Atomic & Molecular Physics MCQ Quiz - Objective Question with Answer for Atomic & Molecular Physics - Download Free PDF

Calculation:

Br⁺ has electronic configuration: [Ar] 4s² 3d¹⁰ 4p⁴. Optical active electrons are equivalent electrons. The p⁴ configuration has exactly the same spectral representation as p², but p⁴ has inverted order of energy levels.

This means: highest S corresponds to lowest energy, highest L corresponds to lowest energy, and highest J corresponds to lowest energy.

So, ³P₂ has the lowest energy (here S and D have singlet state).

Explanation:

The splitting of spectral lines in the normal Zeeman effect is given by the formula:

which translates into wavelength splitting as:

From the equation, we see that splitting is directly proportional to the charge-to-mass ratio e/me​.

• If e/mee/m_e​  increases, then splitting increases.
• If e/mee/m_e​ decreases, then splitting decreases.
• If e/mee/m_e​ remains the same, splitting remains the same.
• The material of the light source does not affect the normal Zeeman effect since it depends only on fundamental constants.

Thus, option '1' is correct: Increase proportionally (if e/mee/m_e​ increases).

Concept:

Energy Difference in Rotational States:

• In a diatomic molecule like HD, the energy difference between the two lowest rotational states is determined by the formula:
• Where is Planck's constant, is the reduced mass, and is the mean distance between the atoms.
• The reduced mass for the HD molecule is given by:
• , where m_H is the mass of hydrogen and m_D is the mass of deuterium.
• Substitute the given values into the formula to calculate the energy difference.

Given data:

• Distance between atoms,
• Mass of hydrogen atom,
• Mass of deuterium atom,

Now, the energy difference between the two lowest rotational states is approximately:

Solution:

Hence, the correct answer is:

• Option 2:

Concept:

Spin-Orbit Interaction and Energy Splitting:

• The spin-orbit interaction is the interaction between the electron's spin angular momentum and its orbital angular momentum. This interaction leads to the splitting of energy levels in atoms.
• The Hamiltonian for spin-orbit interaction is given by:
• The energy splitting between levels is determined by the quantum numbers associated with orbital and spin angular momenta: .

For the levels and , the energy splitting is given by:

Energy splitting =

The correct expression for the splitting between the and levels is .

Solution:

Hence, the correct answer is:

• Option 1:

Concept:

Absorption Spectrum and Moment of Inertia:

• The frequency of absorption ( ν  ) for a rotating molecule is related to its moment of inertia ( I ) by the formula: .
• Here, h is Planck's constant, c is the speed of light, and I is the moment of inertia.
• The absorption frequency ( ν ) is inversely proportional to the moment of inertia ( I ), so we can calculate the ratio of moments of inertia by taking the ratio of the frequencies.

Given:

• ν' for

The ratio of moments of inertia can be calculated as:

Thus, the ratio of moments of inertia is approximately 1.046 when considering the reverse ratio.

Solution:

Hence, the correct answer is:

• Option 3: 1.046

Explanation:

• The given energy levels and their respective spin-parity values indicate a 'rotational spectrum' of a 'deformed nucleus'.
• The fact that the energy levels are increasing but not uniformly, and considering the spin and parity, strongly suggests a pattern consistent with a nuclear deformation leading to a rotational energy level scheme.
• Furthermore, the parity remains positive across transitions, following the (2n)+ rule typical for even-even deformed nuclei. So together, these characteristics point to an even-even deformed nucleus exhibiting a rotational spectrum.

CONCEPT:

Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting (or absorbing) particles result in different Doppler shifts, the cumulative effect of which is the emission (absorption) line broadening.

Mathematically, the Doppler width of an optical line Δ ν₀  = 

Where, ν₀ = central frequency, k_B = Boltzmann's constant, T = temperature

m = mass and c = velocity of light in vacuum

EXPLANATION:

Doppler shift is: 

⇒ Δ ν0  = 

Hence the correct answer is option 2.

Explanation:

If only the normal or "weak field" Zeeman effect is being considered, where splitting is independent of the number of available orbitals and only dependent on the change in the magnetic quantum number (m = -1, 0, +1), then the transition would indeed split into three lines.

This occurs because the orbital angular momentum can adjust to the magnetic field, and the result is three transitions with Δm = -1, 0, +1 respectively that correspond to the three spectral lines

CONCEPT:

Spectral term: It is a symbol that represents the state of an atom.

An atomic state with spin 's' angular momentum 'l' and total momentum j canbe represented as

⇒ Spectral term = 2s+1LJ 

Here, L corresponds to a symbol. like l = 0 corresponds to 'S', 

l = 1 corresponds to 'P', l = 2 corresponds to 'D' and l = 3 corresponds to 'F' etc.

