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Vacuum Technology
Calculations in Chemistry
By D. J. Hucknall, A. Morris The Royal Society of Chemistry
Copyright © 2003 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-1-84755-227-3
CHAPTER 1
Principles
1.1 INTRODUCTION
The purpose of this chapter is to state some of the basic ideas and assumptions that underpin vacuum technology.
To familiarise the reader with the scope of vacuum technology, it begins with a summary of the vacuum pressure range and typical applications in the chemical sciences.
The equation of state for a perfect gas is presented and expressions arising from this for pure gases and gas mixtures are given. The kinetic theory of gases, which is a useful model of perfect gases, is introduced and two particularly useful results are emphasised. These are the mean free path ([bar.l]) and the mean or thermal velocity ([bar.c]). Of particular importance is [bar.c]/4, which is numerically equal to the volume rate of flow per unit area and which can be used to determine quantities such as area-related pumping speeds, conductances, etc.
The transport properties, particularly viscosity and diffusion, of a perfect gas are discussed and the concepts of gas dynamics are briefly mentioned, Such methods can be applied to flowing gas in, for example, pipework or nozzles and jets.
1.2 SURVEY
Typical vacuum processes and plants can be classified according to the pressure regions in which they operate. These regions are shown overleaf.
1.3 PRESSURE
The term 'vacuum' is applied to pressures below, often considerably below, atmospheric pressure.
The object of vacuum technology is to reduce the number density of gas particles in a given volume of a system. At constant temperature, this
Rough
Medium
High
UHV-XHV
vacuum vacuum vacuum
Total
pressure
/mbar 10-3-1 1-10-3 10-3<10-7
Typical applications -10-7 -<10-10
Rough
Chemical technology (unit operations such as degassing,
drying, filtration)
Medium
Chemical technology (distillation)
Chemical Vapour Deposition (CVD)
Sputtering processes
High
Physical Vapour Deposition (PVD) (coating of various
substrates with a range of materials)
GC-MS
Kinetic studies
UHV–XHV Surface science (e.g.AES, XPS, EELS)
Fusion research
Synchrotron radiation sources (various applications)
always corresponds to a reduction in gas pressure. The system pressure is just a convenient index of number density.
Pressure is the result of molecules, within a fluid, colliding with the walls of the containing vessel. Its magnitude depends on the force of the impacts exerted perpendicular to a defined area. The relationship between pressure (p), force (F) and area (A) is:
p = F/A (1.1)
A force of 1 Newton (1 N = 1 kg.m s-2) on an area of 1 m2 exerts a pressure of 1 Pascal (Pa). Acceptable forms are kg.m s-2, kg m s-2, m kg s-2.
Other permissible pressure units are:
bar = 105Pa
millibar (mbar) = 102Pa
Although obsolete, the unit 'Torr' continues to be used. It is the pressure exerted by a mercury column of height 1 mm on the base area of the column at 0°C.
1 torr = 133.322 Pa [congruent to] 1.33 mbar
Commonly used instruments, e.g. Pirani, capacitance diaphragm and ionisation gauges, for pressure measurement in vacuum systems indicate the total pressure. This is equal to the sum of the partial pressures of the individual components in a system. Unless otherwise stated, the term 'pressure' in vacuum technology invariably means total pressure.
The normal state of a gas is determined by the normal temperature (Tn) and normal pressure (pn):
Tn = 0 °C = 273.15 K
pn = 101325Pa = 1013.25mbar
pn is also referred to as 'standard atmospheric pressure'. Atmospheric pressure is therefore regarded as approximately 1000 mbar.
1.4 IDEAL GAS LAW
With very few exceptions, gases and vapours at atmospheric pressure or below behave ideally. An ideal gas is one which obeys exactly the relationship:
pV = nRT (1.2)
where p is the pressure, V the volume, n the amount of substance of the gas, T the absolute temperature and R, the gas constant, is a fundamental constant independent of the nature of the gas.
The amount of substance (n) has units of mol (or mole) or kmol (or kmole). It is related to the number of particles. One mol (or kmol) of a substance contains as many particles (atoms, molecules, ions, etc.) as there are atoms, in 12 g (or 12 kg) of 12C. This is the Avogadro constant (NA).
NA = 6.022136 x 1023mol-1 = 6.02213 x 1026 kmol-1
If the mass of a substance is m and the molar mass of the substance is M, the amount of substance is:
n = m(g)/M(g mol-1)mol (1.3)
The molar gas constant (R) has a value:
R = 8.3145472JK-1mol-1 = 8.3145472kJK-1 kmol-1
In vacuum calculations, other units are often used:
R = 83.14 mbar L mol-1K-1 = 8.314 x 104mbar L kmol-1K-1
Based on the above, the molar volume of a perfect gas at pn and Tn is:
Vm = RTn/Pn (1.4)
= 83.145 mbar L.mol-1K-1 x 273.15K/ 1013.25 mbar
= 22.414L
The Boltzmann constant (k) is given by R/NA. It has a value:
k = 1.380650 x 10-23JK-1
The particle number density (n) can be obtained from:
pV = N/NART = NkT
p = nKT (1.5)
where N is the number of particles under consideration. Under normal conditions:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
At 293 K and pn:
n293K = 2.51 x 1025m-3 = 2.51 x 1022L-1
At 293 K:
In 1 L at 1 mbar, there are 2.478 x 1019 particles
In 1 L at 10-6mbar, there are 2.478 x 1013 particles
In 1 L at 10-10mbar, there are 2.478 x 109 particles
2.6867 x 1025 particles occupy 1 m3 at pn and Tn
1 mole of particles (6.022 x 1023) occupies 22.414 L at pn and Tn.
