**Abstract**

Infrared spectroscopy is an important analytical tool to determine the structure of molecules. Fourier transform infrared spectroscopy was used to study the vibrational and rotational motions of diatomic molecules hydrogen chloride, HCl and deuterated chloride, DCl. The isotopic effect was observed in a spectrum of both HCl and DCl with DCl at a lower wavenumber than HCl which coincided with ^{37}Cl being observed at a lower frequency than ^{35}Cl. The spectra of HCl and DCl were used to create separate plots of the m transition with corresponding wavenumber. Polynomial fit was determined of the plots and then used to calculate ,_{, }D_{e, }α_{e, }and B_{e }by using the harmonic oscillator and rigid rotor models_{. }These constants were then used to determine the moment of inertia, I_{e}, the internuclear separation, r_{e}, force constant, k, anharmonicity, *v _{e}x_{e}*, and equilibrium frequency

*v*. The k and r

_{e}_{e}were unaffected by the isotopic effect with values of 515.20 N/m and 1.31 A for HCl and 515.23 N/m and 1.30 A for DCl. Values for HCl were also determined using computational Gaussian modeling and compared to Literature. Computational and literature values had high correlation with calculated HCl constants.

**Introduction**

Molecules can have three modes of movement; vibration, rotation, and translation. A molecules vibrational and rotational movement is essential in the study of infrared spectroscopy, which measures the absorption of light by a molecule. Absorption of infrared light only occurs when the frequency of the wavelength is the same as the vibrational frequency of a molecule. Diatomic molecules only have one mode of vibration described by the harmonic oscillator,

where *E* is energy, *𝑣* is the vibrational quantum number, *v* is frequency, and *h* is plank’s constant. Rotation of atoms is important in infrared study of molecules because changes in the rotational state affect the molecules vibrational fine structure. Rotation of a diatomic molecule in its simplest form is described by the rigid rotor,

where *J* is the rotational quantum number, *I* is the moment of inertia, and *h* is plank’s constant.

Molecules undergo vibration and rotation simultaneously so Eqs. (1) and (2) are combined to describe the motion of a molecule while also considering anharmonicity and the interaction of vibration and rotation,

where is the anharmonic vibrational frequency correction, *𝐵*_{𝑒}*.* the rotational constant, *,**𝐷*_{𝑒} accounts for centrifugal stretching, and *𝛼*_{𝑒} is the anharmonicity correction to rotation. Molecules are quantized so both *J* and *𝑣* are integers (±0, 1, 2…). At room temperture only the ground state *v=0* is usually populated and Δ*𝑣**=+1* when excited. Rotational states have smaller energy differences than vibrational states soΔ*𝐽**=±1, 2, 3..*.

Figure 1. Possible vibrational and rotational transitions.

Absorption bands vary location and intensity because of rotation-vibration interaction. A spectrum can be split into three branches P, Q, and R. The R branch represents the cumulative energy of the vibrational and rotational transitions and the P branch the difference,

The Q branch is usually not observed because it represents the pure vibrational mode where rotation is Δ*𝐽**=0* in the excited state. Combining Eqs. (4) and (5),

results in the ability to use an infrared spectrum to calculate the constants * *, *B*_{e}*,**𝛼*_{𝑒}*.*, and *D _{e}* of a diatomic molecule. The moment of inertia, Ie, the internuclear distance, re, force constant, k, anharmonicity, v

_{e}x

_{e}, and equilibrium frequency v

_{e }can then be determined by assuming the molecule behaves as a harmonic oscillator and rigid rotor.

Vibration and rotation are contingent on the bonding molecules. Isotopic substitution changes the reduced mass, *𝜇*, which effects the rotational constant, *B _{e}*, and vibrational frequency,

The change in *𝜇* results in a different spectrum for each isotopomer. The force constant, k, and equilibrium bond length, r_{e}, are unaffected since they depend on the character of the chemical bond.

**Experimental**

To obtain measurements a Nicolet iS10 FTIR spectrophotometer was used in high resolution with a salt plate cell. The cell was vacuumed prior to taking experimental measurements and a baseline was obtained. The cell was then filled with HCl gas two times to ensure residual gases were removed and measurements were taken.

