Equilibrium Statistical Mechanics

Transcription

1 Equlbrum Statstcal Mechancs Ref. Davdson, Statstcal Mechancs McGraw-Hll, 96 We wsh to study plasmas n equlbrum for at least two reasons: (a) Many tmes real plasmas are near true equlbrum, at least locally and n regard to some of the propertes. (b) For stuatons outsde equlbrum, where rates are mportant, f the rate of one process s known by measurement or mechanstc calculaton, the rate of ts nverse process can be deduced from equlbrum propertes (mcro reversblty). Statstcal Mechancs bulds on the quantum mechancal notons of orbtal, energy state and excluson prncple, to analyze ensembles of many partcles and deduce ther macroscopc propertes. In the end, Statstcal Mechancs deduces the laws of Thermodynamcs from those of partcle physcs. In addton, t provdes specfc formulae for calculatng thermodynamc quanttes, whch can only be defned and nterrelated by Thermodynamcs alone. Instead of solvng n detal the mult-partcle equatons of moton and then takng the pertnent average, Statstcal Mechancs uses only a few results from Quantum theory and brdges the transton to the macroscopc world by means of a few smple postulates of a statstcal nature. The most mportant of these s the equ-probablty of mcro-states, whch we wll see soon. The plausblty of ths postulate arses from a property of classcal dynamcal systems, namely, that many-degree of freedom systems evolve n tme n a quas-random manner, such that over the long run they spend equal tme n any dynamcal state compatble wth the constrants. A smlar stuaton occurs n nteractng mult scale quantum systems, where perturbatons of sngle-partcle states ensure over the long run that all possble mult-partcle states are occuped the same fracton of the tme. Despte ths plausble foundaton, t s startlng to see the power and quantty of the deductons whch can be obtaned from such smple statements. Ther ablty to re-create the well establshed emprcal scence of thermodynamcs s the best check of the valdty of the statstcal postulates. On the other hand, ther rgorous logcal justfcaton s a dffcult and subtle task whch has kept mathematcans busy for over a century. Defntons for Quantum Statstcal Mechancs: Sngle-partcle orbtal: Set of values of all quantum numbers for a sngle partcle n a gven volume and feld. Many-partcle orbtal: Set of values of all quantum numbers for a gven number of partcles n a gven volume and feld. Mcrostate: Same thng as a many-partcle orbtal. Energy level: Physcally allowed value of -partcle energy. Degeneracy of an energy level: Number of sngle-partcle orbtals whch all have the same energy. More precsely, n any (small) volume n 6-D phase space (x, xw)

2 Energy dstrbuton: Specfcaton of the number of partcles whch have each energy level. Macrostate: Same as energy dstrbuton. Statstcal weght of a macro state: Number of mcro states n t,.e., number of ways n whch the gven number of partcles can occupy the avalable orbtals for a gven energy dstrbuton. System energy: Sum total of the partcle energes. Ferm-Drac statstcs: Specfcatons that no more than one partcle can exst per sngle- partcle orbtal. Also called Paul excluson prncple. Bose-Ensten Statstcs: Specfcaton that any number of partcles may have the same quantum numbers,.e. the same sngle-partcle orbtal. Postulates of Quantum Statstcal Mechancs. All mcro states correspondng to prescrbed system energy, number of partcles, volume and feld, are equally probable.. Partcles are ndstngushable (but mcrostates are not).. The macroscopcally observed value of a quantty Φ can be calculated by equal-weght averagng over all accessble mcrostates. As the number of partcles ncreases, one macrostate becomes much more probable (more mcro states n t) than any other; thus, an alternatve way of calculatng Φ as to assume that only the most probable macrostate exsts. Example : Three energy levels: t o = t = t = Doubly degenerate: g o = g = g = Ferm-Drac statstcs (excluson) 4 partcles Level orbtal orbtal Level orbtal orbtal Level orbtal orbtal E=, W= E=, W=4 E=4, W= E=4, W=4 E=5, W=4 E=6,W= N N N W =, E = (!!!!!!! W = 4, E = (!!!!!!! W =, E = 4 W = 4, E = 4 W = 4, E = 5 W =, E = 6 (!!!!!!!

