CHAPTER 6 IDEAL DIATOMIC GAS

CHAPTER 6 IDEAL DIATOMIC GAS Monatomic gas: Has tanslational and electonic degees of feedom Nuclea patition function can be teated as a constant facto Diatomic gas: Has vibational and otational degees of feedom as well. Electonic enegy state is simila to that of monatomic gas. Nuclea patition function may be combined with the otational one. 1 PRELIMINARY THOUGHTS * How to genealize to diatomic molecules * The geneal pocedue would be to set up the Schodinge equation fo nuclei and n- electons, and solve it fo the set of eigenvalues of the diatomic molecule. This is again too had. We will thus need a good appoximation that allows all degees of feedom to be witten sepaately, like H = Htans + Hot + Hvib + Helec + Hnucl (6-14) which implies that ε = εtans + εot + εvib + εelec + εnucl and q = qtans qot qvib qelec qnucl (6-14 ) Within that appoximation, the patition function of the gas itself will be given by ( qtans qot qvib qelec qnucl ) QNVT (,, ) = (6-17) N! N But we do not know yet whethe (6-14 ) is possible o not. At least we can wite q q q = tans int whee q int includes (ot, vib, elec, nucl) degees of feedom. q int and we hopefully get a good appoximation fo. 1

* What to do? * THE VIBRATIONAL AND ROTATIONAL MOTION: The igid oto-hamonic oscillato appoximation allows the Hamiltonian of the elative motion of the nuclei to be witten as Hot,vib = Hot + Hvib which implies ε = ε + ε and qot,vib = qot qvib ot,vib ot vib THE TRANSLATIONAL MOTION: m m1+ m so that q π( m + m ) kt = h 1 tans 3/ V THE ELECTRONIC PARTITION FUNCTION will be simila to that fo a monatomic gas, except the definition of the gound state. THE NUCLER PARTITION FUNCTION q may not be sepaable fom q because of the symmety equiement. nucl ot THE VIBRATIONAL DEGREE OF FREEDOM -1 The Bon-Oppenheime appoximation The nuclei ae much moe massive than the electons, and thus move slowly elative to the electons. Theefoe the electons can be consideed to move in a field poduced by the nuclei fixed at some intenuclea sepaation. The Schodlnge equation appoximately sepaates into two simple equations fo: (i) motion of the electons in the field of the fixed nuclei, and (ii) the motion of the nuclei in the electonic potential u (), that is, the potential set up by the electons in the electonic state j. Each electonic state of j the molecule ceates its own chaacteistic intenuclea potential. The calculation of fo even the gound state is a difficult n-electon calculation, and so semiempiical appoximations such as the Mose potential ae often used. See Figue 6-1. uj ( )

The elative motion of the two nuclei in the potential consists of otay motion about the cente of mass and elative vibatoy motion of the two nuclei. It tuns out that the amplitude of the vibatoy motion is vey small, and so it is a good appoximation to conside the angula motion to be that of a igid dumbbell of fixed intenuclea distance. In addition, the intenuclea potential uj () can be expanded about e : uj ( ) e du ( ) u ( ) = u ( ) + ( ) + ( ) + 1 e e e d u d = d e = e 1 = u ( e) + k ( e) + (6-3) The paamete k is a measue of the cuvatue of the potential at the minimum and is called the foce constant. A lage value of k implies a stiff bond; a small value implies a loose bond. The enegy and degeneacy of an hamonic oscillato ae [cf Eq. (1-31)] 3

1 ( ) εvib = hν n + ωn = 1 n = 0, whee 1,, (6-8) whee ν 1 k = π µ 1/ mm 1 µ = m + m 1 (6-9) Fo a molecule to change its vibational state by absobing adiation it must (1) change its dipole moment when vibating and () obey the selection ule n =± 1. The fequency of absoption is, then, seen to be ν εn + 1 εn 1 k = = h π µ 1/ (6-13) Equation (6-13) pedicts that the vibational spectum of a diatomic molecule will consist of just one line. This line occus in the infaed, typically aound 1000 cm -1, giving foce constants k of the ode of 105 o 106 dynes/cm. (See Poblem 6-5.) Table 6-1 gives the foce constants of a numbe of diatomic molecules. 4

