Dr. Philip D. Rack Vacuum Technology Page 1 Kinetic Theory of Gas Dr. Philip D. Rack Assistant Professor Department of Materials Science and Engineering University of Tennessee 603 Dougherty Engineering Building Knoxville, TN 37931-2200 Phone: (865) 974-5344 Fax (865) 974-4115 Email: [email protected]Dr. Philip D. Rack Vacuum Technology Page 2 Vacuum Basics Gas Volume % Pressure (Pa) N 2 78 79,117 O 2 21 21,233 CO 2 0.033 33.4 Ar 0.934 946.4 Atmospheric Pressure = 101,323.2 Pa (760 torr) (133Pa = 1 torr)
Transcript
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Dr. Philip D. Rack
Vacuum Technology
Page 1
Kinetic Theory of Gas
Dr. Philip D. RackAssistant Professor
Department of Materials Science and Engineering
University of Tennessee603 Dougherty Engineering Building
Atmospheric Pressure = 101,323.2 Pa (760 torr)(133Pa = 1 torr)
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Dr. Philip D. Rack
Vacuum Technology
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Vacuum BasicsVacuum Pressure Range
(Pa)Low 105 > P > 3.3x103
Medium 3.3x103 > P >10-1
High 10-1 > P > 10-4
Very High 10-4 > P > 10-7
Ultra High 10-7 >P>10-10
Extreme Ultra-high
10-10 > P
Dr. Philip D. Rack
Vacuum Technology
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Kinetic Picture of an Ideal Gas• Volume of gas contains a large number of molecules• Adjacent molecules are separated by distances that are
large relative to the individual diameters• Molecules are in a constant state of motion• All directions of motion are possible (3-dimensions)• All speeds are possible (though not equally probable)• Molecules exert no force on each other except when they
collide• Collisions are elastic (velocity changes and energy is
conserved)
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Vacuum Technology
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Gas Properties• Atmospheric Pressure at Room Temperature
– ~2.5x1025 molecules/m3 (large number!)– average spacing -- 3.4x10-9 ( > molecular diameters of ~2x10-10)
• Very high vacuum at Room Temperature– ~2.5x1013 molecules/m3
– average spacing -- 3x10-5m
Dr. Philip D. Rack
Vacuum Technology
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Velocity Distribution• Maxwell Boltzmann Distribution
velocityvre temperatu T
Constant sBoltzman' k particleeach of mass m
molecules ofnumber total N
ondistributi velocity particle dvdn
:where2
2 )2(22
3
21
2
=====
=
=
−kT
mvev
kTmN
dvdn
π
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Dr. Philip D. Rack
Vacuum Technology
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Temperature/Mass Dependencies• Temperature Dependence • Molecular Mass Dependence
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Vacuum Technology
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• Average particle velocity (Maxwell-Boltzmann)
• ↑Temperature, ↓ mass -- ↑ average particle velocity
Basic Expressions from Maxwell Boltzmann Distribution
particle of mass meTemperatur TConstant sBoltzman' K
velocityaverage:
8 21
====
=
ν
πν
wheremKT
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Vacuum Technology
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Basic Expressions from Maxwell Boltzmann Distribution
• Peak Velocity (set first derivative of distribution = 0)
• Root Mean Square Velocity
• Maxwell-Boltzmann Statistics– vavg = 1.128vp and vrms = 1.225vp
21
2
=mkTvp
2/13
=mkTvrms
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Vacuum Technology
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Maxwell-Boltzmann Velocities
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0 200 400 600 800 1000 1200
Velocity (m/s)
dn/d
V
dN/dVpeakaverageRMS
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Vacuum Technology
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Maxwell-Boltzmann Energy Distribution
• Energy Distribution
• Average Energy = 1/2kT (x3 dimensions) = 3kT/2
• Most probable energy = kT/2
re temperatu TConstant sBoltzman' k
molecules ofnumber total Nondistributi velocity particle
dEdn
:where)(
2 )(2/3
2/1
21
===
=
=−
kTE
ekTEN
dEdn
π
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Vacuum Technology
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Maxwell-Boltzmann Energy Distribution
Peak
Average
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Vacuum Technology
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For molecules traveling with velocity{Vx}, the distance they can travel in time interval ∆t is:
{Vx} ∆t If they move towards the wall of area A and the number density is n (=N/V), the number of molecules that strike the wall in time ∆t is: n•A{Vx}∆t, but half of the molecules move towards the surface, half away from the surface:
(1/2)n•A{Vx}∆t When a molecule collides with the surface, it’s momentum changes from mVx to -mVx (total 2mVx) (m=MW/NA), hence the total momentum change is:
= [(number of collisions)] • (momentum change per collision)= [(1/2)n•A{Vx}∆t] • (2m{Vx})= n•m•A{Vx
2}∆t
Pressure and Molecular Velocity
Dr. Philip D. Rack
Vacuum Technology
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Since force is the rate of change of momentum: f = n•m•A{Vx2}
Pressure is the force per unit area: P = n•m•{Vx2}
The time a molecule spends between collisions is 1/Z.