3 Example : Three energy levels: t o = t = t = Doubly degenerate: g o = g = g = Bose-Ensten statstcs (no excluson prncple) 4 partcles Level orbtal orbtal 4 4 Level orbtal orbtal Level orbtal orbtal E=, W=5 E=, W=8 E=, W=8 E=, W= E=, W=9 etc N N N 4 E =, W = 5 E =, W = 8 E =, W = 8 E =, W = E =, W = 9 E = 4, W = 9 E =, W = 8 macrostate E = 4, W = E = 5, W = E = 6, W = 8 4 E = 4, W = 5 E = 5, W = 8 E = 6, W = 9 E = 7, W = 8 4 E = 8, W = 5 ( 5!!! = 5 = 5) 4!!!!!! ( 4!!! = 4 = 8)!!!!!!! (!! = = )!!!!!!! (!! = = 9)!!!!!! Quantum statstcs of ndependent partcles: Gven an external feld, suppose we have N partcles wth a total energy E prescrbed. We can (from Quantum mechancs) fnd the -partcle energy levels, t, each contanng g orbtals (degeneracy = g ). A macrostate s a specfcaton of the number of partcles N per energy level, wthn n the constrants, N = N levels E = N t levels These are two knds of Q.M. systems of many partcles: (a) Those for whch no two partcles can share an orbtal (Ferm-Drac statstcs) (b) Those where any number of partcles can occupy the same orbtal (Bose-Ensten statstcs)

4 Electrons, protons and neutrons (and many atoms, as long as they hold fractonal spn) obey F.D.; photons (and many atoms and even speces composed by fermons, as long as ther spn s nteger) obey B.E.. Let us calculate the statstcal weght of a macrostate n both cases. For F.D., the number of ways of selectng N orbtals for occupancy out of the g > N n level t s, g g! = N N!(g N )! Then, for any gven choce n level t, there are a smlar number of choces n other levels, and altogether, g! W F.D. = Π N!(g N )! (and g N, of course) For B.E. statstcs, n each level of t the number of ways of dstrbutng N partcles over g orbtals wthout restrcton as to how many per orbtal s the number of combnatons wth repetton of N objects taken from a group of g ; that number s, N + g (N + g )! = N N!(g )! Ths can be demonstrated wth a smple example havng N = 4 g =. Pck one realzaton: X X- XX X XX-X X Box Box Rearrange: XX XX ; X XXX Number of (objects+parttons)=n + (g ) Number of ways to scramble these enttes = (N + (g ))! But objects (N ) can be separately rearranged wthout affectng the count, so dvde by N!. Also, parttons can be rearranged, so dvde by (g )!: W = (N + g )! N!(g )! Altogether then, (N + g )! W B.E. = Π N!(g )! The most probable macrostate s that set of numbers N whch maxmzes W (or better ln W ) subject to gven N, E. Usng Lagrange multplers, and the approxmaton (Strlng s), N N ln(π) ln N N! πn ln N! = + + N ln N N N ln N N (for large N) e 4

5 For B.E. statstcs: φ = ln W αn βe = (N +g ) ln(n +g ) (N +g ) N ln N +N (g ) ln(g )+g αn βt N φ = ln(n + g ) + ln N + α βt = N g g ln + = α + βt ln + = α + βt N N snce both g and N are large numbers. Then we have, N g = α B.E. e e +βe For F.D. statstcs: φ = [g ln g g N ln N + N (g N ) ln(g N ) + g N αn βt N ] φ N = ln N + + ln(g N ) + α βt = g ln = α + βt N Therefore we have (and s easy to see that N < g ), N = g e α e +βe + F.D The classcal lmt or Dlute systems are those for whch g» N, so that only a few of the orbtals n each level are actually occuped. Then, regardless of the knd of statstcs (F.D. or B.E.), only sngle occupancy of orbtals s probable, and one should get a common lmt α+βe from the above two cases. Snce g» N t must be true that e», or α + βt s a postve, large number. Then, α βe N g e e both F.D. and B.E. Ths s called corrected Boltzmann statstcs snce t s smlar (but not dentcal) to the result n classcal mechancs (worked out by Boltzmann), where partcles are dstngushable. Manly due to ther large translatonal degeneracy, gases are n ths category most of the tme. It s shown n Quantum Mechancs that the number of possble states for a gven energy s (L/λ DB ), where L s the box sze (or the potental well sze), and λ DB = h s the mv DeBrogle wavelength. For a plasma at K, λ DB (electrons) 5Å, so the degeneracy s large ndeed. 5

6 Sgnfcance of α, β, W In general, a Lagrange multpler s the senstvty of the maxmzed functon to the correspondng constrant. To see ths, consder, Maxmze f(x ) =,,... n Subject to g j (x ) = c j j =,,...m < n φ = f(x ) λ j (g j c j ) j For maxmzaton, we mpose, φ f g j = λ j = x x x j g j = c j and calculate the correspondng x = x M and λ = λ M. M Suppose we now perturb ( one ) of the c j s, say c n, and solve agan. The maxmum f = f(x ) wll change by dc n, or df = f x x M dc n c j =n g j x g ( j x ) λ j dc n = λ j dc n x j c n c j =n x j c n c j =n X X- X But, snce g = c j, g j c n = δ jn, so df = dc n j λ j δ jn = λ n dc n f.e., λ n = q.e.d. c n c j =n g ( j cn ) cj n Suppose now the constrants c j are allowed to vary along a certan drecton n ther own m-space,.e., dc dc dc m = = = = dt ν ν ν m where dt s an arbtrary parameter. and we want to maxmze f also wth respect to such changes. Clearly, we frst must assume that for any set of c j s the x s are chosen such as to maxmze f for those fxed c j s. Then the change of f MAX due to the dc j s s f MAX df MAX = dc j = dt λ j ν j j ) c j c =j and snce dt s arbtrary, we must have the lnear relaton among the λ s: ν j λ j = j j 6

7 (Note ths s the equaton of the plane normal to dc j /ν j = dt n m-space.) Applyng ths to our maxmzaton of ln W wth fxed N (multpler α), E (multpler β) and V, we calculate, ln W α = N V,E ln W β = E Now, turnng to W, t s an aggregate system property whch s maxmum when the system n equlbrum (maxmum lkelhood), at fxed N, E. From classcal thermodynamcs, the same property s possessed by the Entropy, S of the system, so that we must have S = f(w ), wth f a monotancally ncreasng functon. Now, f we consder two non-nteractng together, we know that S = S + S. But also, snce ther probabltes are ndependent, W = W W, or ln W = ln W + ln W. Hence f must be lnear, and gnorng any constant shft, S = k ln W (k stll undetermned). S S Hence α = ; β = k N k E V,E Now, thermodynamcally, E Where µ s by defnton µ = N S,V V,N S µ S = ; = N T E T V,E V,N de = T ds pdv + µdn (chemcal potental), from here, µ and so α = β = (any statstcs) kt kt Ths relates α and β to known thermodynamc functons (µ, T ). We stll need to dentfy the constant k. V,N Dlute Systems The Partton Functon. The followng apples only to corrected Boltzmann statstcs,.e., α βe µ N = g e = g e kt We can now relate α, β (.e., µ, T ) to the actual constrants N, E: α βe N = N = e g e βe The group Q(β) = g e = e α βe E = t = e g N t e βe can be calculated a pror, once the quantum orbtals mechancs problem of one partcle n the gven volume and feld has been solved. It s called 7

8 the Partton Functon, and t plays an mportant role n chemcal equlbrum and other statstcal mechancs dervatons. In terms of Q, N e βe N = e α Q(β) and then = g N Q(β) therefore, we can wrte, Also, snce, then, whch can be wrtten as, µ = kt ln Q N E = e α g t e βe α Q = e β E N = Q Q β = ln Q β E = NkT ln Q T Systems wth non-nteractng degrees of freedom. In many dlute systems, translaton, rotaton, vbraton, exctaton, etc. nteract very weakly wth each other, so that the quantum mechancs problems can be solved separately, leadng to ndependent sets of translatonal, rotatonal, etc. energy levels and degeneraces, and to a total energy, Then, t = t tr. + t rot. + t vb. + texc. + Q = g e βe = e βe = e βe levels orbtals tr. rot. vb. exc. = e βe.tr. e βe.rot. tr. rot. and so we can calculate separately the varous partton functons, and then multply them, Also then, snce µ = kt ln Q/N Q = Q tr. Q rot. Q vb. Q exc. Q tr. µ = µ tr. + µ rot. + = kt ln kt ln Q rot. N X X- X all others have no N E E tr. + E rot. E tr. ln Q tr. and = ln(q tr. Q rot ) = =, etc. N β N N β 8