- The Vibational Patition Function Since we ae measuing the vibational enegy levels elative to the bottom of the intenuclea potential well, we have 1 εn = n+ hν n= 0,1,, (6-19) The vibational patition function q vib then, becomes q ( T) = e = e e = e vib 1 1 e βεn βν h / βν h n βν h / β hν (6-0) n n= 0 whee we have ecognized the summation above as a geometic seies. This is one of the ae cases in which q can be summed diectly without having to appoximate it by an integal, as we did in the tanslational case in Chapte 5 and shall do shotly in the otational case. The quantity β hν is odinaily lage than 1, but if the tempeatue is high enough, β hν 1, and we can eplace the sum in (6-0) by an integal to get βν h / βν h n kt 0 ( ν ) qvib ( T) = e e dn= kt h (6-1) hν Although we can calculate q ( T) vib exactly, it is wothwhile to compae this appoximation to some othes which we shall deive late on. -3 Contibution to the Themodynamic Enegy dln q v v v Ev = NkT = Nk + v / T dt e Θ Θ Θ 1 (6-) whee Θv hν / kand is called the vibational tempeatue. Table 6-1 gives Θ v fo a numbe of diatomic molecules. The vibational contibution to the heat capacity is Θv / T Ev Θv e = Nk T N T e Θv / T ( 1) (6-3) Notice that as T, Ev NkT and C v Nk, a esult given in many physical chemisty couses and one whose significance we shall undestand moe fully when we discuss equipatition of enegy. 5

-4 The Level Population The faction of molecules in excited vibational states designated by n is f n βν h ( n+ 1/) e = (6-4) q vib This equation is shown in Fig. 6-4 fo B at 300 K. Notice that most molecules ae in the gound vibational state and that the population of the highe vibational states deceases exponentially. Bomine has a foce constant smalle than most molecules, howeve (cf Table 6-1), and so the population of excited vibational levels of B is geate than most othe molecules. Table 6- gives the faction of molecules in all excited states fo a numbe of molecules. This faction is given by βν h ( n+ 1/) e βν h f = = 1 f = e = n> 0 0 n= 1 qvib Θ/ T e (6-5) 6

3 THE ELECTRONIC PARTITION FUNCTION The electonic patition function will be simila to that of monatomic case. But we must choose a zeo of enegy fo the otational and vibational states. The zeo of otational enegy will usually be taken to be the J = 0 state. In the vibational case we have two choices. One is to take the zeo of vibational enegy to be that of the gound state, and the othe is to take the zeo to be the bottom of the intenuclea potential well. In the fist case, the enegy of the gound vibational state is zeo, and in the second case it is hν /. We shall choose the zeo of vibational enegy to be the bottom of the intenuclea potential well of the lowest electonic state. Lastly, we take the zeo of the electonic enegy to be the sepaated, electonically unexcited atoms at est. If we denote the depth of the gound electonic state potential well by, the enegy of the gound electonic state is D e, and the electonic patition function is D e D / kt / kt elec ωe1 ωe q e e ε e = + + (6-18) whee De and ε ae shown in Fig. 6-. We also define D0 = De hν /. As Fig. 6- shows, Do is the enegy diffeence between the lowest vibational state and the dissociated molecule. The quantity D 0 can be measued spectoscopically (by pe-dissociation specta, fo example) o caloimetically fom the heat of eaction at any one tempeatue and the heat capacities fom 0 K to that tempeatue. See Table 6-1 fo. D 0 7

4 THE ROTATIONAL DEGREE OF FREEDOM The enegy eigenvalues of a igid oto is given in (1-3) ε J JJ ( + 1) = J = 0,1,, (6-7) I the degeneacy of a igid oto ω J = J + 1 J = 0,1,, the moment of inetia of the molecule. I = µ e Tansitions fom one otational level to anothe can be induced by electomagnetic adiation. The selection ules fo this ae: (1) The molecule must have a pemanent dipole moment, () The fequency of adiation absobed in the pocess of going fom a level J to J + 1 is given by ν εj+ 1 εj h = = ( J + 1) J = 0,1,, h (6-10) 4π I We thus expect absoption of adiation at fequencies given by multiples of h/4π I and should obseve a set of equally spaced spectal lines, which fo typical molecula values of µ and e will be found in the micowave egion. Expeimentally one does see a seies of almost equally spaced lines in the micowave specta of linea molecules. The usual units of fequency in this egion ae wave numbes, o ecipocal wavelengths. ω (cm ) 1 1 λ ν c = = (6-11) Micowave spectoscopists define the otational constant the enegy of igid oto (in cm -1 ) becomes π B by h/ 8 I c (units of cm -1 ), so that ε = BJ( J + 1) (6-1) J Table 6-1 1ists the values of B fo seveal diatomic molecules. 8

