Atomic Structure

Binding energy is the energy required to remove the electron from the given state. (ii) Ionization Potential % Solution Solution: Ionization potential is the energy required to ionize the atom by removing an electron from the current state. (iii) The Number of Energy States % Solution Solution: The number of possible states for an electron in the nth energy level is given by: For : (iv) The Angular Momentum % Solution Solution: The angular momentum of an electron in the nth orbit is: For : (v) Kinetic Energy % Solution Solution: In hydrogen atoms, kinetic energy in an energy state is numerically equal to the binding energy.

The stationary orbit of an atom refers to the specific orbit of an electron in which the electron does not radiate energy. These orbits are defined by quantized energy levels and are stable, meaning that the electron remains in a given orbit without emitting or absorbing energy. These stationary orbits are described by Bohr’s model of the atom.
According to Bohr’s model, the electron in the hydrogen atom moves in circular orbits around the nucleus under the influence of the Coulomb force between the negatively charged electron and the positively charged proton. The energy of the electron in a stationary orbit can be derived using the following principles: 1. Coulomb Force and Centripetal Force: The centripetal force required to keep the electron in a circular orbit is provided by the electrostatic attraction between the electron and the proton. Using Coulomb’s law, the force is: Where:
- is the charge of the electron,
- is the radius of the orbit,
- is the permittivity of free space.
2. Quantization of Angular Momentum: Bohr postulated that the angular momentum of the electron in a stationary orbit is quantized and given by: Where:
- is the mass of the electron,
- is the speed of the electron,
- is the radius of the orbit,
- is a positive integer (the principal quantum number),
- is Planck’s constant.
3. Expression for Energy: The total energy of the electron in the orbit is the sum of its kinetic energy and potential energy . The potential energy is given by: The kinetic energy is half the magnitude of the potential energy (due to the virial theorem for Coulomb forces): Thus, the total energy is: 4. Relation with Radius: Using the quantization condition , we can derive the radius of the electron’s orbit as: Therefore, the energy of the electron in the -th orbit is: This shows that the energy of the electron is inversely proportional to the square of the radius of the orbit.

Rutherford's -Particle Scattering Experiment:
In Rutherford's experiment, a beam of -particles was directed at a thin gold foil. By observing the scattering angles of the -particles, Rutherford concluded that most of the mass of an atom is concentrated in a tiny, dense nucleus at the centre, and that the majority of the atom's volume is empty space. The -particles were deflected at large angles by the positive charge concentrated in the nucleus. This led to the discovery of the nuclear model of the atom, where electrons orbit around a dense nucleus. Shortcomings of Rutherford's Model:
Rutherford's model, while groundbreaking, had certain flaws:
- It did not explain the stability of the atom. According to classical electromagnetism, the revolving electrons would emit radiation, causing them to lose energy and spiral into the nucleus, leading to atomic collapse.
- It could not explain the observed spectral lines of hydrogen, as predicted by experimental data, where the electron should continuously radiate energy and not remain in discrete energy levels.
Rectification by Bohr's Model:
Niels Bohr improved upon Rutherford's model by introducing the idea that electrons exist in discrete orbits or energy levels. The electrons can only occupy these stable orbits without radiating energy. Radiation occurs only when an electron jumps from one orbit to another. The energy associated with these orbits is quantized, and Bohr used this idea to explain the discrete spectral lines observed in hydrogen.