A molecule of diameter “do” sweeps out a collision cylinder of cross-sectional area: σ = πd0
2, and length {V}∆t, during period ∆t. For two colliding objects we must really take into account their relative speeds (not one fixed, one moving). The collision frequency Z (per unit time) per molecule is = √2•σ{V}•n
Collision Frequency
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Vacuum Technology
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↑ Pressure (↑ particle density) -- ↓ mean free path
The Mean Free Path
Mean free path (l) - average distance a particle travels before it collides with another particle:
density particle gas ndiametermolecular
:2
122
1
==
=
o
o
dwhere
ndπλ
)(6.6)(PaP
mm =λ
(for air at room temperature)
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Vacuum Technology
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• Distribution of free paths
Basic Equations from Kinetic Theory
collision a suffering before x distance a erse that travmolecules ofnumber N
particle of mass meTemperatur TConstant sBoltzman' K
density particle :2
21
====
=Γ
nwhere
mKTnπ
Area
( ) 21
2 mkT
P
π=Γ
From ideal Gas law
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Vacuum Technology
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Monolayer Formation TimesThe inverse of the Gas impingement rate (or flux) is related to the Monolayer coverage time (tc). If a surface has ~ 1015sites/cm2
At 300K and 1 atm, if every Nitrogen molecule that strikes the surfaceremains absorbed, a complete monolayer is formed in about t = 3 ns.If P = 10-3 torr (1.3 x 10-6 atm), t = 3x10-3 sIf P = 10-6 torr (1.3 x 10-9 atm), t = 3 sIf P = 10-9 torr (1.3 x 10-12 atm), t = 3000 s or 50 minutes
tc = 1015 /sΓ,where S is the sticking coefficientΓ is the particle flux
Requirement for Experiment in Vacuum: Clean surface quickly becomes contaminated through molecular collision, ∴ p must be less than about 10-12 atm (7.67x10-5 torr).
10-10 to 10-11 torr (UHV-ultra high vacuum) is the lowest pressureroutinely available in a vacuum chamber.
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Vacuum Technology
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Vacuum Technology
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Boyle’s Law (1622)• P∝1/V (T and N constant)
P
V
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Vacuum Technology
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Amontons’ Law (1703)• P∝T (N and V constant)
T
P
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Vacuum Technology
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Charles’ Law (1787)• V∝T (P and N constant)
T
V
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Vacuum Technology
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Dalton’s Law (1801)• Dalton’s Law of Partial Pressures• Pt = n1kT + n2kT + n3kT + ... nikT
– where Pt is the total pressure and ni is the number of molecukles of gas i
• Pt = P1 + P2 + P3 … Pi– where Pt is the total pressure and Pi is the
partial pressure of gas i
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Vacuum Technology
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Avagadro’s Law (1811)• P∝N (T and V constant)
N
P
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Vacuum Technology
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Low Pressure Properties of Air
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Vacuum Technology
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Gas Transport Phenomena• Viscosity -- due to momentum transfer via molecular
collisions (development of a force due to motion in a fluid)
surfaces twobetwen theposition this
at velocity gas theof change of rate dydU
viscosityoft coefficien plane z-in x area surface A
direction-in x force :
xz
=
==
=
=
η
η
x
xz
x
Fwhere
dydU
AF
z
y
x
U1Moving Surface
U
Fixed Surface
Axz
2
1
U1 < U2
Dr. Philip D. Rack
Vacuum Technology
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Gas Transport Phenomena• Viscosity
– Kinetic Theory
– More Rigorous Treatment
νλη nm31
=
νλη nm4999.0=
) y(when )4(4999.022
3
21
λπ
η ≥=od
mkT
Viscosity ∝ (mT)1/2 and do2
and independent of P (only true for y ≥ λ)
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Vacuum Technology
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Gas Transport Phenomena• Viscosity
– for λ >> y (free molecular viscosity)
surface) plate on the atoms of slip the to(related 1
viscositymolecular free 4
force viscous
:4
1
≈
=
=
=
β
β
kTPmvAFwhere
UkTPmv
AF
xz
x
xz
x
Viscosity ∝ Pressure
Dr. Philip D. Rack
Vacuum Technology
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Gas Transport Phenomena
λ < dd
λ > dd
Viscosity controlled by particle-particle collisions
Viscosity controlled by particle-wall collisions
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Vacuum Technology
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Gas Transport Phenomena• Heat Flow (y ≥ λ)
z
y
x
Hot Surface (T2)
T
Cold Surface (T1)
Axz
2
1
T1 < T2gradient re temperatu
dydT
olumeconstant vat heat specific c ty conductiviheat K
flowheat H:
v
=
===
=
=
vc
wheredydTAKH
η
Heat Flow ∝ (mT)1/2 and do2
and independent of P (only true for y ≥ λ)
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Vacuum Technology
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Gas Transport Phenomena• Heat Flow (y ≥ λ) more detailed analysis of K (cf slide
#31)– Simplified
– DetailedvcK η=
lumecostant voat heat specificpressurecostant at heat specific
