# Vacuum Technology. Kinetic Theory of Gas. Dr. Philip D. Rack

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1 Kinetic Theory of Gas Assistant Professor Department of Materials Science and Engineering University of Tennessee 603 Dougherty Engineering Building Knoxville, TN Phone: (865) Fax (865) Page Vacuum Basics Gas Volume % Pressure (Pa) N 78 79,7 O,33 CO Ar Atmospheric Pressure = 0,33. Pa (760 torr) (33Pa = torr) Page

2 Vacuum Basics Vacuum Pressure Range (Pa) Low 0 5 > P > 3.3x0 3 Medium 3.3x0 3 > P >0 - High 0 - > P > 0-4 Very High 0-4 > P > 0-7 Ultra High 0-7 >P>0-0 Extreme Ultrahigh 0-0 > P Page 3 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) Page 4

3 Gas Properties Atmospheric Pressure at Room Temperature ~.5x0 5 molecules/m 3 (large number!) average spacing x0-9 ( > molecular diameters of ~x0-0 ) Very high vacuum at Room Temperature ~.5x0 3 molecules/m 3 average spacing -- 3x0-5 m Page 5 Velocity Distribution Maxwell Boltzmann Distribution dn N m mv (kt ) = v e dv π kt dn = particle velocity distribution dv N = total number of molecules m = mass of each particle k = Boltzman's Constant T = temperature v = velocity 3 Page 6

4 Temperature/Mass Dependencies Temperature Dependence Molecular Mass Dependence Page 7 Basic Expressions from Maxwell Boltzmann Distribution Average particle velocity (Maxwell-Boltzmann) 8 KT ν = πm ν = average velocity K = Boltzman's Constant T = Temperature m = mass of particle Temperature, mass -- average particle velocity Page 8

5 Basic Expressions from Maxwell Boltzmann Distribution Peak Velocity (set first derivative of distribution = 0) kt v p = m Root Mean Square Velocity / 3kT v rms = m Maxwell-Boltzmann Statistics v avg =.8v p and v rms =.5v p Page 9 Maxwell-Boltzmann Velocities dn/dv dn/dv peak average RMS Velocity (m/s) Page 0

6 Maxwell-Boltzmann Energy Distribution Energy Distribution dn de = N E ( kt ) e 3/ π E / ( kt) dn = particle velocity distribution de N total number of molecules k = = Boltzman's Constant T = temperature Average Energy = /kt (x3 dimensions) = 3kT/ Most probable energy = kt/ Page Maxwell-Boltzmann Energy Distribution Peak Average Page

7 Pressure and Molecular Velocity For molecules traveling with velocity{v x }, the distance they can travel in time interval t is: {V x } 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{V x } t, but half of the molecules move towards the surface, half away from the surface: (/)n A{V x } t When a molecule collides with the surface, it s momentum changes from mv x to -mv x (total mv x ) (m=mw/n A ), hence the total momentum change is: = [(number of collisions)] (momentum change per collision) = [(/)n A{V x } t] (m{v x }) = n m A{V x } t Page 3 Pressure and Molecular Velocity Since force is the rate of change of momentum: f = n m A{V x } Pressure is the force per unit area: P = n m {V x } Generalizing: {V }= {V x } + {V y } + {V z } = 3 {V x }, P = (/3)n m{v } Generally V RMS is used here v rms P=nkT (where n=n/v) 3kT = m / atm = 03 mbar =.03 bar = 760 mmhg atm = 760 torr = 0,35 Pa = 0,35 Nm - Page 4

8 Collision Frequency A molecule of diameter d o sweeps out a collision cylinder of cross-sectional area: σ = πd 0, 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 = σ{v} n The time a molecule spends between collisions is /Z. Page 5 The Mean Free Path Mean free path (l) - average distance a particle travels before it collides with another particle: λ = d o = molecular diameter n = gas πd o n particle density λ( mm ) = 6.6 P( Pa) (for air at room temperature) Pressure ( particle density) -- mean free path Page 6

9 Basic Equations from Kinetic Theory Distribution of free paths N = N' e x λ (random walk distribution) N' = number of molecules in a volume N = number of molecules that traverse a distance x before suffering a collision N/N' (% of particles) x/lambda 63% suffer collision 0<x<λ 37% suffer collision λ<x<5λ only 0.6% travel farther than 5λ Page 7 Flux Area Particle Flux or Impingement Rate nν Γ = 4 Γ = particle flux n = particle density ν = average velocity KT Γ = n πm n = particle density K = Boltzman's Constant T = Temperature m = mass of particle From ideal Gas law Γ = P ( πmkt ) Page 8

