Dr. Kasra Etemadi September 21, 2011

Transcription

1 Dr. Kasra Etemadi September, 0

2 - Velocity Distribution -Reaction Rate and Equilibrium (Saha Equation 3-E3 4- Boltzmann Distribution 5- Radiation (Planck s Function 6- E4

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4 z r dxdydz y x Applets

5 f( x r dxdydz m f ( x ( kt / mx exp( kt x y z x

6 f( x r dxdydz m f ( x ( kt / mx exp( kt f( = f x (. f y (. f z ( x f( Maxwell-Boltzmann distribution function f ( m 4 ( kt 3/ exp( m kt Velocities

7 Distribution Functions f (r, f (r, drd= normalized f( Maxwell-Boltzmann distribution function f ( m 4 ( kt 3/ exp( m kt Velocities - n(r = n total. f (r, d Number Densities - <(r> = n total.. f (r, d Aerage Velocities - <p(r> = n total. m. f (r, d Aerage Momentum - <E(r> = n total. ½ m. f (r, d Aerage Energy

8 f ( m 4 ( kt 3/ exp( m kt f( T T T<T<T3 T3 Maxwell-Boltzmann distribution function

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10 exp( ( 4 ( 3/ kt m kt m f ( d f m kt d f 8 ( 0 /, ( ( rms d f / 0 / 3 ( ( ( m kt d f rms

11 f( m rms kt m ( m / 8kT m 3kT rms ( m /

12 Energy Distribution Function w / w f ( w ( kt 3/ exp( kt w 0 wf ( w dw 3 kt W=/m

13 Find the fraction N /N of the total number of particles N haing speeds aboe a gien speed V in a system of particles described by the Maxwell-Boltzmann distribution. Solution: N V m 3/ m / kt dn f ( d 4 ( e V V V N N m f ( 4 ( kt 3/ kt m exp( kt du d m kt d dn Nf ( d u m kt N V N 4 U u e u du

14 U u V du e u N N } { 4 } ] {[ 4 U u u U U u U u V du e du e Ue du e ue N N U u U U u U V du e Ue du e Ue N N 0 0 } { 4 ( U erf Ue N N U V V kt m U x x dx e x erf 0 ( Error function

15 The fraction N /N of the total number of particles N haing speeds aboe a gien speed V in a system of particles described by the Maxwell-Boltzmann distribution. N V N Ue U erf ( U Error Function erf ( x 0 x e x dx x erf(x U m kt V

16 Dr. Kasra Etemadi September, 0

17 Chemical Kinetics: Definition At what rate does a chemical reaction undergo changes under a gien set of conditions? Reaction Rate: a A b B products Kinetic order determined experimentally Rate k m A B n d A dt Rate constant ~ T & chemical potential Concentrations Total order = m n

18 Chemical Equilibrium Rate forward = Rate backward Example: CH 4 H O > 3 H CO The law of Mass Action will be: H CO CH H O K eq / 4 aa bb pp qq K eq Z Z p a Z Z q b exp( 0 kt

19 n It describes the relationship between the electron, ion, and neutral number densities e A A : e n n a ion ( m kt e h 3 3/ Z Z ion a exp( e kt n e number density of electrons n ion number density of ions n a number density of atoms m e mass of electron k Boltzmann constant T Temperature h Planck s constant e Electron charge Z ion Partition function of ion Z a Partition function of atom a E i -E i E ionization Ionization potential of neutral atoms Lowering of ionization potential E ionization ( see page 35 a

20 E ionization Lowering of ionization potential E ionization = n e /3 [ev] Unit of n e is [m -3 ] a E ionization E ionization

21 Inglis & Teller Theory of Lowering of ionization Potential (only for hydrogen atoms log n * logn e n e n * Electron density [/cm3] Energy leel cut-off

22 Pressure Boltzmann s Constant Temperature P = n k T Number Density of particles atm = 760 Torr (mm Hg = 035 N/m (Pascal bar = 0 5 Pa k = J/K Boltzmann s constant

23 p = n k T = (n e n ion n a k T For singly ionized plasma p = (n e n a k T ; n e = n ion For multi-temperature plasmas p = n e kt e n ion kt i n a kt a

