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Ideal Gas Law

The ideal gas law can be written as

$\displaystyle PV \eqsp nRT \eqsp NkT \protect$ (B.45)

P &=& \mbox{total pressure (Pascals)}\\
V &=& \mbox{volume (c...
...ture}\index{absolute temperature\vert textbf} (degrees Kelvin).}
The alternate form $ PV=NkT$ comes from the statistical mechanics derivation in which $ N$ is the number of gas molecules in the volume, and $ k$ is Boltzmann's constant. In this formulation (the kinetic theory of ideal gases), the average kinetic energy of the gas molecules is given by $ (3/2) kT$. Thus, temperature is proportional to average kinetic energy of the gas molecules, where the kinetic energy of a molecule $ m$ with translational speed $ v$ is given by $ (1/2)mv^2$. 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 $ m$ of the gas in volume $ V$ is given by $ m=nM$, where $ M$ is the molar mass of the gass (about 29 g per mole for air). The air density is thus $ \rho=m/V$ so that we can write

$\displaystyle P \eqsp \frac{R}{M} \rho T.

That is, pressure $ P$ is proportional to density $ \rho$ at constant temperature $ T$ (with $ R/M$ 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 $ P_0$. The physical pressure is then $ P=P_0+p$, where $ p$ is the usual acoustic pressure-wave variable. That is, we are only concerned with small pressure perturbations $ p$ in typical audio acoustics situations, so that, for example, variations in volume $ V$ and density $ \rho$ 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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