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 > 04 Very High 04 > P > 07 Ultra High 07 >P>00 Extreme Ultrahigh 00 > 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 (3dimensions) 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 x09 ( > molecular diameters of ~x00 ) Very high vacuum at Room Temperature ~.5x0 3 molecules/m 3 average spacing  3x05 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 (MaxwellBoltzmann) 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 MaxwellBoltzmann Statistics v avg =.8v p and v rms =.5v p Page 9 MaxwellBoltzmann Velocities dn/dv dn/dv peak average RMS Velocity (m/s) Page 0
6 MaxwellBoltzmann 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 MaxwellBoltzmann 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 crosssectional 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 = 03 torr (.3 x 06 atm), t = 3x03 s If P = 06 torr (.3 x 09 atm), t = 3 s If P = 09 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 (7.67x05 torr). 00 to 0  torr (UHVultra 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 particleparticle collisions d λ > d Viscosity controlled by particlewall 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 particleparticle collisions d λ > d Heat Flow controlled by particlewall 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 xy plane at t=0, z=0 z x y +dz z=0 dz Page 37 d λ < d Diffusion controlled by particleparticle collisions d λ > d Diffusion controlled by particlewall collisions Page 38
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