can be neglected. Notable exceptions include brass instruments which can achieve nonlinear sound-pressure regions, especially near the mouthpiece [198,52]. Additionally, the aeroacoustics of air jets is nonlinear [196,530,531,532,102,101].

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.