How is the mean free path of gas molecules calculated?

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Definitions of related concepts

Leak rate q_L (mbar ·l · s^–1)

According to the definition formulated above, it is easy to understand that the size of a gas leak, i.e. movement through undesired passages or “pipe” elements, will also be given in mbar · l · s^–1. A leak rate is often measured or indicated with atmospheric pressure prevailing on the one side of the barrier and a vacuum at the other side (p < 1 mbar). If helium (which may be used as a tracer gas, for example) is passed through the leak under exactly these conditions, then one refers to “standard helium conditions”. For more information, see the section on leak detection.

Outgassing (mbar · l)

The term outgassing refers to the liberation of gases and vapors from the walls of a vacuum chamber or other components on the inside of a vacuum system. This quantity of gas is also characterized by the product of p · V, where V is the volume of the vessel into which the gases are liberated, and by p, or better Δp, the increase in pressure resulting from the introduction of gases into this volume.

Outgassing rate (mbar · l · s^–1)

This is the outgassing through a period of time, expressed in mbar · l · s^–1.

Outgassing rate (mbar · l · s^–1 · cm^–2) (referenced to surface area)

In order to estimate the amount of gas which will have to be extracted, knowledge of the size of the interior surface area, its material and the surface characteristics, their outgassing rate referenced to the surface area and their progress through time are important.

Mean free path of the molecules λ (cm) and collision rate z (s^-1)

The concept that a gas comprises a large number of distinct particles between which – aside from the collisions – there are no effective forces, has led to a number of theoretical considerations which we summarize today under the designation “kinetic theory of gases”.

One of the first and at the same time most beneficial results of this theory was the calculation of gas pressure p as a function of gas density and the mean square of velocity c2 for the individual gas molecules in the mass of molecules mT:

(1.14)

where

(1.15)

The gas molecules fly about and among each other, at every possible velocity, and bombard both the vessel walls and collide (elastically) with each other. This motion of the gas molecules is described numerically with the assistance of the kinetic theory of gases. A molecule’s average number of collisions over a given period of time, called the collision index z, and the mean path distance which each gas molecule covers between two collisions with other molecules, called the mean free path length λ, are described as shown below as a function of the mean molecule velocity c- the molecule diameter 2r and the particle number density molecules n – as a very good approximation:

where

(1.16)

and

(1.18)

Thus, the mean free path length λ for the particle number density n is, in accordance with equation (1.1), inversely proportional to pressure p. Thus, the following relationship holds, at constant temperature T, for every gas

(1.19)

λ ⋅ p = const (1.19)

Table III and fig 9.1 are used to calculate the mean free path length λ for any arbitrary pressures and various gases. The equations in gas kinetics which are most important for vacuum technology are also summarized in Table IV.

Table III Mean free path l Values of the product c* of the mean free path λ ( and pressure p for various gases at 68°F or 20°C (see also Fig. 9.1)

Fig 9.1 Variation of mean free path λ (cm) with pressure for various gases

Table IV Compilation of important formulas pertaining to the kinetic theory of gases

Impingement rate z_A(cm^–2 ⋅ s^–1) and monolayer formation time τ (s)

A technique frequently used to characterize the pressure state in the high vacuum regime is the calculation of the time required to form a monomolecular or monoatomic layer on a gas-free surface, on the assumption that every molecule will stick to the surface. This monolayer formation time is closely related with the impingement rate zA. With a gas at rest, the impingement rate will indicate the number of molecules which collide with the surface inside the vacuum vessel per unit of time and surface area:

(1.20)

If a is the number of spaces, per unit of surface area, which can accept a specific gas, then the monolayer formation time is

(1.21)

Collision frequency z_v (cm^–3 · s^–1)

This is the product of the collision rate z and the half of the particle number density n, since the collision of two molecules is to be counted as only one collision: