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Ideal Gas Law

The ideal gas law can be written as

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

where

\begin{eqnarray*}
P &=& \mbox{total pressure (Pascals)}\\
V &=& \mbox{volume (cubic meters)}\\
n &=& \mbox{number of \emph{moles}\index{moles\vert textbf} of gas in the volume $V$}\\
R &=& 8.3143 \eqsp \mbox{\emph{ideal gas constant}\index{ideal gas constant\vert textbf} (SI units)}, and\\
T &=& \mbox{\emph{absolute temperature}\index{absolute temperature\vert textbf} (degrees Kelvin).}
\end{eqnarray*}

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 [199,52]. Additionally, the aeroacoustics of air jets is nonlinear [197,532,533,534,102,101].


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA