Gaseous systems composed of molecules with internal motion. Diatomic molecules.

Gaseous systems composed of molecules with internal motion In most of our studies so far we have consider only the translation part of the molecular motion. Though this aspect of motion is invariably present in a gaseous system, other aspects, which are essentially concerned with the internal motion of the molecules, also exist. It is only natural that in the calculation of the physical properties of such a system, contributions arising from these motions are also taken into account. In doing so, we shall assume here that a) the effects of the intermolecular interactions are negligible and

b) the nondegeneracy criterion (7.1) is fulfilled; effectively, this makes our system an ideal, Maxwell- Boltzmann gas. Under these assumptions, which hold sufficiently well in a large number of practical applications, the partition function of the system is given by (7.2) where (7.3)

The factor in brackets is the transitional partition function of a molecule, while the factor j(T) is supposed to be the partition function corresponding to the internal motions. The latter may be written as (7.4) where i is the molecular energy associated with an internal state of motion (which characterized by the quantum numbers i), while gi represents the degeneracy of that state. The contributions made by the internal motions of the molecules to the various thermodynamic quantities of the system follow straightforwardly from the function j(T). We obtain

Fint= - N kT lnj int= - kT lnj Eint=NkT2 (7.5) (7.6) (7.7) (7.8) (7.9)

j(T)=jelec(T) jnuc(T) jvib(T) jrot(T) Thus the central problem in this study consists of deriving an explicit expression for the function j(T) from a knowledge of the internal states of the molecules. For this purpose, we note that the internal state of a molecule is determined by: electronic state, state of nuclei, vibrational state and rotational state. Rigorously speaking, these four modes of excitation mutually interact; in many cases, however, they can be treated independently of one another. We then write j(T)=jelec(T) jnuc(T) jvib(T) jrot(T) (7.10) with the result that the net contribution made by the internal motions to the various thermodynamic quantities of the system is given by a simple sum of the four respective contributions.

Monatomic molecules At the very outset we should note that we cannot consider a monatomic gas except at temperatures such that the thermal energy kT is small in comparison with the ionization energy Eion; for different atoms, this amounts to the condition: T<<Eion/k104-105 oK. At these temperatures the number of ionized atoms in the gas would be quite insignificant. The same would be true for atoms in excited states, for the reason that separation of any of the excited states from the ground state of the atom is generally comparable to the ionization energy itself. Thus, we may regard all the atoms of the gas to be in their (electronic) ground state.

Now, there is a well-known class of atoms, namely He, Ne, A, Now, there is a well-known class of atoms, namely He, Ne, A,..., which, in their ground state, possess neither orbital angular momentum nor spin (L=S=0). Their (electronic) ground state is clearly a singlet: ge=1. The nucleus, however, possesses a degeneracy, which arises from the possibility of different orientations of the nuclear spin. (As is well known, the presence of the nuclear spin gives rise to the so-called hyperfine structure in the electronic state. However the intervals of this structure are such that for practically all temperatures of interest they are small in comparison with kT.) If the value of this spin is Sn, the corresponding degeneracy factor gn=2Sn+1. Moreover, a monatomic molecule is incapable of having any vibrational or rotational states

The internal partition function (7 The internal partition function (7.10) of such a molecule is therefore given by Fint= - N kT lnj int= - kT lnj Eint=NkT2 j(T)=ggr.st.=ge gn=2Sn+1 (7.11) Equations (7.4-7.9) then tell us that the internal motions in this case contribute only towards properties such as the chemical potential and the entropy of the gas; they do not make contribution towards properties such as the internal energy and the specific heat. In other cases, the ground state of the atom may possess both orbital angular momentum and spin (L0,S0- as, for example, in the case of alkali atoms), the ground state would then possess a definite fine structure.

The intervals of this structure are in general, comparable with kT; hence, in the evaluation of the partition function, the energies of the various components of the fine structure must be taken into account. Since these components differ from one another in the value of the total angular momentum J, the relevant partition function may be written as (7.12) The forgoing expressions simplifies considerably in the following limiting cases: kT >> all J ; then (7.13)

kT<< all J ; then (7.14) where J0 is the total angular momentum, and 0 the energy of the atom in the lowest state. In their case, the electronic motion makes no contribution towards the specific heat of the gas. And, in view of the fact that both at high temperatures the specific heat tends to be equal to the translational value 3/2 Nk, it must be passing through a maximum at a temperature comparable to the separation of the fine levels. Needless to say, the multiplicity (2Sn+1) introduced by the nuclear spin must be taken into account in each case.

Diatomic molecules Now, just as we could not consider a monatomic gas except at temperatures for which kT is small compared with the energy of ionization, for similar reasons one may not consider a diatomic gas except at temperatures for which kT is small compared with the energy of dissociation; for different molecules this amounts once again to the condition: T<<Ediss/k104-105 oK. At this temperatures the number of dissociated molecules in the gas would be quite insignificant.

Cv=(Cv)elec+(Cv)vib+(Cv)rot At the same time, in most cases, there would be practically no molecules in the excited states as well, for the separation of any of these states from the ground state of the molecule is in general comparable to the dissociation energy itself. The heat capacitance of the diatomic gas is consist from three parts Cv=(Cv)elec+(Cv)vib+(Cv)rot (7.15) Let us consider them consequently. In the case of electron contribution the electronic partition function can be written as follows (7.16)

In particular we obtain for the contribution towards specific heat where g0 and g1 are degeneracy factors of the two components while  is their separation energy. The contribution made by (7.16) towards the various thermodynamic properties of the gas can be readily calculated with the help of the formula (7.4-7.9). In particular we obtain for the contribution towards specific heat (7.17) We note that this contribution vanishes both for T<</k and for T>>/k and has a maximum value for a certain temperature /k; cf. the corresponding situation in the case of monatomic atom.

Let us now consider the effect of vibrational states of the molecules on the thermodynamic properties of the gas. To have an idea of the temperature range, over which this effect would be significant, we note that the magnitude of the corresponding quantum of energy, namely , for different diatomic gases is of order of 103 oK. Thus we would obtain full contributions (consistent with the dictates of the equipartition theorem) at temperatures of the order of 104 oK or more, and practically no contribution at temperatures of the order of 102 oK or less. We assume, however, that the temperature is not high enough to excite vibrational states of large energy; the oscillations of the nuclei are then small in amplitude and hence harmonic.

The energy levels for a mode of frequency  are then given by the well-known expression (n+1/2) h/2. The evaluation of the vibrational partition function jvib(T) is quite elementary. In view of the rapid convergence of the series involved, the summation may formally be extended to n=. The corresponding contributions towards the various thermodynamic properties of the system are given by eqn.(4.64 -4.69). In particular, we have (7.18) We note that for T>>v the vibrational specific heat is very nearly equal to the equipatition value Nk; otherwise, it is always less than Nk. In particular, for T<<v , the specific heat tends to zero (see Figure 7.1); the vibrational degrees of freedom are then said to be "frozen".

Figure 7.1 The vibrational specific heat of a gas of diatomic molecules. At T=v the specific heat is already about 93 % of the equipartition value.

Finally, we consider the effect of the states of the nuclei and the rotational states of the molecule: wherever necessary, we shall take into account the mutual interaction of these modes. This interaction is on no relevance in the case of the heternuclear molecules, such as AB; it is, however, important in the case of homonuclear molecules, such as AA. In the case of heternuclear molecules the states of the nuclei may be treated separately from the rotational states of the molecule. Proceeding in the same manner as for the monatomic molecules we conclude that the effect of the nuclear states is adequately taken care of through degeneracy factor gn. Denoting the spins of the two nuclei by SA and SB, this factor is given by

gn= (2SA+1)(2SB+1) (7.19) As before, we obtain a finite contribution towards the chemical potential and the entropy of the gas but none towards the internal energy and specific heat. Now, the rotational levels of a linear "rigid" with two degrees of freedom (for the axis of rotation) and the principle moments of inertia (I, I, 0), are given by (7.20) here I=M(r0)2 , where M=m1m2/(m1+m2) is the reduced mass of the nuclei and r0 the equilibrium distance between them. The rotational partition function of the molecule is then given by

The summation (7.21) is the replaced by integration: For T>>r the spectrum of the rotational states may be approximated by a continuum. The summation (7.21) is the replaced by integration: (7.22)

The rotational specific heat is the given by (CV)rot=Nk (7.23) which is indeed consistent with equipartition theorem. A better evaluation of the sum (7.21) can be made with the help of the Euler-Maclaurin formula Putting one can obtain (7.24)

which is the so-called Mulholland's formula; as expected, the main term of this formula is identical with the classical partition function (7.22). The corresponding result for the specific heat is (7.25) which shows that at high temperatures the rotational specific heat decreases with temperatures and ultimately tends to the classical value Nk.

Fig.7.2. The rotational specific heat of a gas of heteronuclear diatomic molecules. Thus, at high (but finite) temperatures the rotational specific heat of diatomic gas is greater than the classical value. On the other hand, it must got to zero as T 0. We, therefore, conclude that it must pass through at least one maximum. (See Figure 7.2)

In the opposite limiting case, namely for T<<r , one may retain only the first few terms of the sum (7.21); then (7.26) whence one obtains, in the lowest approximation (7.27) Thus, as T 0, the specific heat drops exponentially to zero (Fig. 7.2). Now we can conclude that at low temperatures the rotational degrees of freedom of the molecules are also "frozen".