4-1 Heteonuclea Diatomic Molecules Fo heteonuclea diatomic molecules, the calculation of the otational patition function is staightfowad. The otational patition function is given by ot βε ( 1) ( ) J BJ J + ωj (6-6) q ( T) = e = J + 1 e β J J= 0 (Note: this is a summation ove levels athe than ove states.) B Θ, the chaacteistic tempeatue of otation. (Table 6-1) k Unlike the vibational case, this sum cannot be witten in closed fom. APPROXIMATIONS TO THE PARTITION FUNCTION At high enough tempeatues, ( Θ / T is quite small at odinay tempeatues fo most molecule) Θ J( J+ 1) Θ J( J+ 1) T qot ( T) = (J + 1) e dj = e d[ J( J + 1) ] = Θ 0 0 8π IkT = Θ h T (6-7,8,9) This is called the high-tempeatue limit. (It is eally ε ( ) / kt = Θ J + 1 / T that must be small compaed to one, and this of couse cannot be tue as J inceases. Howeve, by the time J is lage enough to contadict this, the tems ae so small that it makes no diffeence.) Fo low tempeatues o fo molecules with lage values of Θ, e.g., HD with Θ = 4.7 K, one can use the sum diectly. Fo example, q ( ) 1 3 5 7 ot T e e e Θ / T 6 Θ / T 1 Θ / T = + + + + (6-30) is sufficient to give the sum to within 0.1 pecent fo Θ > 0.7T. Fo intemediate tempeatues i.e., Θ < 0.7T, but not small enough fo the integal to give a good appoximation, we need some intemediate appoximation. The eplacement of a sum by an integal can be viewed as the fist of a sequence of appoximations. The full scheme is a standad esult of the field of the calculus of finite diffeences and is called the Eule-MacLauin summation fomula. It states that if f ( n ) is a function defined on the integes and continuous in between, then 9

b b j f( b) + f( a) B j ( j 1) ( j 1) f ( n) = f( n) dn+ + ( ) f ( a) f ( b) ( j )! (6-31) n= a a j= 1 ( k whee f ) ( a) is the kth deivative of f evaluated at a. The B j 's ae the Benoulli numbes, B1 = 1, B 1 =, B3 = 1, Befoe applying this to [cf. Eq. (6-0)] q ( T) ot 6 30 4, let us apply it fist to a case we can do exactly. Conside the sum 3 α j 1 1 1 α α e = = + + + (6-33) α 1 e α 1 70 j= 0 Since 3 1 1 1 1 α α e = α α α = α + + α + 6 3 1 1 70 + (6-34) We see that these two expansions ae the same. If α is lage, we can use the fist few tems of (6-33); othewise, we use the Eule-MacLauin expansion in α. Applying this fomula to q ( T) ot gives (see Poblem 6-9): q ot 3 T 1 Θ 1 Θ 4 Θ ( T) = 1+ + + + (6-35) Θ 3 T 15 T 315 T which is good to within one pecent fo Θ < T. Fo simplicity we shall use only the hightempeatue limit in what we do hee since Θ T fo most molecules at oom tempeatue (cf. Table 6-1). THERMODYNAMICS ASSOCIATED WITH ROTATION The otational contibution to the themodynamic enegy is ( lnt ) ln q + T T ot Eot = NkT NkT NkT and the contibution to the heat capacity is + (6-36) 10

CV,ot = Nk+ (6-37) The faction of molecules in the J-th otational state Θ ( 1)/ ( 1) J J + N T J J + e = (6-38) N q ( T) ot Figue 6-5 shows this faction fo HCl at 300 o K. Contay to the vibational case, most molecules ae in excited otational levels at odinay tempeatues. We can find the maximum of this cuve by diffeentiating (6-38) with espect to J to get J max 1/ 1/ kt 1 kt T = = B B Θ 1/ We see then that Jmax moment of inetia of the molecule since inceases with T and is invesely elated to 1 B. I B, and so inceases with the 4- The Symmety Requiement fo a Homonuclea Diatomic Molecule Fo homonuclea diatomic molecules, the calculation of the otational patition function is not quite so staightfowad. 11

