An H2 molecule is in its vibrational and rotational ground...

An $\mathrm{H}_{2}$ molecule is in its vibrational and rotational ground states. It absorbs a photon of wavelength 2.2112$\mu \mathrm{m}$ and jumps to the $v=1, J=1$ energy level. It then drops to the $v=0, J=2$ energy level while emitting a photon of wavelength $2.4054 \mu \mathrm{m} .$ Calculate (a) the moment of inertia of the $\mathrm{H}_{2}$ molecule about an axis through its center of mass and perpendicular to the $\mathrm{H}-\mathrm{H}$ bond, (b) the vibrational frequency of the $\mathrm{H}_{2}$ molecule, and $(\mathrm{c})$ the equilibrium separation distance for this molecule.

Key Concepts

Quantization of Rotational Energy Levels

In diatomic molecules, the rotational energy levels are quantized and typically described by the expression E_rot = J(J+1)(?^2/(2I)), where J is the rotational quantum number and I is the moment of inertia. This quantization explains why molecules only absorb or emit energy in discrete amounts when undergoing rotational transitions, a foundational concept in molecular spectroscopy.

Moment of Inertia in Diatomic Molecules

The moment of inertia is a measure of the distribution of mass relative to the axis of rotation and is crucial for understanding rotational motion. For a diatomic molecule, it is calculated using I = ?r^2, where ? is the reduced mass of the molecule and r is the separation between the nuclei. This concept links the molecular structure directly to its rotational energy levels.

Quantization of Vibrational Energy Levels

Vibrational energy levels in diatomic molecules are also quantized and are commonly approximated by the quantum harmonic oscillator model. The energy levels are given by E_v = ??(v + 1/2), where v is the vibrational quantum number and ? is the angular frequency of vibration. This quantization is fundamental for analyzing vibrational spectra and understanding molecular dynamics.

Vibrational Frequency and Force Constant Relation

The vibrational frequency of a diatomic molecule is determined by its force constant and reduced mass, following the relation ? = ?(k/?). This concept connects the mechanical properties of a bond, such as stiffness (force constant), to the observed vibrational motion, allowing researchers to infer molecular bonding characteristics from spectroscopic data.

Photon Absorption and Emission in Molecular Transitions

Molecular transitions between quantized energy states involve the absorption or emission of photons, where the energy of the photon corresponds to the energy difference between the initial and final states. This relationship, expressed as E = hc/?, enables the determination of molecular parameters such as rotational constants and vibrational frequencies from the observed wavelengths in spectroscopic experiments.

Transcript

00:01 Hi everyone in this problem it is given h2 molecule is in vibrational and rotational ground state is in its vibrational and rotational ground state when it absorbs the photon of wavelength 2 .211 micrometer then it jumps to vibrational quantum number one and rotational quantum number one in the part a we have to calculate the moment of inertia of s2 molecule if s2 molecule drops to vibrational quantum number zero and rotational quantum number two in second part we have to calculate vibrational frequency and in c part equilibrium separation are we start solving one by one in the first part energy of vibrational and rotational quantum number you may write new plus half h into f plus h cross square two i j into j plus 1 for new skull to 0 and j is called to 0 its energy will be 0 0 0 half h f for new is skull to 1 and j is called to 1 its energy will be 3 by 2 h into f plus h square by i am highlighting these energy and energy for new is called to zero and j is called to two so this will be half h into f plus three h square by i now do the calculation e11 minus e02 is called to h into f i equal to sc by lambda so from here you may find this quantity h into f plus i will be 6 .626075 10 to the power of minus 34 into c that is 2 .99725 10 to the power of minus 34 into c that is 2 .99725 the power 8 divided by wavelength of photon absorbed by it 2 .2 -112 10 to the power of minus 6.

06:42 On solving it, you will get this quantity to be 8 .9 -83573 into 10 to the power of minus 20 joules.

07:26 Now e11 minus e02 is equal to hf minus s c by lambda now substitute the value again we have to do the calculation very closely because the difference in two parameter is very very small that's why we need to calculate very close to it so from here hf minus i you will get 8 .258 -28 -284 into 10 to the power of minus 20 joules.

09:13 By solving equation 1 and 2, moment of inertia of scl molecule from equations 1 and 2, moment of inertia can be obtained 4 .6.

09:39 Into 10 to the power 48 kg meter square...

Please give Ace some feedback