1.5 MIXTURES OF GASES: PARTIAL PRESSURES
If n mol of a gas (nA) is injected into a volume (V), then the pressure (pA) of A is given by:
pA = nA(RT/V) (1.6)
If, instead of A, n mol of gas B (nB) is injected into volume V, then the pressure (pB) of B is given by:
pB = nB (RT/V)
If B was injected into V already containing A then, according to Dalton's law of partial pressures, the total pressure (p) is given by:
p = pA + pB = (nA + nB) (RT/V)
For several components (A, B, C, D, etc.):
p = pA + pB + pC + pD + ...
This law can also be expressed in terms of component mole fractions. The mole fraction (x) of a component is given by:
xcomponent = ncomponent/ntotal (1.7)
where ntotal is the total number of moles in the system. Thus, in a mixture of gases (A, B, C, D, etc.) containing nA, nB, nC, nDetc., the mole fraction of A is:
xA = nA/ntotal (ntotal = nA + nB + nC + ...) (1-8)
Therefore, the partial pressure of a component i in n mol of a perfect gas mixture with a total pressure p is:
pi = ni(RT/V) = xin(RT/V) = xip (1.9)
Example 1.1
A 22.4L vessel contains 2 mol H2 and 1 mol N2 at Tn. What are the mole fractions and the partial pressures of the components in the vessel?
Total no. of moles in vessel = 3
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Since the molar volume (Vm) at pn and Tn is 22.414L then:
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Similarly, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
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Example 1.2
A 500 cm3 vessel contains a H2/N2 mixture at 500 mbar and Tn. If the partial pressure of H2, is 200 mbar what are the mole fractions of H2 and N2?
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Example 1.3
4 L of methane has a pressure of 500 mbar at 20 °C. What is the mass of the gas?
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Example 1.4
An excellent example of a gas mixture is atmospheric air. At sea level, the composition of dry air, in volume YO, is approximately as follows:
N2, 78.1; O2, 20.9; Ar, 0.9; CO2, 0.03; remainder 0.07.
What is the partial pressure of each when the total pressure is equal to pn and the temperature is Tn? Calculate the mean molar mass of air.
Based on the above, 100L of dry air contains approximately 78.1 L N2, 20.9 L O2, 0.9 L Ar, 0.03 L CO2 and 0.07 L of traces of Kr, He, CH4, etc.
At pn, Tn, the molar volume of a perfect gas is 22.414L mol-1. The molar composition of 100 L of dry air is, therefore:
N2 = 78.1/22.414 = 3.484 mol
O2 = 20.9/22.414 = 0.933 mol
Ar = 0.9/22.414 = 0.040 mol
CO2 = 0.00134 mol
remainder = 0.00312 mol
Total no. of moles = 4.4615 mol
The corresponding mole fractions are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The corresponding partial pressures at pn are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Mean molar mass:
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1.6 KINETIC THEORY OF GASES
A very useful model for the behaviour of an ideal gas is as follows:
• In a volume V, there are N particles of mass m.
• Each particle has a velocity [??] (strictly, this is a vector with components cx, cy and cz).
• Values of c cover the complete range between 0 and ∞.
• Particles can be regarded as point masses which exert no force on each other beyond a separation distance r >R (R is known as the interactive radius). At r< R there is an infinite repulsive force between the particles involved. They therefore behave like perfectly elastic spheres of radius R/2.
• On impact with each other, the velocities of the partners change in both value and direction.
• Particles colliding with the container walls are elastically reflected, creating gas pressure (p).
Because they undergo frequent collisions, particles will not have a constant velocity or move in specific directions such as the x-, y- or z-direction. The Maxwell–Boltzmann frequency distribution is used to describe the non-uniform distribution of particle velocities (c) brought about by collisions.
From the Maxwell–Boltzmann distribution, various velocities may be defined. These are:
The most probable velocity (cp = [square root of 2RT/M]m s-1 (1.10)
The mean velocity or thermal velocity (cav, [bar.c]) = [square root of 8RT/πM]m s-1 (1.11)
The root mean square velocity (rms) or effective velocity (ceff) = [square root of 3RT/M] m s-1 (1.12)
A particularly useful quantity in the case of a Maxwell distribution is:
Mean x-component of velocity = [bar.c]/4 (1.13)
= [square root of RT/2πM] m s-1 (1.14)
= 36.38 [square root of T/M] m s-1 (1.15)
[bar.c]/4 is equal to the mean normal component of velocity relative to any reference surface. It is numerically equal to the maximum area-related pumping speed (volume rate of flow per unit area) of an ideal pump for a gas at temperature T.
Example 1.5
Calculate the mean velocity of Ar atoms at 20°C.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Example 1.6
The interior of a spherical vessel of radius r = 0.5 m is covered with a monomolecular layer of gas particles each having a cross-sectional area (Apart) = 10-19m2.
What would be the pressure increase if all the particles were desorbed at a temperature of 300°C?
Pressure increase (Δp) is given by:
Δp = ΔnkT = N/V kT
where Δn = the number of particles in the gas phase after desorption.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Example 1.7
Calculate the effective speed of (i) N2 molecules at 1600°C. (ii) H2 molecules at 4K.
(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Example 1.8
At what temperature would:
(i) Ar atoms have a mean velocity ([bar.c]) = 500 m s-1?
(ii) 4He atoms have a mean velocity of 150 m s-1?
(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Wall flux density (area-related collision rate). A measure of the rate of flow of matter from one part of a system to another is given by the flux (flow rate per unit area). For the transportation of matter, this is given by:
jN = n[bar.c]/4 (1.16)
where n is particle number density (n = pkT = N/V), N is number of particles and [bar.c]/4 is mean normal component of velocity (mean or thermal speed of the gas in the x-direction).
(Continues...)
Excerpted from Vacuum Technology by D. J. Hucknall, A. Morris. Copyright © 2003 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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