HCl was then reacted with D_{2}O for 30 min to make DCl by the reaction of

*𝐷*_{2}*𝑂**+**𝐻𝐶𝑙* *→**𝐻*_{2}*𝑂**+**𝐷𝐻𝑂**+**𝐷𝐶𝑙*

The D_{2}O vial was cooled in dry ice to separate D_{2}O/H_{2}O from the mixture and allow HCl and DCl to fill the cell. The FTIR spectrum of the cell filled with HCl and DCl gas was taken. HCl was pumped out of the system and crystallized by a liquid nitrogen trap to prevent toxic HCl gas from entering the atmosphere.

Table 1. Reagents used

Reagent | Formula | Molecular Weight (g mol^{-1}) |
Density (g mL^{-1}) |
CAS Number |

Hydrochloric Acid | HCl | 36.46 | 1.18 | 7647-01-0 |

Deuterium Chloride | DCl | 37.47 | – | 7698-05-7 |

**Results**

*1.1 HCl infrared spectroscopy study*

Infrared spectroscopy is a vital tool in determining quantum properties of molecules. HCl constants were determined from an IR spectrum.

Figure 3 shows the IR spectrum of HCl with rotational-vibrational effects. Absorption peak intensity increases to a maximum and then decreases moving towards. This pattern occurs due to the quantum states available and the population of atoms at that state. The distance between absorption bands, in the P and R branch is expected to be 2B_{e} and 4B_{e} in the zero gap (Q Branch). From the spectra it is seen that moving away from the Q branch absorption bands in the P branch move apart and in the R branch closer together. The deviation from the predicted pattern occurs due to rotational-vibrational coupling and centrifugal distortion. Actual values can be found in Table 5A under the appendix and loosely followed the expected spacing trend of 2B_{e} and 4B_{e}. B_{e} is discussed in the next section and can be found in Table 2. The maxima of each absorption band was plotted against the m transition number shown in Figure 4*.*

The second and third order polynomials were found from the data set in Figure 4 using Origin. Using the F-test it was determined that values obtained from the second order polynomial are not significantly different from values obtained from the third order polynomial. The third order polynomial was used for subsequent calculations of frequency , rotational constant* B _{e}*, centrifugal stretching

*𝐷*

_{𝑒}, and the rotational anharmocity constant

*𝛼*

_{𝑒}. It was determined that

_{ }is 2885.4 ±0.2 cm

^{-1}using the third order polynomial in Figure 4. The literature value for is 2885.1 cm

^{-1}which falls outside of the calculated error, however, there is only a 0.01% difference for the values. Multiple linear regression was performed to obtain constants for HCl.

*B*and

_{e}*𝛼*

_{𝑒}

*.*were calculated using Eq. 6 and determined to be 10.63 ± 0.09 cm

^{-1}and 0.304 ± 0.004 cm

^{-1}, respectively. The literature value for

*B*of 10.54 cm

_{e}^{-1}falls within the error of the calculated value and has a percent difference of 0.08%.

*𝛼*

_{𝑒 }literature value of 0.311 cm

^{-1}is within the experimental error calculated. The

*D*was calculated from high m transitions due to m

_{e}^{3}dependence and found to be (7.25±0.02) x10

^{-4}cm

^{-1}which has an 8.2% difference with a literature value of 5.2 x10

^{-4}cm

^{-1}. Although calculated and

*D*did not correlate with the literature, these values are assumed accurate since they are in the same order of magnitude with relatively small percent differences. Calculated and experiment values summarized in Table 2.

_{e}Using constants found from the third order polynomial, the anharmonic vibrational frequency correction , the equilibrium vibrational frequency* v _{e}*, force constant k, internuclear distance r

_{e}, and moment of inertia, I

_{e}were calculated. The

*v*and were calculated from Eqs. 10 and 11 under the appendix, to be 2989.66 cm

_{e }^{-1 }and 52.12 cm

^{-1}, respectively. Both

*v*and

_{e }*correlated to literature values of 2990.95 cm*

_{ }^{-1}and 52.82 cm

^{-1}. The HCl k was found by treating the vibrational transition from the ground to first excited state as a harmonic oscillator. The k was found from Eq. 9 under the appendix to be 515.20 N/m which has a 0.07% difference with the literature value of 516.82 N/m. The r

_{e}was calculated by taking HCl to resemble the rigid rotor model using Eq. 8 under the appendix to calculate experimental r

_{e}at 1.31

*Å*. The r

_{e}compared to the literature value of 1.27

*Å*had a 0.8% difference. The I

_{e }was calculated to be 2.80 x 10

^{-47}kg m

^{2}from Eq. 7 under appendix, which correlated with the literature value of 2.64 x 10

^{-47}kg m

^{2}at a 1.4% difference. A summary of all HCl constants can be found in Table 2.