9 α βe Entropy of a dlute system. We have S = k ln W, ln W = lm (ln W F D ) and N = g e. N <<g [ ] ln W F D = g ln g g J N ln N + ; N (g N ) ln(g N ) + g J ; N ( ln(g N ) = ln g + ln N ) ln g N g N «g g [ ] N ( N ) ln W = mmmm g ln g N ln N (g N ) ln g = N ln N ++ln g g g «( S = k N ln N ) = k N ( + α + βt ) = k[n( + α) + βe] g or Hence, an alternatve defnton of the chemcal potental s G µ S ( = k µ ) N + E kt T µn = E + NkT T S The Gbbs free energy. By defnton, the Gbbs free energy G s G E + P V T S Now, then, and snce, Then, dg = de + P dv + V dp T ds SdT de = T ds P dv + µdn dg = SdT + V dp + µdn N T,P Ths has the advantage that (T, P ) are ntensve varables. We can buld up the G of a system by addng new molecules (dn) whle mantanng the same T and P. Snce µ s also ntensve, µ = µ(t, P ), so µ s constant durng ths buldng up, and we obtan smply, G = µn Notce we cannot go from µ = ( E/ N) V,S to E = µn, snce when we keep V, S constant and vary N, p wll also vary, hence µ wll too. Equaton of state of dlute systems. We have calculated for a dlute system, µn = E + NkT T S Snce µn = G, we have, G = E + NkT T S 9

10 But G n general s E + P V T S, so we conclude, P V = NkT Now, by comparng to the deal gas law, R P V = N T we dentfy Boltzmann s constant, R 8.4J/(mol. K) J k B = = =.8 6. (Molecules/mol.) K N A In retrospect, the reason we arrve at P V = NkT s that we are assumng non-nteractve systems, where the energy levels and g s are calculated for each partcle as f t were alone n the box. Statstcal mechancs can also be appled to more complex systems, of course, but the methods are somewhat more dffcult (canoncal and grand-canoncal ensemble, as opposed to out mcro-canoncal ensemble). N A Mult-component systems. Suppose there are speces A, B, C, non-reactng for now. The total W s the product of W = W A W B W C and then ln W = ln W A + ln W B + ln W C. There are number constrants and one E constrant: N A = Formng agan, N A B N j j Nk C k A B C j j k k j k N B = N C = E = t N A + t N B + t N C φ = ln W A + ln W B + ln W C α A N A α B N B α C N C βe A and we dfferentate relatve to each N, each N B C j, and each N k. We then get for each speces a F.D. or B.E. (or corrected Boltzmann) dstrbuton wth a dfferent α for each, but wth a common β. Followng the same steps as before, we can agan relate these to the µ s and the T : µ A µ B µ C α A = α B = α C = β = kt kt kt kt and can also prove that G = N A µ A + N B µ B + N C µ C P V = (N A + N B + N C )kt

11 and (for dlute systems) Q A Q B Q C µ A = kt ln, µ B = kt ln, µ C = kt ln N A N B N C Reactng Systems. Suppose now that A, B, C can nterconvert accordng to the reacton, ν A A + ν B B ν C C Ths means that N A, N B, N C are not fxed, but that, f they change accordng to ths reacton, dn A dn B dn C = = (a drecton n N A, N B, N C space) ν A ν B ν C But we have proven before that for the object functon (ln W or S n our case) stll to be maxmum, there must exst a relatonshp among the multplers of the form, or, n terms of the µ = kt α, ν A α A + ν B α B = ν C α C ν A µ A + ν B µ B = ν C µ C st form of the law of mass acton In terms of the N s and Q s, Q A Q B Q C Q A Q B Q C ν A ln + ν B ln = ν C ln = N A N B N C N A N B N C or, In terms of P s, N ν A N ν B ν A ν B ν C Q ν A Q ν B = nd form of the law of mass acton A B A B N ν C Q ν C C C ν A ν B ν A ν B n A n B q A q B N j Q j ν C = ν C wth n j =, q j = n q V V C C kt kt kt P A = N A P B = N B P C = N C V V V kt Q A ν A kt Q B ν B P ν A P ν B V V A B = rd form of the law of mass acton P ν C C kt Q C V ν C The mportance of ths one s that the RHS wll be shown to depend on T only. It s called the Equlbrum constant K P (T ) for the reacton. The zero of energy. For a sngle speces, or for non-reactng speces, the t s can be measured from arbtrary levels; a shft t ' = t t merely makes a new p.f. Q ' = t E /kt Q and a new ' µ = µ t. However, when speces can nterconvert, they generally lberate or absorb