10 Monolayer Formation Times The inverse of the Gas impingement rate (or flux) is related to the Monolayer coverage time (t c ). If a surface has ~ 0 5 sites/cm t c = 0 5 /sγ, where S is the sticking coefficient Γ is the particle flux At 300K and atm, if every Nitrogen molecule that strikes the surface remains absorbed, a complete monolayer is formed in about t = 3 ns. If P = 0-3 torr (.3 x 0-6 atm), t = 3x0-3 s If P = 0-6 torr (.3 x 0-9 atm), t = 3 s If P = 0-9 torr (.3 x 0 - atm), t = 3000 s or 50 minutes Requirement for Experiment in Vacuum: Clean surface quickly becomes contaminated through molecular collision, p must be less than about 0 - atm (7.67x0-5 torr). 0-0 to 0 - torr (UHV-ultra high vacuum) is the lowest pressure routinely available in a vacuum chamber. Page 9 Page 0

11 Boyle s Law (6) P /V (T and N constant) P V Page Amontons Law (703) P T (N and V constant) P T Page

12 Charles Law (787) V T (P and N constant) V T Page 3 Dalton s Law (80) Dalton s Law of Partial Pressures P t = n kt + n kt + n 3 kt +... n i kt where P t is the total pressure and n i is the number of molecukles of gas i P t = P + P + P 3 P i where P t is the total pressure and P i is the partial pressure of gas i Page 4

13 Avagadro s Law (8) P N (T and V constant) P N Page 5 Low Pressure Properties of Air Page 6

14 Viscosity -- due to momentum transfer via molecular collisions (development of a force due to motion in a fluid) Fx du = η Axz dy Fx = force in x - direction A xz = surface area in x - z plane η = coefficient of viscosity du = rate of change of the gas velocity at dy this position betwen the two surfaces y z Moving Surface U Fixed Surface U < U A xz x U Page 7 Viscosity Kinetic Theory η = nmνλ 3 More Rigorous Treatment η = nmνλ (4mkT ) η = 3 π d o (when y λ) Viscosity (mt) / and d o and independent of P (only true for y λ) Page 8

15 Viscosity for λ >> y (free molecular viscosity) Fx Pmv U = Axz 4kT β Fx A xz = viscous force Pmv = free molecular viscosity 4kT β (related to the slip of atoms on the plate surface) Viscosity Pressure Page 9 d λ < d Viscosity controlled by particle-particle collisions d λ > d Viscosity controlled by particle-wall collisions Page 30

16 Heat Flow (y λ) dt H = AK dy H = heat flow K = heat conductivity = ηc c v = specific heat at constant volume dt = temperature gradient dy v y z T Hot Surface (T ) Cold Surface (T ) T < T A xz x Heat Flow (mt) / and d o and independent of P (only true for y λ) Page 3 Heat Flow (y λ) more detailed analysis of K (cf slide #3) Simplified K = ηc v Detailed K = (9γ 5) ηcv 4 where c : P γ = cv c p = specific heat at costant pressure c = specific heat at costant volume v γ =.4 (diatomic molecule) =.667 (monatomic molecule) γ γ =.333 (triatomic molecule) Page 3

17 Heat Flow (λ >> y) E = αλp( T 0 E = heat flow 0 T ) α = accomodation coefficient (how effective the surfaces transfer and absorb energy) Λ = free - molecular heat conductivity (how effective the molecules absorb and transfer energy) Heat Flow Pressure Page 33 d λ < d Heat Flow controlled by particle-particle collisions d λ > d Heat Flow controlled by particle-wall collisions Page 34

18 Diffusion D dn Γ = D dx D = diffusion coefficient dn = concentration gradient dx kt 8 + m m π = 3πn( d + d ) D 0 0 kt 4 πm = 3πnd 0 dn Γ = D dx D = diffusion coefficient dn = concentration gradient dx (interdiffusion of two gases) (self diffusion) D (T/m) / and /nd 0 suggests that as n 0, D (only good when λ < d or y) Page 35 Diffusion (λ >> d) D = rv 3 r = radius of pipe or chamber v = thermal velocity Knudsen diffusion coefficient Gas diffusion is limited by collisions with container wall Page 36

19 Diffusion Ndz dn = ( πdt) z (4Dt) e dn = number of molecules located between z and z + dz z0 f = erfc ( Dt) f = fraction of molecules that are located between z and 0 0 z =.3( Dt) 0 z = minimum distance that 0% of the molecules have diffused after a time t N molecules x-y plane at t=0, z=0 z x y +dz z=0 -dz Page 37 d λ < d Diffusion controlled by particle-particle collisions d λ > d Diffusion controlled by particle-wall collisions Page 38

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