## What is Pressure?

Pressure is defined as force per unit surface area. Where does this force originate from in a container filled with gas? Gas consists of a large number of particles (atoms, molecules) which are permanently in motion. When they collide with a surface, their recoil produces a force. Pressure is the sum of these forces produced by particles on unit surface area. The unit, in which the pressure is measured, is the Pascal (Pa = N/m2, Newton per square meter). Another valid unit of pressure is the bar (1 bar = 105 Pa). The standard atmospheric pressure (dry air at sea level) is 1.013 bar. This value comes from the older, and no longer used unit the Torr (760 Torr = 760 mm mercury column).

## What is Vacuum?

Vacuum is produced by removing gas from a container. It is not possible to remove all the gas from a container by pumping; particles still remain in the container, producing a pressure. Zero pressure is therefore not possible in practice. The pressure relative to the absolute vacuum is called the absolute pressure. Thus there is still a positive pressure even in a good vacuum. Atmospheric pressure is about 1 bar absolute, and manometers which zero at atmospheric pressure and show negative in vacuum, measure a relative pressure against atmospheric pressure. Note that the negative values do not represent a negative absolute pressure.

A gas having a negative pressure relative to the atmosphere is called vacuum. By definition there are four levels of vacuum, depending on the size of the negative relative pressure:

rough vacuum
fine vacuum
high vacuum
ultra-high vaccum
1013 - 1 mbar
1 - 10-3 mbar
10-3 - 10-7 mbar
<10-7 mbar
1.013 x 105 - 100 Pa
100 - 0.1 Pa
0.1 - 10-5 Pa
<10-5 Pa

## Composition of Air

Air is a mixture of different gases. More than 99% of air consist of nitrogen and oxygen.

The rest are other gases.

Gas
Nitrogen
Oxygen
Argon
Carbon dioxide
Neon
Helium
Krypton
Hydrogen
Xenon
Percent
79,119%
21,224%
0,946%
~0,0334%
0,00184%
0,00053%
0,000116%
0,00005%
0,0000088%

Table 1

## What is Partial Pressure?

Partial pressure is produced by a single gas in a mixture of gases. The table below shows the percentage composition of air. At a pressure of 1.013 bar the oxygen content is 21.224%, the partial pressure of oxygen in air is = 0.21224 bar or 212.24 mbar. The table can therefore be interpreted for partial pressures of the gas mixture in air

Gas
Nitrogen
Oxygen
Argon
Carbon dioxide
Neon
Helium
Krypton
Hydrogen
Xenon
Partial pressure
791.19 mbar
212.24 mbar
9.46 mbar
3.34 x 10-1 mbar
1.84 x 10-2 mbar
5.3 x 10 -3 mbar
1.16 x 10-3 mbar
5.0 x 10 -4 mbar
8.8 x 10 -5 mbar

Table 2
The table shows the partial pressures of the gases in air in relation
to the Standard pressure of dry air at sea level = 1013 mbar.

## Dalton's Law

Dalton's law gives the relation between the total pressure of a gas in a container and the partial pressure of its constituents.

The sum of all partial pressures in a mixture of gases is equal to the total pressure.

$p_t_o_t_a_l=p_1+p_2+p_3+{_\dots}+p_n$

Note that:

The gas mixture in a closed volume does not separate. In an air-to-helium mixture, the helium does not rise to the top of the container even though it is lighter than the other gases. This is because of the thermal speed of the particles there are inter-particle collisions in the gas as well as with the container walls and therefore always a mixing process, called "diffusion"

## What is Gas?

Gas is a substance, where the particles (molecules and atoms) can freely move. In thermodynamic equilibrium these particles are uniformly distributed in space, so that the pressure, partial pressures and gas composition are the same at all points of the container.

A solid is a substance, the particles are fixed in their positions, they can only vibrate and rotate. A liquid is a substance, where particles can freely slide on each other but cannot separate from each other, hindered by intermolecular forces.

## What is Vapour Pressure?

When the pressure above a liquid is reduced at constant temperature, it evaporates and the resulting gas is called a vapour. This vapour, like other gases, has a pressure called vapour pressure. The vapour can become a liquid again. This is called condensation. All vapours have a saturation pressure which is the pressure of the vapour in equilibrium with its liquid. When the pressure in a system is higher than the saturation pressure, the vapour condenses, causing the pressure to return to the saturation value. Vice versa, as long as there is still a liquid in the system, the pressure cannot be reduced (by pumping) lower than the saturation pressure, because the liquid continues to evaporate.