24 n n n a f (T e ion f(t is Saha Equation; p and T are gien n e =n ion n a ne f ( T } pf ( T ne f ( T ne 0 p=(n e n a kt kt Assume an initial alue for n e n e find E ionization Find a Sole Eq aboe new n e New iteration

25 Equilibrium s non-equlibrium Plasma Number densities Momentum Energy Charge densities Velocities Potential Electronic ibrational rotational etc. Kinetic Velocities Saha-Eggert Equation Maxwell Distribution Boltzmann Distribution Maxwell Distribution

26 Dr. Kasra Etemadi September, 0

27 - Kinetic Theory of Gases f ( x m ( kt / exp( mx kt - Boltzmann Distribution n i g Z i a n a E exp( kt i

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30 Consider a system with three particles (e.g. H each haing four equidistance energy leels. 3E E E 0 E

31 We introduce an energy equialent to 3E in the Box Energy=3E

32 3E E E 0 3E E E 0 3E E E 0

33 n i g Z i a n a E exp( kt i n i E i g i Z a n a T k Number densityof atom in i th energy state Energy state of the i th energy leel Degeneracy Partition function Total number of atoms Temperature Boltzmann Constant

34 Z a n ionization i g i exp( E kt i

35 3E E E 0 3E E E 0 3 Degeneracy, g i 3E E E 0 3E 3E 3E E E E E E E 0 0 0

36 System of three particles Four different combinations 3E has 4 energy leels. Third energy leel has a degeneracy of 3E 4 E E E 0 3 n= 3 4

37 ionization n i i i a kt E g Z exp( E 0 E E 3E 3 4 kt E kt E kt E kt a e e e e Z 3 0 x e kt E 3 x x x Z a

38 Z a n i x x x g Z i a =3 n a Energy Conseration E exp( kt i 3 3E E 3 E n 0 n E n3 E n4 3 E x 3 x 0 E x Z a 0 When we look at one particle, this is the probability of being in that energy state n.58 %4 n n n 3 4 %4 %7 %7

39 n i g Z i a n a E exp( kt i n i n 0 T=constant E i

40 Dr. Kasra Etemadi September 3, 009

41 hc E(, T 5 e hc kt Steffan-Boltzmann Law: E T 4 Wien Displacement Law: Max 30 T 7 ( is in Å

42 Continuous Line 4000 Å 7500 Å Continuous Spectrum Absorption Spectrum of Hydrogen Emission Spectrum of Hydrogen Colors

43 electron - electron -, ( ( / 8 T T N T e T c ff T C fb i i e T e T Z g T, ( (, (, (

44 Absorption Emission E=h E=h

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46 Ej E=E ji =h ji Ei E E E total n 4 n 4 E j a A g Z ji j a E ji n 4 E exp( kt E continum j j A ji h A h ij ji ji

47 Plasmas Optically Thin Optically Thick

48 Maxwellian Distribution f ( n i nen n a i g Z i a m 4 ( kt Boltzmann Distribution Saha Equation n Planck s Function a E exp( kt (mekt 3 h hc E(, T 5 e 3/ 3/ hc kt i exp( Zi exp( Za m kt ea kt T Maxwellian T Boltzmann T Saha T Planck

49 Complete Thermodynamic Equilibrium T Maxwellian = T Boltzmann = T Saha = T Planck Local Thermodynamic Equilibrium T Maxwellian = T Boltzmann = T Saha Partial Local Thermodynamic Equilibrium T Maxwellian =T Saha

50 Example For an atmospheric pressure hydrogen plasmas haing a temperature of 0,000 K determine the following: number density of atoms, ions and electrons Maximum number densities of atoms, ions and electrons that can contribute to excitation of hydrogen atoms from ground state. Density distribution of energy states of hydrogen atomes Intensity [J/cm-3] of hydrogen spectral line at 5.7 Å (first excited to ground state, g u =8, Z H =, A ul =4.699e8 s-

51 Project (Deadline October 5, 0 Prepare a Powerpoint presentation (PPP with the format similar to final presentation (see the Presentation Format as described in the EE403/503 website. Transfer this file to the your memory space assigned for your project. Put a link on top of your index.html file to access the PPP file. You still need to continue on your project with new research materials.