12 defnte amounts of energy n the process. If A + B + ΔE C then n the state n whch t A = and t B =, we must say that C has an energy ΔE, not arbtrary anymore. In all, we can assgn arbtrary zero levels of energy only to a set of non-nterconvertble atoms or partcles. The zeros of all the others then follow from ther energes of formaton. In chemcal thermodynamcs, the conventon s to assgn zero enthalpy to the pure speces at 98 K, atm, n the natural state (O, H, C (graphte),...). Then, for nstance, snce O + O O + 59Kcal, the enthalpy per mole of O at 98 K, atm s, The Translatonal Partton Functon 59 kcal ( per atom, J). mole 6 Consder a sngle partcle n a rectangular box, wth sdes L x, L y, L z. The frst task s to solve the Schrödnger equaton n order to obtan the allowable energes and quantum numbers (hence the degeneraces). The general Schrödnger equaton s (wth I = h/π), I ψ T p I = H (ψ T ) ; H = + V ; p = \ () t m Assume separaton of the tme dependence: Substtute and dvde by ψ T : ψ T (t, x) = Π(t)ψ(x) () I dπ = H (ψ) = t (4) π dt ψ where t s the (so far arbtrary) separaton constant. Ths gves the two equatons, and so, (E/ )t Ce dπ t + π = (4a) dt I H (ψ) = tψ Substtute n (6), dvde by ψ: I X xx Y yy Z zz t = (8) m X Y Z (4b) π = Ce, ψ (E/ )t T = ψ(x) (5) For our case, there s no potental energy nsde the box: V (x) = for ( < x, y, z < L x,y,z ), so (4b) reduces to, I \ ψ = tψ m or, I \ ψ + tψ = (6) m Assume next a soluton that s also separable n (x, y, z): ψ(x, y, z) = X(x)Y (y)z(z) (7)

13 Each of the terms X xx, etc. must be separately a constant, and we therefore obtan X equatons, wth, m X xx + t x X = I m Y yy + t y Y = (9) I m Z zz + t z Z = I t x + t y + t z = t () The partcle s confned by the box, and so we must have ψ = at each wall. Ths gves the boundary condtons: X() = X(Lx ) = Y () = Y (L y ) = () Z() = Z(Lz ) = The solutons that are zero at x = y = z = are, mt x mt y mt z X = A x sn x ; Y = A y sn y ; Z = A z sn z () I I I and mposng zero value at x = L x, y = L y, z = L z gves the condtons, mt x L x = n x π (n x =,,,...) I mt y = π =,,,...) () I L y n y (n y mt z L z = n z π (n z =,,,...) I Where the numbers (n x, n y, n z ) dentfy a quantum state, and are the quantum numbers for ths problem. Fnally, mposng t x + t y + t z = t gves, ( ) π I n n x y nz t = + + (4) L L L m x y z whch gves all the possble energy levels of the partcle. Clearly, there are many possble combnatons of n x, n y, n z gvng the same energy t (degenerate states), and the degeneracy wll ncrease wth the sze of the box. We can now calculate the translatonal partton functon as, Q tr. = e E/kT (5) n x n y n z In ths form, snce the summaton ncludes all the quantum numbers, the degeneracy factors are not needed. In detal, π n n x y n + + z mkt L L L Q tr. x y z = e (6) n x n y n z

14 π The summaton can be approxmated as an ntegral, provded «. Recallng that mkt L the DeBrogle wavelength for a partcle wth average thermal energy s, h πi πi λ DB = = = (7) p m kt mkt m DB We see that the condton to go to a contnuum vew n the (n x, n y, n z ) space s <<, or L a matter wave much shorter than the contaner sze. Notce ths s a weaker condton than that we already assumed for a dlute (Boltzmann) system: λ DB «n / (dstance between partcles), and so the contnuum approxmaton s well justfed. We then wrte, = ˆ ˆ ˆ n π n x y n + + z mkt Lx Ly Lz Q tr. dn x dn y dn z e (8) ˆ π ˆ π n ˆ x mkt mkt π n y z L n L mkt e x dn e y x dny e L z dn z Change varables to n x = mkt Lx ξ, n y = mkt Ly η, n z = mkt Lz ζ and use π π π e ξ π dξ =, etc. We obtan, ( ) Q tr. L x L y L z π = (mkt ) / (πi) λ Note that πi = h, and L x L y L z = V (the box volume). So, Q tr. / πmkt = V (9) h Ths can be wrtten n terms of λ DB as Q tr. = (π/) / V/λ DB, whch gves Q tr. an nterpretaton: (roughly) the number of DeBrogle boxes that ft nto the volume. Notce Q tr. s proportonal to volume. The other peces, Q rot, Q vb, etc. are ndependent of V, and so Q V altogether, and q = V Q = q(t ). Ths proves that the equlbrum constant K P, as derved prevously, s ndeed only a functon of temperature. 4

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