The Vibration-Rotation Spectrum of Gaseous HCl

As a first approximation, the rotational and vibrational energy states of a diatomic molecule can be represented as a linear combination of harmonic oscillator (HO) vibrational and rigid-rotor (RR) rotational energies. Accepting that the vibrational quantum number change is +1, and that the rotational quantum number J must change by +1 or -1 , the calculated absorption spectrum is given by (cm^-1):

The spectrum is described as having three regions:

1. the P branch, in which the rotational quantum number J changes by -1;
2. the Q branch, in which J does not change
3. the R branch, in which J changes by +1.

Since the vibrational quantum number has changed by +1 under our conditions, the P branch lines lie to the low energy side of the Q branch, while the R branch lines lie to the high energy side. The Q branch line itself is not observed since the transition is forbidden. The above approximations (HO/RR) lead to the prediction of a series of equally spaced lines, a gap (the Q branch), and another series of equally spaced lines. The intensities of the lines are dependent upon the facility with which the HCl molecules interact with radiation, and upon the equilibrium populations of the rotational states. The first requires a quantum mechanical analysis of the transitions, while the second factor may be derived from the Maxwell-Boltzmann statistics.

Examination of an actual HCl spectrum (Fig 1) shows that although the general features predicted by Eqs. (1) and (2) are observed, the lines are not equally spaced in the P and R branches, suggesting that the HO/RR model is not completely satisfactory. The assumption of harmonic oscillator behavior implies that the potential function is a parabola, whereas we know that the actual potential function is more like the empirical Morse potential. An improved expression for the rotational and vibrational energy levels of a diatomic molecule can be obtained by solving the Schrdinger equation in which the anharmonic nature of the potential is explicitly recognized. Inclusion of the 3rd and 4th derivatives of the potential expansion leads to the following expression for the internal energy levels of a diatomic molecule (see, for example, I. Levine, Quantum Chemistry vol II: Molecular Spectroscopy, chapter 3, and secs. 1 through 10 of chapter 10 in McQuarrie, Quantum Chemistry):

The first term on the right hand side of Eq.(3) is the electronic energy plus the nuclear repulsion contribution. The second term is the HO approximation to the vibrational energy. The third term is the correction to the vibrational energy, and causes the vibrational level spacings to decrease with increasing energy. The term n₀c₀ is known as the anharmonicity constant. In Eq. (3), n₀ is the frequency (in Hz) for the molecule vibrating about its equilibrium internuclear separation r_e and is given by

where k is the force constant of the chemical bond, and m is the reduced mass. Also note that frequency and the wave number (defined as the inverse of the wave length, usually in cm^-1) of the absorbed radiation are related by the expression

Returning to Eq.(3), the fourth term is seen to be the RR approximation to the rotational energy, and the fifth term can be regarded as arising from the fact that the rotational and vibrational energies are not independent; this correction depends upon both the rotational and vibrational quantum numbers. The constant a_e is known as the rotational-vibrational coupling constant. The last term can be regarded as a correction to the rotational energies arising from centrifugal distortion. This last correction, known as the centrifugal distortion constant D_e, is very small and may not be supported by our spectral data. The following expressions will help clarify Eq. (3) (the subscripted e represents quantities evaluated at the equilibrium internuclear distance):

The spectral lines for the improved energy expression are calculated from Eq. (3) by assuming that the vibrational quantum number n changes by +1. For rotational quantum number J decreasing by 1 (P branch), the energy difference equation becomes:

For J increasing by 1 (R branch), the equation is:

These two equations can be fit separately , but by defining a variable z such that in the P branch z = -J and in the R branch z = J + 1, the single equation

can be written. Note that the variable z is referred to the rotational level J from which the transition originates. We will use Mathcad to evaluate the parameters of Eq. (10). The independent variable is z and the dependent variable is the energy of the transition in cm. Eq. (10) corresponds to the linear regression function, Eq. (12), and therefore values of w₀ , B_e, and a_e can be found using our experimental data.

Experimental

We will generate HCl gas from the reaction of concentrated sulfuric acid (H₂SO₄) with sodium chloride (NaCl). The HCl generator is set up in the hood next to the vacuum rack. The apparatus must be clean and dry before it can be used. Concentrated sulfuric acid is somewhat hard on the stopcock grease used in the apparatus, so be sure that the grease is fresh prior to commencing work. Start by adding a small amount of H₂SO₄ from the separation funnel to the NaCl; vent the reaction vessel to the hood periodically so as to displace air from the vessel. DO NOT add a large amount of H₂SO₄, and DO NOT let pressure build in the reaction vessel. Keep the hood sash pulled down during this procedure. Obtain an IR cell from the dessicator; handle the cell very carefully. We will then use the vacuum rack to pump the IR cell down for a few seconds, and then connect the IR cell to the HCl generator and fill the cell. Use the Perkin-Elmer FTIR spectrophotometer in Sc 260 (the Organic Spec lab) to collect the spectrum of the gaseous HCl from 2600-3200 cm^-1.

After you have collected the spectrum, place the IR cell in the hood and let it vent for a while before placing it back in the dessicator. IN THE HOOD, slowly add some NaHCO₃ to the reaction vessel to neutralize any remaining H₂SO₄. After this has been accomplished, disassemble the apparatus, CLEAN IT THOROUGHLY, and place the pieces in the hood to dry.

Data Analysis and Report

The spectrum is presented as per cent transmittance versus wave number value. Prepare an array containing the J value of each line, its energy in wave numbers, and in a third column, enter the value of z for each transition. The digitizer in Mathcad will allow you to pick the energies directly off the plot of the spectrum and save these quantities into an array. Perform the requisite analyses using Mathcad, and include a table presenting the value and standard deviation of each parameter. Compare your results to published experimental and theoretical values.

Calculate:

1. The force constant k representing the 'strength' of the chemical bond
2. The equilibrium bond length r_e
3. The fraction of the total energy change which is due to the rotational contribution for the first line in the R branch
4. The fraction of HCl molecules which are in the first excited vibrational state at room temperature. Give the ground vibrational state a population of 1.
5. The fraction of HCl molecules which are in rotational states J = 1, 5, and 13 at room temperature. Give the ground rotational state a population of 1.

The fundamental vibrational frequency w₀ and anharmonicity constant w₀c₀ for HCl can be obtained as follows. We will utilize Table 10-3 and Eq. 10.44 from McQuarrie, "Quantum Chemistry". Table 1 lists vibrational band heads w for HCl as follows:

Table 1. Vibrational Band Heads for HCl

Eq. 10.44 relates the vibrational band heads to the equilibrium vibrational frequency, the anharmonicity constant, and the vibrational quantum number as follows:

Linearize the above equation and calculate and report values of w₀ and w₀c₀.

Present these results in tabular form and compare your results to published experimental values where applicable.