In a container, in which a liquid and its vapour are present and the pressure is equal to the saturation vapour pressure, evaporation and condensation occur simultaneously. Water and its vapour pressure in vacuum systems need our special attention, because water is always present as humidity in the air and water is difficult to remove from vacuum systems.

We call gases vapour, when they can condense at normal temperatures. Strictly speaking all gases can condense, it depends only how much the temperature is reduced.

Vapour pressure of water at different temperatures.

Temp in C
100
50
25
0
-40
-78,5
-196
Pressure in mbar
1010.8
130.3
22.6
7.3
0.13
7.7 x 10-3
1.3 x 10-9

Table 3

Vapour pressure of some liquids at 20° C

Liquid
Methyl alcohol
Ethyl alcohol
Acetone
Water
Tetrachlorcarbon
High Vacuum-Diffusion Pump Oil
Pressure in mbar
127.7
58.3
245.8
23.37
121.0
10-7 bis 10-9

Table 4

## What is mean free path?

The mean free path (MFP) is the average distance which a molecule of a gas- or vapour travels before it collides with another molecule. At high vacuum, these distances become very large, so the particles collide only with the walls of the vacuum chamber.

Pressure
Quantity per cm3
MFP
1013 mbar
3 x 1019
6,8 x 10-5 mm
1 x 10-3 mbar
3 x 1013
6,8 cm
1 x 10-9 mbar
3 x 107
6,8 km

The mean free path is proportional to the pressure. At atmospheric pressure the particles collide every 10 thausandths of a millimetre, but at a pressure of 109 mbar they collide only after 68 km. The following formula gives the mean free path for air at 200C at pressure P mbar.

$MFP = {6,8 \cdot 10^{-3}\,mbar \cdot cm \over p_{mbar}}$

## The mean free path for some gases at 20° C and different pressures

Pressure (in Pa)
Pressure
(in mbar)
mean free path
Air
Argon
CO2
Hydrogen
Water vapour
Helium
Nitrogen
Neon
Oxygen
10-6
10-8

km
6,8
7,2
4,5
12,5
4,2
19,6
6,7
14,0
7,2
10-3
10-5

m
6,8
7,2
4,5
12,5
4,2
19,6
6,7
14,0
7,2
1
10-2

mm
6,8
7,2
4,5
12,5
4,2
19,6
6,7
14,0
7,2
103
10

μm
6,8
7,2
4,5
12,5
4,2
19,6
6,7
14,0
7,2
105
1000

nm
68
72
45
125
42
196
67
140
72

Table 5
1 nm [Nanometer] = 10-9 m

Equal volumes of gases at the same temperature and pressure contain equal numbers of gas molecules.

## What is a Mole?

A Mole (abbreviated to "mol") is a unit of quantity. The definition is:

The weight of each gas in a volume of 22.415 litres (at 0°C and 1,013 bar) is equal to the relative mass of one molecule of this gas.

Example: 1 mol Helium = 22.415 litre has a weight of 4 gm.

Units of quantity can be expressed in different units:

Volume V
Mass M
Number of particles N
Amount of substance n
if it is a gas: "p·V - Value" p·V
[V]
[M]
[N]
[n]
[p·V]
m3, l, cm3
kg, g
1
kmol, mol
Pa·m3 = N·m, mbar·l

The characterisation of an amount of gas as p x V-value is mostly used in vacuum technology.

## What is a leak rate?

A leak rate is an amount of substance, which passes in a certain time through a leak. As we have seen, there are several units for quantity, so the leak rates can also be expressed in different units. The symbol for the leak rate is "q".

Normally used are:

The volume leak rate q =
The mass leak rate q =
Pa·m3/s, mbar·l/s
g/a (gram per year)

Each gas in 1 mol (= 22.415 litre) at 0°C and 1,013 bar (760 Torr) has 6.023 x 1023 particles.

## The relative masses of molecules

In the past these masses have been called "atomic weight or molecular weight", but this was misleading, because these numbers are only ratios between atoms or molecules.

The relative mass of molecules M (also called molar Mass) at SI-units is defined in kg/kmol. One hypothetical standard atom has the relative mass of 1/12th of the 12 C atom. The relative mass of a Helium atom is then:

M = 4 kg / kmol
or
M = 4 g / mol

## Boyle's Law

When the volume of a gas is decreased (for example by a piston), the pressure increases in the same ratio, if the temperature does not change. Compressing the volume to a half of its size doubles the pressure.