The total wave function of the molecule, that is, the electonic, vibational, otational, tanslational, and nuclea wave function, must be eithe symmetic o antisymmetic unde the intechange of the two identical nuclei. It must be symmetic if the nuclei have integal spins (bosons), o anti symmetic if they have half-integal spins (femions). This symmety equiement has pofound consequences on the themodynamic popeties of homo nuclea diatomic molecules at low tempeatues. We shall discuss the intechange of the two identical nuclei of a homonuclea diatomic molecule in this section, and then apply the esults to the calculation of in the next. q ot An exchange of the nuclei = (1) an invesion of all the paticles, electons and nuclei, though the oigin, & () an invesion of just the electons back though the oigin. Let us wite ψ total exclusive of the nuclea pat as ψ ' total = ψ tans ψ ot ψ vib ψ elec (the pime indicates the nuclea contibution being ignoed) ψ tans depends only upon the coodinates of the cente of mass of the molecule, and so this facto is not affected by invesion. ψ vib depends only upon the magnitude of e, and so this pat of the total wave function is unaffected by any invesion opeation. ψ elec unde the invesions in both Steps (1) and () above depends upon the symmety of the gound electonic state of the molecule. The gound electonic state of most molecules (designated by + g ψ ot theefoe contols the symmety of ψ total. Only Step (1) affects ψ ot. tem symbol) is symmetic unde both of these opeations. The effect of this invesion is to change (,, ) θ φ to (, π θ, φ π) + that descibe the oientation of the diatomic molecule. One can see this eithe analytically fom the eigenfunctions themselves o pictoially fom the otational wave functions shown in Fig. 6-6. (Notice that the igid oto wave functions ae the same functions as the angula functions of the hydogen atom.) + When the gound electonic state is symmetic, ψ g total emains unchanged fo even J and changes sign fo odd J. This esult applies to the total wave function, exclusive of nuclea spin. 1

NB: J=0,, is symmetic with espect to (, θ, φ ) (, π θ, φ π) antisymmetic. A molecule such as H : +, whee as J=1, is Thee ae two nuclei each with spin of 1. The two nuclei will have total (I+1)(I+1)=4 spin functions: αα, ββ, and ( αβ + βα ) /, and ( αβ βα ) / Since nuclei with spin 1/ act as femions, the total wave function must be antisymmetic in the exchange of these two nuclei. The fist thee spin functions ae symmetic, and theefoe necessaily couple with odd values of J to satisfy the equied anti-symmety. The last spin function is anti-symmetic, theefoe mut couple with even values of J to satisfy the equied anti-symmety Since thee symmetic nuclea spin functions be combined with the odd J levels to achieve the coect oveall antisymmety fo + g electonic states, the odd J levels have a statistical weight of 3 and even J levels have a weight of 1. This leads to the existence of otho- (paallel nuclea spins) states and paa- (opposed nuclea spins) states in H. This weighting of the otational states will be seen shotly to have a pofound effect on the low-tempeatue themodynamics of H. 13

Moe geneal case: Total # of spin states : Fo nuclei of spin I, thee ae 1 + 1 spin states fo each nucleus. Let the eigenfunctlons of these spin states be denoted by α1, α, α I + 1. (Fo H, I = 1/, so thee ae spin states,α and β ) Total nuclea wave functions: Fo diatomic molecule, thee ae ( ) (I1+ 1) (I + 1) = I + 1 nuclea wave functions to include in total I + 1 =4 nuclea spin functions.) The anti-symmetic nuclea spin functions ae of the fom αi(1) α j() αi() α j(1), 1 i, j I + 1. Thee ae I + 1C = ( I + 1)( I) / = I(I + 1) such combinations fo the numbe of antisymmetic nuclea spin functions. (Fo H, I + 1 I = 1 is anti-symmetic choice.) ( ) ψ (Fo H, ( ) ( ) ( ) ( )( ) The symmetic nuclea spin functions : All the emaining nuclea spin functions I + 1 I I + 1 = I + 1 I + 1 ae symmetic. (Fo H, ( I + 1)( I + 1) = 3 ae antisymmetic choices.) The summay fo + g states; half-integal spin (femions-anti-symmetic) I(I + 1) antisymmetic nuclea spin functions couple with even J ( I + 1)( I + 1) symmetic nuclea spin functions couple with odd J integal spin (bosons-symmetic) I(I + 1) antisymmetic nuclea spin functions couple with odd J. ( I + 1)( I + 1) symmetic nuclea spin functions couple with even J These combinations of nuclea and otational wave functions poduce the coect symmety equied of the total wave function unde intechange of identical nuclei. Remembe that all of these conclusions ae fo Poblem 6-6 fo a discussion of O.) + g electonic states, the most commonly occuing gound state. (See Even though we have consideed only diatomic molecules hee, the esults of this section apply also to linea polyatomic molecules such as CO, H C. Fo example, the molecules HC l C l H and DC l C l D have thei otational states weighted in a simila way as H and D, Figue 6-7 shows the vibation-otation spectum of H C. The altenation in the intensity of these otational lines due to the statistical weights is vey appaent. 14

