Mean Molecular Weight


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1 Mean Molecular Weght The thermodynamc relatons between P, ρ, and T, as well as the calculaton of stellar opacty requres knowledge of the system s mean molecular weght defned as the mass per unt mole of materal, or, alternatvely, the mean mass of a partcle n Atomc Mass Unts). Recall that a mole of any substance contans N A = atoms. Thus, the number densty of ons s related to the mass densty, ρ, by n = ρ = ρn A µm a µ 5.1.1) where m a s the mass that s equvalent to 1 A.M.U. If the mass fracton of speces s x, thets number densty s = x ρn A A 5.1.2) where A s the atomc weght of the speces. The number densty of all ons n a volume of gas s then n I = x = ρn A A or n I = ρn A µ I where µ I = x A ) ) To compute the contrbuton of massless) electrons to the mean molecular weght, let Z be the atomc number of speces, and f be the speces onzaton fracton,.e., the fracton of electrons of that are free. The number densty of electrons s therefore ) x n e = ρn A f Z 5.1.4) A
2 or n e = ρn A µ e where µ e = Z x f A ) ) Note that n the case of total onzaton f = 1), ths equaton smplfes greatly. Snce Z /A = 1 for hydrogen, and 1/2 for everythng else, µ e = X Y + Z) ) 1 = X + ) 1 1 X) = X 5.1.6) From the defntons above, the total number densty of partcles s n = n I + n e = ρn A µ where the mean molecular weght s defned as µ = [ 1 µ I + 1 µ e ] )
3 The Ionzaton Fracton The calculaton of mean molecular weght requres knowledge of the chemcal composton of the materal and the onzaton fracton. To calculate onzaton fracton, one needs the Saha equaton. In general, the Saha equaton can be used to compute onzaton fractons over most of the star. It does, however, requre that the gas be n thermodynamc equlbrum. Ths s true throughout the star, as at hgh denstes, collsons wll control the level populatons. Ths approxmaton only breaks down the solar corona, where the denstes become very low. The Saha equaton also breaks down the centers of stars, where hgh denstes cause the onzaton energes of atoms to be reduced. Obvously, f the mean dstance between atoms s d, then there can be no bound states wth rad greater than d/2.) In the case of the hydrogen atom, the Bohr radus of level s a n = n + 1) 2 h2 m e e 2 = n + 1) 2 cm Thus, f the partcle densty s ρ µm a 0.3 µ g cm 3 4/3π2a 0 ) 3 then all the hydroges necessarly pressure onzed. In practce, the Saha equaton begns to break down at nuclear dstances of 10 a 0, whch corresponds to µ gcm 3. To correct for ths effect, the Saha equatos normally used untl t begns to show decreasng onzaton fractons toward the center of the star. When ths happens, complete onzatos assumed.
4 To derve the Saha equaton, begn by consderng the Boltzmann equaton, whch states that the number of atoms n level relatve to level j s = ω e χ j/k T 5.2.1) n j ω j where ω s the statstcal weght of the level.e., the number of separate, ndvdual states that are degenerate n energy), and χ j s the dfference n energy between the two levels. The number of atoms n level relatve to the number n all levels s thus n = ω ω 0 e +χ 0/kT + ω 1 e +χ 1/kT + ω 2 e +χ 2/kT +... = ω e χ /kt ω 0 + ω 1 e χ 1/kT + ω 2 e χ 2/kT +... n = ω e χ /kt u 5.2.2) where χ s the energy dfference between the th level and the ground state. The varable u s the partton functon for the atom or on). Because u s a functon of temperature, t s sometmes wrtten ut). Now let s generalze ths equaton to electrons n the contnuum. Let be the number of atoms n all levels defned as n above), and let state +1 be that where an excted electros n the contnuum wth momentum between p and p + dp. The Boltzmann equaton then gves d+1 = dω +1 u exp χ + p 2 ) /2m e k T where χ s the energy needed to onze the ground state of the atom, and dω +1 s the statstcal weght of the onzed state.
5 Now consder that dω has two components: one from the on ω +1 ), and other from the free electron dω e ). The former s just the statstcal weght of the ground state of the on, whle the latter can be computed usng the excluson rule. Snce each quantum cell n phase space can have only two electrons t spn up and spn down), then the number of degenerate states n a volume h 3 s Thus dω e = 2 d3 x d 3 p h 3 d+1 = 2 dv d3 p h 3 = 2 h 3 dv 4πp2 dp 5.2.3) = 8πp2 ω +1 h 3 u T) exp χ + p 2 ) /2m e dv dp k T The number of electrons n volume dv = 1/n e, so the total number of electrons n all contnuum states s therefore +1 = ω +1 8π ) u T) n e h 3 e χ /kt p 2 exp p2 dp 2m e kt or, f we let x 2 = p 2 /2m e kt, then +1 = ω +1 u T) = ω +1 u T) 8π n e h 3 e χ /kt 0 8π n e h 3 e χ /kt 2m e kt) 3/2 0 2m e kt) x 2 e x2 2m e kt) 1/2 dx 0 x 2 e x2 dx = ω +1 u T) 8π n e h 3 e χ /kt 2m e kt) 3/2 π 4 +1 = 2 n e ω +1 u T) 2πm e kt) 3/2 h 3 e χ /kt 5.2.4)
6 Fnally, note that for the calculaton above +1 represents those atoms of speces that have one electron the contnuum state,.e., onzed. It does not consder atoms of +1 that are themselves excted. In other words, +1 n 5.2.4) only ncludes onzed atoms n ther ground state.) To nclude all the excted states of +1, we must agan sum the contrbutons n exactly the same way as we dd n 5.2.2). Thus, the statstcal weght n 5.2.4) should be replaced by the partton functon, and +1 = 2 n e u +1 T) u T) 2πme kt h 2 ) 3/2 e χ /kt 5.2.5) Ths s the Saha equaton, whch relates the number of atoms n onzaton state + 1 to the number onzaton state. Note that f need be, we can substtute the electron pressure for the electron densty usng P e = n e kt, and wrte the Saha equaton as +1 P e = 2 u +1T) u T) ) 3/2 2πme k T) 5/2 e χ /kt h ) The sense of these equatons s ntutve: the hgher the temperature, the greater the rato, but the hgher the densty or pressure), the lower the rato due to the greater possblty for recombnatons).
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