The Rotation / Vibration Spectrum of HCl

Index

Introduction
Quantum Vibration and Rotation
Electric Dipole IR Transitions in Molecules
Experimental Procedure
Calculations and Discussion

Infra-Red Transitions

Since both rotational and vibrational motions are simultaneously occurring in the diatomic, the energy level scheme for two adjacent vibration levels (spaced, say, 3000 cm^-1 apart) where

ca. 10 cm^-1 in the two states, would be as shown in Fig. 1 (Transitions between levels indicated there will be referred to later.). Each vibrational state has a set of rotational levels. These rotational levels are not exactly the same since

varies slightly with v, the quantum number labeling the vibrational state. (This latter effect is called vibration-rotation coupling)

Figure 1: Transitions between the lowest vibrational state, denoted v=0, and the next highrest vibrational state (v=1). Each vibrational state has many rotational levels denoted by a rotational quantum number (J' for the upper vibrational state, and J" for the lower). The spacing between rotational levels of the same vibrational state has been greatly exagerrated for clarity.

Equations (1) and (3) may be combined and if expressed as Term Values of a given state and level of the molecule as labelled by v and J, T(v,J) [in units of cm^-1 then

However, these are only a first approximation to the true Term Values since there are small additional terms to be added to account for departures from our simplest models of vibration and rotation:

Here

subfix _e means a quantity defined at the equilibrium distance R_e of the molecule

the term in

_ex_e is a small correction for departure from harmonic vibrational motion (i.e., a non-parabolic potential energy). This correction is termed the anharmonicity of the bond. Anharmonicity must occur in all chemical bonds since at a certain energy all chemical bonds break.

the term D_e corrects for the non-rigidity of the bond which can be expected to stretch as the molecules rotates, moreso at the high J values (i.e., centrifugal distortion)

the term

_e is due to the interaction between rotation and vibration; since R changes during the vibrational motion, so does I and therefore

Quantum mechanics also stipulates that only the certain transitions shown in Fig. 1 are "allowed" via electric dipole radiation. The selection rules for this process are such that both v and J change by 1 quantum in allowed transitions. (v=+/-1, J=+/-1) Only molecules with permanent electric dipole moments may exhibit infrared absorption, v=+1,or emission, v=-1. J=0 is allowed only for electronic transitions in some diatomic molecules and non-linear polyatomic molecules. Traditionally, the J transitions are grouped together as branches (see Fig. 1):

J = + 1, R branch
J = - 1, P branch
J = 0, Q branch (forbidden here)

To a first approximation the length of the arrows in Fig. 1 (proportional to the frequency of the transition) differ by 2 among the P branch transitions and among the R branch transitions, but with 4 between the central P(1) and R(0) lines. If the forbidden Q branch transition v=0, J"=0 to v=1, J'=0 (call it _o) were included in the Figure then all spacings would be 2. If the very small term involving D_e is neglected in Eq. (4), the combined rotation-vibration transitions in Fig. 1 can be written

For least squares fitting the P and R equations can be conveniently combined by letting m=(J"+1) in the R branch and (m = -J") in the P branch, yielding

(Note that there is no value corresponding to _o (m=0) for this molecule.) For example, the transitions in Fig. 1 can be assigned values of m from -5 to +5 (with 0 missing) and a plot of (m) versus m will yield _e and _e. Then by use of Eq. (7) one can also find _o. A typical spectrum is given in Fig. 2.

Relative Intensities of Aborption Lines

As illustrated in Fig. 2, all absorption lines are not expected to have the same intensity. The relative intensities are governed largely by the populations of molecules occupying the various rotational levels in the ground v=0 vibrational state (where the rotational levels are designated by J"):

where
"_Rot=hc_eJ"(J"+1).

This is the Boltzmann distribution, where the factor 2J"+1 corresponds to the degeneracy g_J" of a given rotational level. One can see that there would be an exponential decrease in intensity with increasing J" except for the partially compensating factor of g_J" causing a maximum in the relative intensities.

Isotopic Substitution

Both the vibrational frequency , (Eq. (2)), and the rotational constant (through the moment of inertia, I) are affected by a change in the reduced mass of the molecule. Changing the mass by the introduction of an isotope, such as D for H in HCl, will not change the force constant k or the equilibrium bond distance R_e since the character of the chemical bond is unaffected. Then from Eq. (2).

Note that Eq. (9) is a first approximation, anharmonicity will change the calculated ratio slightly. HCl (or DCl) prepared in the laboratory already contains two kinds of isotopomers, H³⁵Cl and H³⁷Cl, since the natural abundance of isotopes of chlorine is 75.5% ³⁵Cl and 24.5% ³⁷Cl, but the isotope shift between the chlorine isotopes is much smaller than that induced by deuteration as can be seen by the much smaller reduced mass ratio of (H³⁷Cl) / (H³⁵Cl) relative to (D³⁵Cl) / (H³⁵Cl) (which is approximately 2).

Since the two dominant parameters, and , in Eq. (4) are different for the different isotopes, one can expect a different spectrum for each isotopomer. One can, in first approximation, calculate the shifts involved from Eqs. (9) and (10). In the case of a mixture of H³⁵Cl and H³⁷Cl (or D³⁵Cl and D³⁷Cl), the chlorine isotope effect is small and the spectrometer used in the measurements may have insufficient resolution to detect the small splittings ( <1cm^-1) in the absorption lines in Fig. 2 due to this effect.

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This document originates from the notes of W. Weltner, Jr.

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