HANAN ABDULLA ISSA ID.# Dr. Abdulla Obeidat

Transcription

1 eous systems Composed Molecules with Internal Motion HANAN ABDULLA ISSA ID.# Dr. Abdulla Obeidat

2 INTRODUCTION olecules in a room temperature gas can be considere tion of non-interacting particles. In statistical mecha eans that the overall partition function Q N (V,T of N ules is separable into a product of single particle par ion Q(V,T : Q ( V, T Q V T N! N [ N (, ]...[] energy of each gas molecules can be separate ional and internal energies; this leads to a separab tition function: T j ( T j ( T j ( T j ( T j ( T trans elec nuc rot vib

3 Monatomic Molecules toms in a monatomic gas have no vibrational or rot of freedom, so the single partition function is; ( V, T j ( V, T j ( T j ( T trans elec nuc...[ e case of(ls0,the ground state; ge. And the causes the degeneracy of nucleus; (gn Sn. j( T j elec ( T j nuc ( T g e g n (S n...[5 s case; the nuclear spin gives rise to the hyperfine st lectronic states with very small intervals compared to

4 of the gas; while the internal energy and the specif change. es: He,Ne,Ar molecules. e case(l0,s 0;the ground state have no fine str. So the internal partition function will be : j( T g e g n (S (S n...[6] ntropy and the chemical potential of the gas will be c s: Alkali Metals.

5 as intervals comparable with the thermal energy kt. e electronic partition function will be: j elec ( T (J e ε J / kt...[7] ome exception to this rule are Oxygen atoms and H F,Br,Cl,. which have two atomic states, P 3/ an ed by a rather small energy Δ0.5, 0., 0.46, and 0 l, Br, and I, respectively. Thus, for the halogen atoms j elec ( T 4( e βδ...[8] ave for the ground nuclear level :

6 (ε J /kt <<; then j ( T (J ( L (S...[7. a] elec 0 r (ε J /kt >>; then j elec ( T (J 0 e ε / kt 0...[7. b] both high and low temperatures; the specific heat i e translational value (Cv3/Nk. So, electronic mo as doesn t contribute to the specific heat of the gas onclude that for monatomic gases most thermod rties are given by the following equations;

7 A NkT int ln j int Energy: ( T...[0] U int T T A T int NkT j int T j int ( T...[] Entropy: U A U S int int int int T T Nk ln j int ( T...[ ] Specific heat capacity: ( C v int U int T Nk jint ( T T v T j ( T T int...[3] Chemical potential:

8 Diatomic Molecules e we will use the result of two useful approximations Oppenheimer approximation; which is used to separat onic and nuclear motion in the quantum mechanical wa pproximation states that the electrons respond rapi clear motion and therefore represent a potential in t of which nuclei move. We are concerned here with nu n in a diatomic molecules. second approximation that separates rotational and rational motion is the rigid rotator approximation. internal partition function for diatomic molecule is g T j ( T j ( T j ( T J ( T...[

9 contribution of the nuclear spin to the partition fun arise from the degeneracy of the nuclear spin state. heteronuclear molecules j ( T (S (S...[6. a] nuc A B homonuclear molecules j ( T (S nuc A...[6. b]

10 The Electronic Partition Function Δβ ctronic energy separation from the ground state usually very large, so for most cases ge. An ortant exception arise in the case of atoms and ecules having electronically degenerate ground tes, in this case jelec(t ge me atoms and molecules have low lying electron ed states. (At high temperatures, all atoms and mole thermally accessible excited states. diatomic molecules, the electronic partition function by:

11 al temperatures, all four states ermally accessible. If we denote rgies of the two levels as E/0 /Є, the partition function for the NO molecule is: g j ylevels e βε e βε e 78/

12 in for Diatomic molecules, thermodynamic propertie lculated from equations [0]-[4]. The specific hea :... / ( ( / 0 / 0 Δ Δ Δ kt kt elec v e g g e g g kt Nk C high temperatures (kt>> and low temperatures ( contribution vanishes and maximum for eratures (kt~.

13 he Vibrational Partition Function s influenced by the centrifugal force due to molecula ion. e vibrational motion of the molecule can be assumed e harmonic oscillator, and the vibrational partition fu iatomic molecule is: j vib ( T e Θ e v / T Θ v / T Θ Sinh v T...[9] e Θvħω/k, is the characteristic vibrational tempera n all thermodynamic properties can be calculated. F ic heat, we have: ( C v vib Nk Θ v e Θ T v ; Θ v hω...[0]

14 rature region T>> Θ v, is then ( C v Nk, while for, as illustrated in figure[].

15 high temperatures, vibrations with large n are excite use anharmonicity and interaction between vibratio onal modes must be taken into account. In this case: E n n hω χ n hω ( n 0,,.....[] Where X<<, indicates the degrees of anharmonicity. the specific heat will be proportional to the temperat the gas: ( Cv vib 4 Nk u u 4χ u 40 u [] ere u ħω/kt.

16 Heteronuclear Molecules necessary to write the allowed rotational energy leve ntum mechanical form: ε rot ( l l h, l I 0,,......[ 3] is the degeneracy for each state, I is moment of in rotational partition function is: j rot ( t σ σ l 0 l 0 (l (l exp exp { l( l h IkT { l( l θ T }...[ 4] Θr is the characteristic rotational temperature r }

17 c molecules are highly excited. The table below shows eristic vibrational Θr and Θv temperature for some c molecules. Molecule Θv /k Θr /k H HD D N O CO NO

18 eplacement for the sum in equation [4] by an integra j ( t (l exp{ l( l θ T} dl rot r σ 0 exp{ yθ T} dy r σ 0 T T exp( yθ T...[5] r 0 σθ σθ r r y l(l, and σ for heteronuclear molecules. is high temperature limit, the rotational energy is: ε jr ( t T rot NkT NkT...[6] rotational specific heat is: ( C v rot Nk

19 mula that a better high-temperature approximation i j rot ( T T θ r 3 5 θ r T 4 35 θ r T......[7] use of equation [7] is to gives additive rot tions to thermodynamic properties. And Cv; the rot heat becomes: ( C v rot Nk 45 θ r T θ r T [8] r >>T, the first four terms in equation[4] are already t to give the sum to 0. percent:

20 e [3]: otational contribution to Cv for a heteronuclear d ules as a function of T/Θr.

21 re [4]: general features of the temperature dependence of capacity of diatomic molecules. Each mode becomes

22 [5]: ational vibrational specific heat Cp of the diatomic and DT.

23 Homonuclear Molecules is case, the state of the nuclear spin will determine t rotational wave function of the molecule because of symmetry of the molecule. given j, the nuclear spin wave function must be eithe ric or antisymmetric depending on the parity of l and cs of the nuclei. For two nuclei, each having the spin S re (Sn²states in total, among which (Sn(Sn ric and Sn(Sn are antisymmetric with respect to perm clear spin variables. can obtain the nuclear rotational partition function:

24 S (S r ( S (S r n n even n n odd... osons with an integral spin, (Sn0,.,. ( S (S r S (S r n n even n n odd.. r even and r odd are: even l 0,,.. (l exp. { l( l Θ / T}...[3. a] r r odd l,3,.. (l exp. { l( l Θ / T}...[3. b] r

25 ydrogen molecules (lodd, Para-hydrogen molecules, Ortho-deuterium molecules (leven and Para- deute es (lodd. e case of fermions (as in H, the ortho-to-para r : n F. D. ( S r A S r A even odd ; S A...[3. a] e case of Bosons (as in D, the corresponding ratio w n B. E. ( S r S r A A odd even ; S A...[3. b] w temperature, the ratio n tends to zero in the c ns and to infinity in case of Bosons. Then H gas is while D gas is wholly ortho-.

26 the theoretical results do not generally agree imental results. sion find out that the samples of hydrogen, or de prepared and kept at room temperature (above Өr, the to-para is very nearly equal to (Sn/ Sn. And rature is lowered, one would expect the ratio will cha ance with equation [3], but it doesn t do so??!!

27 transition probability of a molecule is very small, bec e flipping of the spin of one of the nuclei. eriods involved in this process are of order of a year nnot expect to get the equilibrium ratio n during the available. at low temperature, we have a non-equilibrium mixt dependent substances. efore, the partition function in equation [30] can and we have for the specific heat:

28 ...[33 ( (.. C S S C S S odd n n even n n E B Where; ]...[34. ln a r T T T Nk C even even ]...[34. ln b r T Nk C odd odd...[33 ( (.. C S S C S S odd n n even n n D F

29 [6]; with Ce ven and C odd for hydrogen.

30 olyatomic Molecule polyatomic molecules the electronic partition functi by j elec j trans g e0 ( V, T e β E e π mkt h...[35] translational partition function for polyatomic molec he same form as that for diatomic or monatomic ules: 3/ near molecules there are (3n-5 vibrational degrees of fre or non-linear molecules, there are (3n-6 vibrational deg. V

31 j vib ( T 3n 6 i e Θvi / T e Θvi / T Θ vi hω k i...[36] e ωi is the frequency of each normal mode in the mo efore the vibrational specific heat is the sum of the ibutions obtained from each normal mode: C vib Nk 3 5 or n 3n 6 i Θ T vi ( e e Θvi / T Θvi / T...[37] re [7] below shows two examples of normal mod r molecule (CO and non-linear molecule (HO.

32 e Three normal Four normal modes o

33 es is given by: j rot nuc ( T g n j classic rot σ ( T g n π σ Θ A T Θ 3 B Θ C / ; Θ i h I i k...[38] or linear molecules: T j rot ( T σθ r he rotational specific heat is given by: T T T ln j classic rot ( T 3 Nk ( for non linear molecu

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