$p1 \cdot V1 = p2 \cdot V2$

also well known as:

$p \cdot V = \textup {constant}$

When a vacuum is produced and the pressure is decreased by a large amount, the gases expand to very large volumes. This occurs also with contamination on the inner walls of the vacuum chamber, which evaporates (for example fingerprints).

with equal numbers of molecules (temperature kept constant)

for the same number of molecules (constant temperature)

## Charles' Law

The volume of a gas changes with temperature if the pressure is kept constant. When the gas becomes colder, the volume decreases. As it becomes warmer, the gas expands.

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$

T in Kelvin (see scale of absolute temperature below).

When the temperature doubles, the volume increases similarily by the factor of 2, provided that the pressure stays constant.

with equal numbers of molecules and constant pressure

With a closer look at Charles` law, one can find a more detailed relationship between temperature and volume, known as the law of Gay-Lussac:

## The Law of Gay-Lussac

When the temperature of a gas changes from 0°C by one degree, the volume changes by 1/273 of the original value:

$V = V_0 + \frac{\Delta T}{273k} \times V_0$

or

$V = V_0 \left(1+ \frac{\left \Delta T}{273k} \right)$

From this law, Lord Kelvin founded the absolute scale for the temperature, in which 0 Kelvin = absolute zero = 273.15 degrees Celsius.

Thermodynamic Temperature T
in Kelvin [K]
373.15
273.15
263.15
10
0
Celsius Temperature in [0°C]

100
0
-10
-263.15
-273.15
Boyles' Law
describes the relationship between volume and pressure, where temperature and mass (number of particles) must stay constant.
Charles' Gesetz
describes the relationship between volume and temperature, where pressure and mass stay constant.
describes the relationship between volume and mass, where temperature and pressure stay constant.

The above three law´s combine to give the universal gas law:

## The universal Gas Law

$p \cdot V = \nu \cdot R \cdot T$

Greek ν(Ny) is the symbol for the quantity in mol or kmol.

The universal gas law is not restricted like Charles´, Boyle´s and Avogadro´s laws.

The temperatures in Charles law and in the universal gas law are inserted as absolute temperatures in Kelvin.

P =
V =
T =
ν =
R =
pressure
volume
Temperatur
quantity
universal gas constant
N/m2= Pa
m3
K
mol

1 bar = 105Pascal
cubic metre
Kelvin

## The Universal Gas Constant

$R = \frac{p \cdot V}{\nu \cdot T}= 8,314 J/(\textup{mol} \cdot \textup K)$

1 J (Joule) = 1 Nm = 1 Ws

For calculations, the following, equivalent values are often used:

$R = 83,14 \, mbar \cdot l / mol \cdot K$

$R = 8,314 \cdot 10^{4} \, mbar \cdot l / kmol \cdot K$

$R = 8,314 \cdot 10^{3} \, Pa\:m^3 / kmol \cdot K$

## Individual gas constants for some gases (J/kg x K)

Gas
Air
Argon
Helium
Hydrogen
Nitrogen
Oxygen
Water vapour
R - 12
R - 22
Ri
287
208
2079
4116
297
260
461
68.7
96.1

If many calculations are made for the same gas, it is often easier to use the individual gas constant Ri

$p \times V = \nu \times Ri \times T$

Α α
Β β
Γ γ
Δ δ
Η η
Ζ ζ
Η η
Θ θ
Ι ι
Κ κ
Λ λ
Μ μ
Alpha
Beta
Gamma
Delta
Eta
Zeta
Eta
Theta
Jota
Kappa
Lambda
My
Ν ν
Ξξ
Ο ο
Π π
Ρ ρ
Σ σ
Τ τ
Υ υ
Φ φ
Χ χ
Ψ ψ
Ω ω
Ny
Ksi
Omikron
Pi
Rho
Sigma
Tau
Ypsilon
Phi
Chi
Psi
Omega

## The Individual Gas Constant

If the universal gas constant is divided by the relative mass number (M) of the molecule of the gas in question, the result is the individual gas constant Ri for this gas.

$Ri = \frac{R}{M} = \frac{p \times V}{\nu \times M \times T} = J/kg \times K$