*1.2 Computational HCl Study*

Gaussian computational package was used to determine the potential energy surfaces, Figure 5, by implementing Self-Consistent Field (SCF), Second-order Møller-Plesset Perturbation Theory (MP2), and Couple Cluster with Single, Double and approximate Triple excitations (CCSD(T)).

Variation among the methods is due to assumptions made in each. SCF takes the average of the electrons and ignores electron correlation. MP2 includes electron correlation effects by Rayleigh-Schrödinger perturbation theory to the second order. Among the theories used in CCSD(T) uses SCF and constructs multi-electron wavefunctions. Figure 5 to obtain the potential energy surfaces, CCSD(T) is the most accurate and was used to calculate HCl constants. Computational constants determined by CCSD(T) for HCl correlated to both experimental and literature values with the highest percent difference at 8.2% for *D _{e}* between computational and experimental

_{. }The lowest percent difference was

*v*at 0.05%. The high correlation among the values shows the effectiveness of computational calculations using CCSD(T). A summary of all calculated computational HCl constants can be found in Table 2. Computational information under the appendix calculations.

_{e }Table 2. HCl constants with experimental calculated using the third order polynomial of Figure # compared to literature values with the percent difference and calculated computational values.

Constant |
Experimental |
Literature^{7} |
Percent Difference [Expt. Lit.] |
Computational |
Percent Difference [Expt. Comp.] |

(cm^{-1}) |
2885.4 ±0.2 | 2885.1 | 0.01% | – | – |

B_{e} (cm^{-1}) |
10.63 ±0.09 | 10.591 | 0.08% | 10.54 | 0.2% |

α_{e} (cm^{-1}) |
0.304 ±0.004 | 0.302 | 0.2% | 0.311 | 0.6% |

D_{e} (cm^{-1}) |
(7.25 ±0.02) x10^{-4} |
5.3 x10^{-4} |
8.2% | 5.2 x 10^{-4} |
8.2% |

v(cm_{e }^{-1}) |
2989.66 | 2990.95 | 0.01% | 2996.31 | 0.05% |

v (cm_{e}x_{e}^{-1}) |
52.12 | 52.82 | 0.3% | 52.3 | 1.3% |

k (N/m) | 515.20 | 516.82 | 0.08% | 518.6 | 0.6% |

r_{e }(Å) |
1.31 | 1.27 | 0.8% | 1.28 | 0.09% |

I_{e }(kg m^{2}) |
2.80 x 10^{-47} |
2.64 x 10^{-47} |
1.4% | 2.65 x 10^{-47} |
0.2% |

* *

*1.3 Isotopic Substitution Study*

The absorbance peak observed for H_{2}O in Figure 6 is not relevant to this experiment and can be disregarded. Figure 6 shows the isotopic effect of ^{1}H,^{ 2}H (D), ^{35}Cl, and ^{37}Cl. From the spectrum it is seen that DCl absorbed energy at a lower frequency (2000-2200 cm^{-1}) than HCl (2600-3100 cm^{-1}). The lower absorption frequency of DCl occurred due to a change in the reduced mass, Table 6A under the appendix, from 1.62612 x 10^{-27} to 1.904413 x 10^{-27} for HCl and DCl, respectively. It was determined that increasing the mass of an isotope resulted in absorption at a lower frequency. A shift in absorption frequency is also observed for ^{35}Cl and ^{37}Cl. Viewing a close-up of the spectrum in Figure 6 shows that there are two peaks present at each absorption band with ^{37}Cl absorbing at a lower wavenumber than ^{35}Cl. The shift of ^{37}Cl/^{35}Cl is small compared to the one observed for D/H which is due to a larger ratio of for hydrogen than chlorine with values of 1.944 and 1.00, respectively.

