Pfeiffer Vacuum

1.2.2 General gas equation

Each material consists of atoms or molecules. By definition, the amount of substance is indicated in moles. One mole of a material contains 6.022 · 1023 constituent particles (Avogadro constant. This is not a number but a physical magnitude with the unit mol-1). 1 mole is defined as the amount of substance of a system which consists of the same number of particles as the number of atoms contained in exactly 12 g of carbon of the nuclide 12C.

Under normal conditions, i.e. a pressure of 101,325 Pa and a temperature of 273.15 K (equals 0°C), one mole of an ideal gas fills a volume of 22.414 liters.

As early as 1664, Robert Boyle studied the influence of pressure on a given amount of air. The results confirmed by Mariotte in experiments are summarized in the Boyle-Mariotte law:

\[p\cdot V = \mbox{const.}\]

Formula 1-4: Boyle-Mariotte law [4]

Expressed in words the Boyle-Mariotte law states that the volume of a given quantity of gas at a constant temperature is inversely proportional to the pressure – the product of the pressure and volume is constant.

Over a hundred years later, the temperature dependence of the volume of a quantity of gas was also identified: the volume of a given quantity of gas at a constant pressure is directly proportional to the absolute temperature or

\[V = \mbox{const.}\cdot T\]

Formula 1-5: Gay-Lussac’s law

Subjecting a given quantity of gas successively to a change in pressure and a change in temperature results in

\[\frac{p\cdot V}T=\mbox{const.}\]

This still applies for a given quantity of gas. The volume of gas at a given temperature and a given pressure is proportional to the quantity of material $\nu$. We can therefore write:

\[\frac{p\cdot V}T=\nu\cdot\mbox{const.}\]

The quantity of material is determined by weighing. We can express the quantity of gas by the ratio of mass divided by the molar mass. The constant const. refers here to 1 mole of the gas in question, and it is referred to as the gas constant $R$. As a result, the state of an ideal gas can be described as follows as a function of pressure, temperature and volume:

\[p\cdot V=\frac m M \cdot R \cdot T\]

Formula 1-6: General equation of state for ideal gases [5]

$p$ Pressure [Pa]
$V$ Volume [m3]
$m$ Mass [kg]
$M$ Molar mass [kg kmol-1]
$R$ General gas constant [kJ kmol-1 K-1]
$T$ Absolute temperature [K]

The amount of substance $\nu$ can also be indicated as the number of molecules in relation to the Avogadro constant.

\[p\cdot V=\frac N {N_A} \cdot R \cdot T = N \cdot k \cdot T \quad\mbox{where } k=\frac R {N_A}\]

Formula 1-7: Equation of state for ideal gases I

$N$ Number of particles    
$N_A$ Avogadro constant = 6.022 · 1023 [mol-1]
$k$ Boltzmann constant = 1.381 · 10-23 [J K-1]

If both sides of the equation are now divided by the volume, then we obtain

\[p=n\cdot k\cdot T\]

Formula 1-8: Equation of state for ideal gases II

$n$ Particle number density [m-3]