4-3 The Rotational Patition Function of a Homonuclea Diatomic Molecule The esults of the pevious section show that fo homonuclea diatomic molecules with nuclei having integal spin, otational levels with odd values of J must be coupled with the I(I + 1) antisymmetic nuclea spin functions, and that otational levels with even values of J must be coupled with the ( I + 1)( I + 1) symmetic nuclea spin functions. Thus we wite q ( T) = ( I + 1)(I + 1) (J + 1) e ot,nucl J odd J even + I(I + 1) (J + 1) e Θ J( J+ 1)/ T Θ J ( J+ 1)/ T (6-40) Likewise, o molecules with nuclei with half intege spins, q ( T) = I(I + 1) (J + 1) e ot,nucl J even J odd Θ J( J+ 1)/ T + ( I + 1)(I + 1) (J + 1) e Θ J ( J+ 1)/ T (6-41) Notice that in this case the combined otational and nuclea patition function does not facto into qot qnucl. This is a situation in which we cannot ignoe qnucl. Fo most molecules at odinay tempeatues, Θ T, and we can eplace the sum by an integal. We see then that 1 1 (6-4) Θ J( J+ 1)/ T (J + 1) e dj = 0 Jeven Jodd Jall Θ T 15

and so both (6-40) and (6-41) become q ot,nucl (I + 1) T ( T) = Θ (6-43) which can be witten as q ( T) q ( T) ot nucl whee T qot ( T) = qnucl( T) = (I 1) Θ + (6-44) valid when Θ / T 0.. To compaed to the esult fo a heteonuclea diatomic molecule: ( ) T qot T = Θ The facto of that appeas above in the high-tempeatue limit takes into account that the molecule is homonuclea, and so its otational patition function is given by (6-40) o (6-41) instead of (6-6). This facto of is called the symmety numbe and is denoted by σ. It legitimately appeas only when Θ / T 0.. Undestanding the oigin of this fact then, we can wite 8π IkT 1 (6-45) Θ J( J+ 1)/ T qot ( T) (J + 1) e σh σ J = 0 Θ T whee σ = 1 fo heteonuclea molecules, and σ = fo homonuclea diatomic molecules. Remembe that this is applicable only to the high-tempeatue limit o its Eule-MacLauin coection. A simila facto will appea fo polyatomic molecules also. 4-5 Nuclea Patition Function & Related Issues Thee ae some inteesting systems in which Θ / T is not small. Hydogen is one of the most impotant such cases. Each nucleus in H has nuclea spin 1/, and so ot,nucl Θ J ( J+ 1)/ T Θ J( J+ 1)/ T (6-46) Jeven Jodd q ( T) = (J + 1) e + 3 (J + 1) e The hydogen with only even otational levels allowed (antisymmetic nuclea spin function o "opposite " nuclea spins) is called paa-hydogen; that with only odd otational levels allowed 16

(symmetic nuclea spin function o "paallel " nuclea spins) is called otho-hydogen. The atio of the numbe of otho-h molecules to the numbe of paa-h molecules is 3 (J + 1) e Notho J odd = N (J + 1) e paa J even Θ J ( J+ 1)/ T Θ J ( J+ 1)/ T Figue 6-8 shows the pecentage of p-h vesus tempeatue in an equilibium mixtue of othoand paa-hydogen. Note that the system is all paa- at O K and 5 pecent paa- at high tempeatues. Figue 6-9 illustates an inteesting situation that occus with low-tempeatue heat capacity measuements on H. Equation (6-46) can be used to calculate the heat capacity of H, and this is plotted in Fig. 6-9, along with the expeimental esults. It can be seen that the two cuves ae in geat disageement. These calculations and measuements wee made at a time when quantum mechanics was being developed, and was not accepted by all scientists. Fo a while, the disageement illustated in Fig. 6-9 was a blow to the poponents of the new quantum mechanics. It was Dennison* who finally ealized that the convesion between otho- and paa-hydogen is extemely slow in the absence of a catalyst, and so when hydogen is pepaed in the laboatoy at oom tempeatue and then cooled down fo the low-tempeatue heat capacity measuements, the oom-tempeatue composition pesists instead of the equilibium composition. Thus the expeimental data illustated in Fig. 6-9 ae not fo an equilibium system of otho- and paahydogen, but fo a metastable system whose otho-paa composition is that of equilibium oomtempeatue hydogen, namely, 75 pecent otho- and 5 pecent paa-. If one calculates the heat capacity of such a system, accoding to 3 1 C = C + C 4 4 ( otho-) ( paa-) V V V 17