The spectrum in Figure 6 also shows the abundance of atoms since the absorbance is directly correlated to the concentration. From the spectrum it is seen that more ^{35}Cl than ^{37}Cl was present which correlates to reported amounts of chlorine isotopes at 75.8% and 24.2%^{4}. There was also more HCl than DCl present when the spectrum was taken seen by the low absorbance of DCl compared to HCl.

The m transition numbers with their corresponding wavenumbers were plotted to make Figure 8. Using the F-test it was determined that values obtained from the second order polynomial are not significantly different from values obtained through the third order polynomial. Third order polynomial was used for subsequent calculations of frequency , *B _{e}*,

*𝐷*

_{𝑒}, and

*𝛼*

_{𝑒}

*.*The

_{ }is 2090.6 ±0.1 cm

^{-1}, the

*B*is 5.23 ±0.05 cm

_{e}^{-1}, the

*𝛼*

_{𝑒 }is 0.114 ±0.004 cm

^{-1}, and the

*D*is (2.67 ±0.02) x10

_{e}^{-4 }cm

^{-1}. The lower value of

*B*of DCl compared to HCl represents that the absorption peaks are expected to be closer together in the DCl spectra which is observed in Figure 7 and reported in Table 5A in the appendix. The spectra of DCl also shows divergence from the 2

_{e}*B*and 4

_{e}*B*distance that was expected for .

_{e}Using constants found from the third order polynomial, the *,,* the *v _{e}*, the k, the r

_{e}, and the I

_{e}were calculated for DCl. The is 26.80 cm

^{-1}for DCl compared to 52.12 cm

^{-1}for HCl representing that DCl needed a smaller vibrational anharmonicity correction term. The

*v*was found to be 2144.18 cm

_{e}^{-1}. It was expected that r

_{e}would be the same for both HCl and DCl which was found to be true with r

_{e }of 1.30

*Å*for DCl compared to 1.31

*Å*HCl which has a 0.2% difference. The k, which also doesn’t depend on

*𝜇*only had a 0.001% difference with 515.23 and 515.20 N/m for DCl and HCl, respectively. The I

_{e}was found to be 5.36 x10

^{-47}which was larger than the I

_{e}calculated for HCl. Calculated values are summarized in Table 3.

Table 3. Calculated Constants of HCl and DCl

Constant |
Experimental HCl |
Experimental DCl |

µ (amu) | 1.63 x10^{-27} |
3.16 x10^{-27} |

(cm^{-1}) |
2885.4 ±0.2 | 2090.6 ±0.1 |

B_{e} (cm^{-1}) |
10.63 ±0.09 | 5.23 ±0.05 |

α_{e} (cm^{-1}) |
0.304 ±0.004 | 0.114 ±0.004 |

D_{e} (cm^{-1}) |
(7.25 ±0.02) x10^{-4} |
(2.67 ±0.02) x10^{-4} |

v(cm_{e }^{-1}) |
2989.66 | 2144.18 |

v (cm_{e}x_{e}^{-1}) |
52.12 | 26.80 |

I_{e }(kg m^{2}) |
2.80 x 10^{-47} |
5.36 x10^{-47} |

r_{e }(Å) |
1.31 | 1.30 |

k (N/m) | 515.20 | 515.23 |

Table 4 Calculated ratios of Isotopes HCl and DCl

B_{e}^{*}/B_{e} |
µ/µ^{*} |
(µ/µ^{*})^{1/2} |
ν̃_{e}^{*}/ν̃_{e} |

0.49 | 0.51 | 0.72 | 0.72 |

The rigid rotor prediction is proven to be accurate through comparison of B_{e}^{*}/B_{e }to µ/µ^{*} which are similar at 0.49 and 0.51. The ratio for the harmonic oscillator of (µ/µ^{*})^{1/2}= ν̃_{e}^{*}/ν̃_{e} was also found to be accurate with values of 0.72 for each. These ratios of HCl and DCl prove that the rigid rotor and harmonic oscillator are fairly accurate at predicting isotopic behavior.

