The radiation absorbed by ^12 C^16 O during a vibrational transition occurs at 2168 cm^-1. (Sect

The radiation absorbed by ^12 C^16 O during a vibrational...

The radiation absorbed by $^{12} \mathrm{C}^{16} \mathrm{O}$ during a vibrational transition occurs at $2168 \mathrm{cm}^{-1}$. (Section 10.5 ) (a) Caiculate the ground state vibrational energy in $\mathrm{kJmol}^{-1}$ (b) What is the ratio of the number of molecules in the first vibrational excited state $(\mathrm{v}=1)$ compared with the ground state $(\mathrm{v}=0)$ at $298 \mathrm{K} ?$ (c) Calculate the force constant of the bond, assuming $^{12} \mathrm{C}^{16} \mathrm{O}$ behaves as a simple harmonic oscillator. (d) Estimate the change in the position of the peak if $^{12} \mathrm{C}$ was replaced by $^{13} \mathrm{C}$

The radiation absorbed by $^{12} \mathrm{C}^{16} \mathrm{O}$ during a vibrational transition occurs at $2168 \mathrm{cm}^{-1}$. (Section 10.5 )
(a) Caiculate the ground state vibrational energy in $\mathrm{kJmol}^{-1}$
(b) What is the ratio of the number of molecules in the first vibrational excited state $(\mathrm{v}=1)$ compared with the ground state $(\mathrm{v}=0)$ at $298 \mathrm{K} ?$
(c) Calculate the force constant of the bond, assuming $^{12} \mathrm{C}^{16} \mathrm{O}$ behaves as a simple harmonic oscillator.
(d) Estimate the change in the position of the peak if $^{12} \mathrm{C}$ was replaced by $^{13} \mathrm{C}$

Key Concepts

Isotopic Substitution Effects

Isotopic substitution involves replacing one atom in a molecule with another isotope, which can lead to changes in the molecular mass while keeping the electronic structure relatively unchanged. This affects the vibrational frequency of a bond because the frequency is inversely proportional to the square root of the reduced mass of the system. Thus, isotopic substitution can shift the vibrational spectra, providing insights into the molecular dynamics and bonding characteristics.

Force Constant

The force constant is a measure of the stiffness of a chemical bond in a molecule, indicative of how strongly the atoms are held together. In the harmonic oscillator model, it appears in the equation for vibrational frequency and determines the energy spacing between vibrational levels. Calculating the force constant helps in understanding the bond strength and dynamics of molecular vibrations.

Boltzmann Distribution

The Boltzmann distribution is a statistical tool that describes the relative populations of different energy states in a system at thermal equilibrium. In the context of molecular vibrations, it calculates the ratio of molecules in excited states versus the ground state at a given temperature. This is crucial for predicting the intensity of spectral lines and understanding thermodynamic properties of molecular systems.

Wavenumber and Energy Relation

Wavenumber, typically expressed in inverse centimeters, represents the number of wave cycles per unit distance and is directly related to the energy of a vibrational transition. The energy associated with a given wavenumber can be calculated using the relation E = hc?, where h is Planck's constant, c is the speed of light, and ? is the wavenumber. This relationship is fundamental in spectroscopy, linking observed spectral features to molecular energy levels.

Vibrational Energy Levels

This concept refers to the quantized energy states in which a molecule can exist due to vibrational motions. In diatomic molecules, for example, the vibrational energy levels are discrete and typically modeled using quantum mechanics. The lowest state is the ground vibrational state, and transitions between these levels can be induced by absorption or emission of radiation at characteristic energies.

Harmonic Oscillator Model

The harmonic oscillator model is a mathematical approximation used to describe the vibrational motion of atoms in a diatomic molecule. It assumes that the potential energy of the bond is a quadratic function of displacement from equilibrium, allowing for a simple calculation of vibrational energies and the force constant. Although real molecular vibrations are anharmonic to some degree, the harmonic approximation provides a useful and often sufficiently accurate framework for many purposes.

Transcript

00:01 So, vibrational spectra are results of transitions that are able to occur between the rotational energy levels of gaseous and liquid molecules.

00:09 So these transitions can occur when a molecule absorbs radiation within the infrared region.

00:15 So in this vast part, we're looking at the zero point vibrational energy.

00:18 This means the energy of a molecule in its ground vibrational level, that is equal to b is equal to zero.

00:24 So we have the energy, the vibration is equal to b, add 1 over 2, multiplied by a molecule.

00:30 By hv in units of joules.

00:33 So our energy of vibration is equal to 1 ,2966 .46 joules per mole.

00:44 And so we're looking at the population ratio, that is defined as the ratio of the number of molecules that exist in the upper levels and the lower levels.

00:51 So we have n upper divided by n lower, is equal to g v equals 1 over g v equals we have zero, that and we have one at the top.

01:11 So that is equal to the exponential of negative delta e over kt.

01:17 So n upper, divided by n lower, is equal to 2 .387 divided by 1 .000 ,000 ,000.

01:29 So the population ratio is defined on the screen, and this ratio indicates that virtually all of the co molecules will be in the ground state, and only three molecules are in the first.

01:40 Excited state.

01:42 So in the sad part here, the vibrational spectrum are results of transitions that are occurring between vibrational energy levels.

01:49 So the transitions occur when infrared radiation can be absorbed...

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