As already stated above the perfect gas is a limiting case which strictly hols only for \(p = 0\) bar where repulsive or attractive forces between particles are always negligible. Famous scientists have contributed to its equation of state by experimental findings:
Boyle’s law (at const. \(n, T\)): \(p \propto 1/ V\)
Charles’s laws (at const. \(n, p\) or at const. \(n, V\)): \(V \propto T\) resp. \(p \propto T\)
Avogadro’s principle (at const. \(p, T\)): \(V \propto n\)
Combined one finds:
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| \begin{equation*} p\; V = n \;R \;T \label{eq:perfect_gas} \end{equation*} | (1.3) |
\(n =\) number of particles in moles; 1 mol = \(6.022 \times 10^{23}\) molecules
\(R =\) gas constant = 8.314 J/(mol K)
\(V_m =\) molar volume of perfect gas = \(V/n = 22.414\) dm\(^3\)mol\(^{-1}\)
Determination of \(R\): a) evaluation of perfect gas law at low pressure, b) measurement of the speed of sound and extrapolation to \(p = 0\) bar
The partial pressures \(p_i\) of perfect gas mixtures just add up, i.e.
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| \begin{equation*} p_{tot} = \sum_i p_i = \sum_i x_i \;p_{tot} = \sum_i \frac{n_i}{n_{tot}} \frac{n_{tot}RT}{V_{tot}} = \sum_i n_i \frac{R\;T}{V_{tot}} \label{eq:partial_pressure} \end{equation*} | (1.4) |
Here the first equals sign holds for any gas while the second equals sign only holds for
a perfect gas. \(x_i = n_i / n_{tot}\) is the fraction of component type \(i\) in the mixture.
This equation reflects Dalton’s law: ”The pressure exerted by a mixture of gases is the sum
of the pressures that each one would exert if it occupied the container ALONE” (only valid for perfect gases).
A second law which has been found experimentally holds for the (inner) energy \(U\) of a perfect
gas:
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| \begin{equation*} U = \frac{3}{2}\; n \;R\; T \label{eq:equipartlaw_perfect_gas} \end{equation*} | (1.5) |
This equation reflects that neither repulsive nor attractive forces act on the molecules; so the energy cannot depend on the distance between particles, i.e. on the volume \(V\), and according to the perfect gas equation not on the pressure \(p\) correspondingly.
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© J. Carstensen (TD Kin I)