**Conclusion**

The vibrational-rotational effects of HCl were explored through FTIR spectroscopy and computational methods then compared to values obtained for DCl using FTIR. Divergence from expected results was mainly due to anharmonicity and centrifugal stretching caused by vibration-rotational interaction. Experimental and computational values for HCl provided reasonable values compared to the literature. The isotopic effect was observed in the spectra of ^{1}H, ^{2}H, ^{35}Cl, and ^{37}Cl with heavier molecules absorbing at a lower frequency due to reduced mass dependence. The force constant and internuclear distance were not affected by the isotopic effect and had similar values calculated for HCl and DCl. The rigid rotor and harmonic oscillator model accurately predicted the ratios of B_{e} and ν̃_{e} of HCl and DCl. Studies of molecular quantum properties are important to understand how molecules will behave under varying conditions.

**Works Cited**

[1] M. Halpern and G.C. McBane, Experimental Physical Chemistry, 3^{rd} ed., W.H. Freedman and Company, New York, 2006. Experiment 34.

[2] Atkins, P., J. De Paula “Physical Chemistry”, 9th ed., W. H. Freeman, New York (2010)

[3] Spiridoula, M.; *Physical Chemistry Laboratory Molecular Constants of HCl using Computational Chemistry*, Handout, Print.

[4] Nave, R. “Rotational Spectra.” *Hyper Physics*. Georgia State University, 2001. Web. 08 Apr. 2014.

[5] Schuder MD, Nesbitt DJ. 1994. *J. Chem. Phys*. 100: 7250-67

[6] Herzberg, G. *Molecular Spectra and Molecular Structure*. New York: Van Nostrand, 1950. Print

[7] Herzberg, G. “NIST Chemistry WebBook.” *The NIST WebBook*. Web. 27 October 2013.

[8] Cooley, J. “Rovibrational Spectroscopy.” *Chemwiki.* UC Davis, Web.

**Appendix**

Table 5A. M transitions with corresponding wavenumber for HCl and DCl used in Figures 3 and 7 and calculated **Δν.**

HCl |
DCl |
||||

m |
v (cm^{-1}) |
Δν (cm^{-1}) |
m |
v (cm^{-1}) |
Δν (cm^{-1}) |

-11 | 2625 | 27 | -9 | 1986 | 12 |

-10 | 2652 | 25 | -8 | 1998 | 12 |

-9 | 2677 | 25 | -7 | 2010 | 13 |

-8 | 2702 | 25 | -6 | 2023 | 11 |

-7 | 2727 | 25 | -5 | 2034 | 12 |

-6 | 2752 | 23 | -4 | 2046 | 11 |

-5 | 2775 | 23 | -3 | 2057 | 12 |

-4 | 2798 | 23 | -2 | 2069 | 11 |

-3 | 2821 | 22 | -1 | 2080 | (Zero Gap) 21 |

-2 | 2843 | 20 | 1 | 2101 | 11 |

-1 | 2863 | (Zero Gap) 43 | 2 | 2112 | 10 |

1 | 2906 | 20 | 3 | 2122 | 9 |

2 | 2926 | 18 | 4 | 2131 | 10 |

3 | 2944 | 19 | 5 | 2141 | 9 |

4 | 2963 | 17 | 6 | 2150 | 9 |

5 | 2980 | 18 | 7 | 2159 | 9 |

6 | 2998 | 16 | 8 | 2168 | 9 |

7 | 3014 | 16 | 9 | 2177 | 9 |

8 | 3030 | 15 | 10 | 2186 | |

9 | 3045 | 14 | |||

10 | 3059 | 11 | |||

11 | 3070 | 15 | |||

12 | 3085 |

Table 6A. Reduced mass and reduced mass ratios for isotopic molecules

Molecule | Reduced Mass, 𝜇 (kg) |
Isotopes | Reduced Mass Ratio |

^{1}H^{35}Cl |
1.62612 x 10^{-27} |
^{1}H^{35}Cl / ^{1}H^{37}Cl |
1.002 |

^{1}H^{37}Cl |
1.62859 x 10^{-27} |
||

^{2}H^{35}Cl |
1.904413 x 10^{-27} |
^{1}H^{35}Cl / ^{2}H^{35}Cl |
1.944 |

^{2}H^{37}Cl |
1.910033 x 10^{-27} |

**Calculations**

Computational calculations for HCl