In order for a molecule to absorb infrared radiation, the vibrational mode must exhibit a change in the dipole moment. HCI in the gas phase is used in Physical Chemistry Lab to demonstrate t vibrational spectroscopy. 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 2600 2650 2700 2750 2800 2850 2900 2950 3000 cm1 3050 3100 Figure 1: The infrared absorption spectrum of gas phase H35 Cl and H37 Cl. These data can be fit using a multiple least squares regression using the following equation where m is used to index each rotational peak in the spectrum. The frequency of each peak is given by (m) = vo+ (2Be-2αe)m - αem² - 4Dem³ (1) where vo is the frequency of the v=0, J" =0 → v=1, J'=0 forbidden transition (i.e. this is the "missing" Q branch), Be is the rotational constant relative to the equilibrium internuclear separation, de is the vibration-rotation coupling constant and De is the centrifugal distortion constant*. Note that high m transitions are the most important for determining De due to the m³ dependence. (a) Estimate the vo value from the spectrum in Figure 1 and use the following equations for a diatomic molecule to determine ve and vexe for HCl. The frequency of the band center is V₁ = ve - 2vexe where ve is the frequency of the molecule vibrating about its equilibrium internuclear separation, re and vexe quantifies the anharmonicity of the vibration. We can make use of the fact that the equilibrium internuclear separation, re, and force constant, k, are not affected by isotopic substitution since they are solely a function of the bonding electrons such that V₁ = ve - 2√ex = ve (#)½ - 2√exe ( *Please note that De is a completely different physical variable in this problem than it was in the previous problem. (2) (3) Now we rearrange equation (2) to get an expression for xe and substitute into equation (3) to get 1/2 V = Ve -2% ()() which reduces to 1/2 V = V + (Vo-ve) (#) Now rearrange equation (5) to get a final expression for ve that can be used to calculate the force constant, k, for HCl using the following equation. Note that the * indicates HCl made of different isotopes. For this calculation, use D35 Cl where v₁ = 2091 cm¯¹. ( (4) (5) (6) (b) Calculate L, the moment of inertia, and re, the internuclear separation for both H35 Cl and H³7 Cl using the following expression for the rotational constant, Be. Be 8m² cle (7) (c) The solutions to the Schrödinger equation for a rigid rotator are the spherical harmonics and the eigenvalues and degeneracies are identical to the atomic orbitals. That said, we don't see a value of /> 3 in the Periodic Table; whereas, in the spectrum given above in Figure 1, you see many J states populated. On Exam 1, Problem 6a, you showed the 21 +1 projections onto the z-axis and the x-y plane of the total angular momentum vector, L² = 1(l + 1) where I = 1. Now do the same thing for the J states of the R branch of HCI.

In order for a molecule to absorb infrared radiation, the vibrational mode must exhibit a change in the dipole moment. HCI in the gas phase is used in Physical Chemistry Lab to demonstrate t vibrational spectroscopy. 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 2600 2650 2700 2750 2800 2850 2900 2950 3000 cm1 3050 3100 Figure 1: The infrared absorption spectrum of gas phase H35 Cl and H37 Cl. These data can be fit using a multiple least squares regression using the following equation where m is used to index each rotational peak in the spectrum. The frequency of each peak is given by (m) = vo+ (2Be-2αe)m - αem² - 4Dem³ (1) where vo is the frequency of the v=0, J" =0 → v=1, J'=0 forbidden transition (i.e. this is the "missing" Q branch), Be is the rotational constant relative to the equilibrium internuclear separation, de is the vibration-rotation coupling constant and De is the centrifugal distortion constant. Note that high m transitions are the most important for determining De due to the m³ dependence. (a) Estimate the vo value from the spectrum in Figure 1 and use the following equations for a diatomic molecule to determine ve and vexe for HCl. The frequency of the band center is V₁ = ve - 2vexe where ve is the frequency of the molecule vibrating about its equilibrium internuclear separation, re and vexe quantifies the anharmonicity of the vibration. We can make use of the fact that the equilibrium internuclear separation, re, and force constant, k, are not affected by isotopic substitution since they are solely a function of the bonding electrons such that V₁ = ve - 2√ex = ve (#)½ - 2√exe ( Please note that De is a completely different physical variable in this problem than it was in the previous problem. (2) (3) Now we rearrange equation (2) to get an expression for xe and substitute into equation (3) to get 1/2 V = Ve -2% ()() which reduces to 1/2 V = V + (Vo-ve) (#) Now rearrange equation (5) to get a final expression for ve that can be used to calculate the force constant, k, for HCl using the following equation. Note that the * indicates HCl made of different isotopes. For this calculation, use D35 Cl where v₁ = 2091 cm¯¹. ( (4) (5) (6) (b) Calculate L, the moment of inertia, and re, the internuclear separation for both H35 Cl and H³7 Cl using the following expression for the rotational constant, Be. Be 8m² cle (7) (c) The solutions to the Schrödinger equation for a rigid rotator are the spherical harmonics and the eigenvalues and degeneracies are identical to the atomic orbitals. That said, we don't see a value of /> 3 in the Periodic Table; whereas, in the spectrum given above in Figure 1, you see many J states populated. On Exam 1, Problem 6a, you showed the 21 +1 projections onto the z-axis and the x-y plane of the total angular momentum vector, L² = 1(l + 1) where I = 1. Now do the same thing for the J states of the R branch of HCI.

Physical Chemistry

2nd Edition

ISBN:9781133958437

Author:Ball, David W. (david Warren), BAER, Tomas

Publisher:Ball, David W. (david Warren), BAER, Tomas

Chapter14: Rotational And Vibrational Spectroscopy

Section: Chapter Questions

Problem 14.60E

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