Ideal Gas Law

The ideal gas law can be written as

$$PV \eqsp nRT \eqsp NkT \protect$$ (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

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

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 [197,534,535,536,102,101].