How does gas behave in a vacuum? A definition of the ideal gas law

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Continuum theory

Model concept: Gas is “pourable” (fluid) and flows in a similar way to a liquid. The continuum theory and the summarization of the gas laws which follows are based on experience and can explain all the processes in gases near atmospheric pressure. Only after it became possible using ever better vacuum pumps to dilute the air to the extent that the mean free path rose far beyond the dimensions of the vessel were more far-reaching assumptions necessary; these culminated in the kinetic gas theory. The kinetic gas theory applies throughout the entire pressure range; the continuum theory represents the (historically older) special case in the gas laws where atmospheric conditions prevail.

Summary of the most important gas laws (continuum theory)

Boyle-Mariotte Law

p · V = const.
for T = constant (isotherm)

Gay-Lussac’s Law (Charles’ Law)

for p = constant (isobar)

Amonton’s Law

for V = constant (isochor)

Dalton’s Law

Poisson’s Law

Avogadro’s Law

Ideal gas Law

Also: Equation of state for ideal gases (from the continuum theory)

van der Waals’ Equation

a, b = constants (internal pressure, covolumes)
V_m = Molar volume
also: Equation of state for real gases

Clausius-Clapeyron Equation

L = Enthalpy of evaporation,
T = Evaporation temperature,
V_m,v, V_m,l = Molar volumes of vapor or liquid

Kinetic gas theory

With the acceptance of the atomic view of the world – accompanied by the necessity to explain reactions in extremely dilute gases (where the continuum theory fails) – the “kinetic gas theory” was developed. Using this it is possible not only to derive the ideal gas law in another manner but also to calculate many other quantities involved with the kinetics of gases – such as collision rates, mean free path lengths, monolayer formation time, diffusion constants and many other quantities.

Model concepts and basic assumptions:

Atoms/molecules are points.
Forces are transmitted from one to another only by collision.
The collisions are elastic.
Molecular disorder (randomness) prevails.

A very much simplified model was developed by Krönig. Located in a cube are N particles, one-sixth of which are moving toward any given surface of the cube. If the edge of the cube is 1 cm long, then it will contain n particles (particle number density); within a unit of time n · c · Δt/6 molecules will reach each wall where the change of pulse per molecule, due to the change of direction through 180°, will be equal to 2 · m_T · c. The sum of the pulse changes for all the molecules impinging on the wall will result in a force effective on this wall or the pressure acting on the wall, per unit of surface area.

image not found: The ideal gas law 1

Ideal gas law as derived from the kinetic gas theory

If one replaces c2 with c2– then a comparison of these two “general” gas equations will show:

The expression in brackets on the left-hand side is the Boltzmann constant k; that on the right-hand side a measure of the molecules’ mean kinetic energy:

Boltzmann constant

Mean kinetic energy of the molecules

In this form the gas equation provides a gas-kinetic indication of the temperature!

The mass of the molecules is

where NA is Avogadro’s number (previously: Loschmidt number).

Avogadro constant

Thus, from the ideal gas law at standard conditions
(T_n = 273.15 K and p_n = 1013.25 mbar):

For the general gas constant:

Definitions of units and basic equations

Particle number density n (cm^-3)

According to the kinetic gas theory the number n of the gas molecules, referenced to the volume, is dependent on pressure p and thermodynamic temperature T as expressed in the following:

(1.1)

n = particle number density
k = Boltzmann’s constant

At a certain temperature, therefore, the pressure exerted by a gas depends only on the particle number density and not on the nature of the gas. The nature of a gaseous particle is characterized, among other factors, by its mass m_T.

Gas density ρ (kg · m^-3, g · cm^-3)

The product of the particle number density n and the particle mass mT is the gas density
ρ:

(1.2)

The ideal gas law equation

The relationship between the mass mT of a gas molecule and the molar mass M of this gas is as follows:

(1.3)

Avogadro’s number (or constant) N_A indicates how many gas particles will be contained in a mole of gas. In addition to this, it is the proportionality factor between the gas constant R and Boltzmann’s constant k:

(1.4)

Derivable directly from the above equations (1.1) to (1.4) is the correlation between the pressure p and the gas density ρ of an ideal gas.

(1.5)

In practice we will often consider a certain enclosed volume V in which the gas is present at a certain pressure p. If m is the mass of the gas present within that volume, then

(1.6)

The ideal gas law then follows directly from equation (1.5):

(1.7)

Here the quotient m / M is the number of moles υ present in volume V.
The simpler form applies for m / M = 1, i.e. for 1 mole:

(1.7a)

The following numerical example is intended to illustrate the correlation between the mass of the gas and pressure for gases with differing molar masses, drawing here on the numerical values in Table IV. Contained in a 2 gallon (10 liter) volume, at 68°F (20°C), will be
a) 1g of helium
b) 1g of nitrogen
When using the equation (1.7) there results then at V = 10l , m = 1g,

In case a) where M = 4 g · mole^-1 (monatomic gas):
In case b), with M = 28 ≠ g mole^-1 (diatomic gas):

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

The result, though appearing to be paradoxical, is that a certain mass of a light gas exerts a greater pressure than the same mass of a heavier gas. If one takes into account, however, that at the same gas density (see Equation 1.2) more particles of a lighter gas (large n, small m) will be present than for the heavier gas (small n, large m), the results become more understandable since only the particle number density n is determinant for the pressure level, assuming equal temperature (see Equation 1.1).

The main task of vacuum technology is to reduce the particle number density n inside a given volume V. At constant temperature this is always equivalent to reducing the gas pressure p. Explicit attention must at this point be drawn to the fact that a reduction in pressure (maintaining the volume) can be achieved not only by reducing the particle number density n but also (in accordance with Equation 1.5) by reducing temperature T at constant gas density. This important phenomenon will always have to be taken into account where the temperature is not uniform throughout volume V.

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