whee CV (otho-) is obtained fom just the second tem of Eq. (6-46), and CV (paa-) is obtained fom the fist tem of Eq. (6-46), one obtains excellent ageement with the expeimental cuve. A cleve confimation of this explanation was shotly afte obtained by Bonhoeffe and Hateck, t who pefomed heat capacity measuements on hydogen in the pesence of activated chacoal, a catalyst fo the otho-paa convesion. This poduces an equilibium system at each tempeatue. The expeimental data ae in excellent ageement with the equilibium calculation in Fig. 6-9. The explanation of the heat capacity of H was one of the geat tiumphs of post- quantum mechanical statistical mechanics. You should be able to go though a simila agument fo D sketching the equilibium heat capacity, the pue otho- and paa- heat capacity, and finally what you should expect the expeimental cuve to be fo D pepaed at oom tempeatue and at some othe tempeatue, say 0 K. (See Poblem 6-17.) In pinciple, such nuclea spin effects should be obsevable in othe homonuclea molecules, but a glance at Table 6-1 shows that the chaacteistic otational tempeatues fo all the othe molecules ae so small that these molecules each the "high-tempeatue limit" while still in the solid state. Hydogen is somewhat unusual in that its otational constant is so much geate than its boiling point. Fo most cases then, we can use (6-45) which, when we use the Eule-MacLauin expansion, becomes 18

q ot 3 T Θ 1 Θ 4 Θ ( T) = 1+ + + + (6-47) σ Θ 3 T 15 T 315 T Usually only the fist tem of this is necessay. Some of the themodynamic functions ae (6-49) E C S ot ot ot 1 Θ 1 Θ = NkT + (6-48) 3 T 45 T 1 Θ = Nk 1 + + (6-49) 45 T σ Θ 1 Θ = Nk 1 ln + (6-50) T 90 T whee all of these fomulas ae valid in the same egion, in which σ itself is a meaningful concept, that is, Θ < 0.T. The tems in Θ / T and its highe powes ae usually not necessay. Note that (6-47) is identical to (6-35) except fo the occuence of the symmety numbe in (6-47). 5 THERMODYNAMIC FUNCTIONS Having studied each contibution to the total patition function q in (6-17), we can wite in the hamonic oscillato-igid oto appoximation 3/ βν h / πmkt 8π IkT e De / kt qv (, T) = V ω h e1e βν h σ h 1 e (6-51) Remembe that this equies that Θ T, that only the gound electonic state is impotant, and that the zeo of enegy is taken to be the sepaated states at est in thei gound electonic states. Note that only q tans is a function of V, which, we have seen befoe, is esponsible fo the ideal gas equation of state. The themodynamic functions associated with (6-51) ae E 5 hν hν / kt De = + + hν / kt NkT kt e 1 kt (6-5) hν / kt CV 5 hν e = + Nk kt e 1 hν / kt ( ) (6-53) 19

3/ 5/ 1+ m kt hν / kt S π( m ) Ve 8 π IkTe hν / kt hν / kt = ln ln ln ( 1 e ) ln e1 Nk ω h + + + (6-54) N σ h e 1 pv = NkT (6-55) 0 3/ ( ) π ( m1+ m) kt 8 kt µ T π IkT hν hν / kt De = ln ln ln ( 1 e ) ln kt + + ω h σ h kt kt e1 (6-56) Table 6-1 contains the chaacteistic otational tempeatues, the chaacteistic vibational 1 tempeatues, and D0 = De hν fo a numbe of diatomic molecules. Table 6-3 pesents a compaison of (6-54) with expeimental data. It can be seen that the ageement is quite good and is typical of that found fo the othe themodynamic functions. 6 Futhe Impovements It is possible to impove the ageement consideably by including the fist coections to the igid oto-hamonic oscillato model. These include centifugal distotion effects, anhamonic effects, and othe extensions. The consideation of these effects intoduces a new set of molecula constants, all of which ae detemined spectoscopically and ae well tabulated. (See Poblem 6-4.) The use of such additional paametes fom spectoscopic data can give calculated values of the entopy and heat capacity that ae actually moe accuate than expeimental ones. Note that extemely accuate calculations equie a sophisticated knowledge of molecula spectoscopy. 0

1