EXPLANATION:

Ground state the energy has to follow the rule of highest S Highest L and lowest J.

Highest S = ∑m_s = ½ + ½ + ½ = (3/2)

Highest L = | ∑m_l | = |2+1+0| = 3 (corresponding symbol 'F')

Lowest J = |L - S| = |3 - 3/2| = 3/2 

∴ Spectral term = ^2s+1L_J = ⁽+1F3/2 = 4F3/2

Hence the correct answer is option 1.

Explanation:

• The hydrogen molecule H₂ consists of two protons, while the deuterium molecule D₂ consists of two deuterons, with deuterium being twice as heavy as hydrogen.
• The energies of the rotational Raman transitions are inversely proportional to the reduced mass of the molecule. As deuterium is twice the mass of hydrogen, the transitions in hydrogen should occur at twice the frequency as those in deuterium.
• This would make the ratio 

CONCEPT:

Rayleigh and Raman scattering:

• Raman spectroscopy is a spectroscopic technique based on Raman scattering.
• When a substance interacts with a laser beam (or any light wave), almost all of the light produced is Rayleigh scattered light (elastic process).
• However, a small percentage (about 0.000001%) of this light is Raman scattered (inelastic process). Raman scattering is a process, where incident light interacts with molecular vibrations in a sample.
• The photons from the laser beam interact with the molecules and excite the electrons in them.
• The excited electrons immediately fall down to the ground level.
• As electrons lose energy and fall down to the ground state, they emit photons.

There are three different scenarios of how light can be re-emitted after energy had been absorbed by an electron: 

• An electron falls down to the original ground state and there is no energy change, therefore light of the same wavelength is re-emitted. This is called Rayleigh scattering.
• After being excited, an electron falls to a vibrational level, instead of the ground level. This means the molecule absorbed a certain amount of energy, which results in light being emitted in a longer wavelength than the incident light. This Raman scatter is called “Stokes”.
• If an electron is excited from a vibrational level, it reaches a virtual level with higher energy. When the electron falls down to the ground level, the emitted photon has more energy compared to the incident photon, which results in shorter wavelength. This type of Raman scatter is called “Anti-Stokes”.

EXPLANATION:

As most of the produced are Rayleigh scattered light so rayleigh's scattered light is the most intense. The intensity of the Rayleigh line is thus always higher than the intensity of the stokes and Anti-stokes lines. Whereas the intensity of stokes is higher than anti-stokes.

So, Thus the correct formation is IR > IS > IAS

Hence the correct answer is option 2.

CONCEPT:

Ionization energy: The minimum energy to remove the electron from the last orbit of an atom. 

Mathematically, Ionization energy = E = 13.6  (for electron)

and E = 13.6  (for muon)

Here, μ = m_em_μ/(m_e + m_μ) = reduced mass

EXPLANATION:

Given, mass of muon = m_μ = 200 m_e

and mass of  proton = m_p = 1836 m_e

⇒ Ionization energy = E = 13.6 = A 

(Where, A = 13.6 )

For Normal Li-atom :- 

⇒ 

So, E ≈ A

For Muonic Li-atom :- 

So, E ≈ 197 A

Thus, The lowest ionization energy for the muonic Li atom is approximately 200 times larger than that of normal Li.

Hence the correct answer is option 3.

Concept:

• In a bound system of two particles of masses m₁ and m₂, the frequency of light absorbed or emitted during an electronic transition is inversely proportional to the reduced mass (μ) of the system, where 
• For the Hydrogen atom (electron-proton system), because the mass of the proton is much larger than that of the electron (mₚ >> mₑ), the reduced mass is essentially the mass of the electron, i.e., 
• For the muon-electron system (a muon and an electron), the reduced mass is . Since the mass of the muon is much larger than the mass of the electron, the denominator (mₑ + mₘ) is essentially the mass of the muon (mₘ).
• Therefore, the reduced mass 

Explanation:

• The frequency for the Hydrogen atom and the muon-electron system are inversely proportional to the respective reduced masses, i.e.,  and 
• Therefore, the difference  is given by:
• Substituting the above values,  
• But we are interested in the reverse, i.e., . 
• So, the value of  .

Explanation:

• The given energy levels and their respective spin-parity values indicate a 'rotational spectrum' of a 'deformed nucleus'.
• The fact that the energy levels are increasing but not uniformly, and considering the spin and parity, strongly suggests a pattern consistent with a nuclear deformation leading to a rotational energy level scheme.
• Furthermore, the parity remains positive across transitions, following the (2n)+ rule typical for even-even deformed nuclei. So together, these characteristics point to an even-even deformed nucleus exhibiting a rotational spectrum.

K E. . Auger electron is K.E = (E_K - E_L) - E_L

= E_K - 2E_L

= (25.4 - 2 × 3.34) keV

= 18.7 keV