According to Bohr's atomic model, the angular momentum of an electron in orbit is quantized and given by:
Where: - is the angular momentum, - is the principal quantum number, - is the reduced Planck's constant, where is Planck's constant.
The angular momentum is also related to the linear momentum of the electron and its radius as:
Where: - is the mass of the electron, - is the linear velocity of the electron, - is the radius of the orbit.
By equating these two expressions for :
Now, using the de Broglie hypothesis, the wavelength of the electron is related to its momentum as:
Thus, the radius of the nth orbit can be expressed as:
The circumference of the nth orbit is:
Substituting the expression for from above:
Now, from the de Broglie relation, we can substitute into the equation:
Thus, we obtain the desired result:

In a hydrogen atom, the electron orbits the nucleus (proton) in a circular path under the influence of the electrostatic force between the electron and proton. The Coulomb force provides the centripetal force for the electron's motion.
The electrostatic force between the electron and proton is given by Coulomb’s law:
Where: - is the charge of the electron, - is the permittivity of free space, - is the radius of the electron's orbit.
The kinetic energy of the electron is given by the relation:
Using the fact that the centripetal force is equal to , and equating the electrostatic force to the centripetal force, we find:
Thus, the correct expression for the kinetic energy of the electron in the hydrogen atom is:
Hence, the correct answer is option (B) .

In the hydrogen atom, the energy levels are given by the formula: Where: - is the energy of the nth level, - is the principal quantum number, - 13.6 eV is the Rydberg energy for hydrogen.
1. Energy Level Diagram for Hydrogen:
The energy levels in the hydrogen atom are shown in the diagram below: In this diagram: - The energy of the electron decreases as it moves closer to the nucleus.
- The ground state corresponds to , and higher levels correspond to .
2. Lyman Series Transitions:
The Lyman series corresponds to transitions where the final state is (the ground state). The first line of the Lyman series occurs when the electron transitions from to . The energy difference between these levels is: The wavelength of this transition is given by the formula: Substituting the values ( , , and ): 3. Balmer Series Transitions:
The Balmer series corresponds to transitions where the final state is . The second line of the Balmer series occurs when the electron transitions from to . The energy difference is: The wavelength of this transition is: 4. Ratio of the Wavelengths:
The ratio of the wavelengths is: Thus, the ratio of the wavelengths of the first line of the Lyman series and the second line of the Balmer series is approximately 0.185.

The energy levels of the hydrogen atom are quantized, and the energy corresponding to a particular energy level is given by the following formula derived from the Bohr model:
Where: - is the energy of the -th energy level,
- is the principal quantum number,
- is the Rydberg energy constant for the hydrogen atom.
The energy difference between the ground state ( ) and the excited state ( ) is given by:
Simplifying:
The energy of the photon absorbed by the atom is equal to the energy difference, so the energy of the photon is .
To calculate the frequency of the photon, we use the relationship between energy and frequency:
Where: - is the energy of the photon,
- is Planck's constant ( ),
- is the frequency of the photon.
We first convert the energy from eV to joules. Since , we get:
Now, solving for the frequency:
Thus, the frequency of the photon is approximately .

Rutherford's Alpha-Particle Scattering Experiment:
In 1909, Ernest Rutherford conducted the famous alpha-particle scattering experiment, in which a beam of alpha particles was directed at a thin gold foil. The most important conclusion from this experiment was the discovery of the atomic nucleus. Rutherford found that most of the alpha particles passed straight through the foil, but a small fraction were deflected at large angles, and some even bounced back. This indicated that the mass of an atom is concentrated in a tiny, dense nucleus at the center, with the rest of the atom being mostly empty space. This led to the development of the Rutherford model of the atom, where electrons orbit around a dense nucleus.
Definitions:
sotopic Nuclei: Nuclei that have the same number of protons but different numbers of neutrons, and thus different atomic masses. For example, (hydrogen) and (deuterium) are isotopes of hydrogen.
Isobaric Nuclei: Nuclei that have the same mass number (total number of protons and neutrons) but different numbers of protons (and hence different elements). For example, and are isobars.
Isotonic Nuclei: Nuclei that have the same number of neutrons but different numbers of protons (and thus different elements). For example, and are isotonic nuclei.
Energy Equivalent of One Atomic Mass Unit (1 u):
The energy equivalent of one atomic mass unit can be found using Einstein's famous equation: Where:
- is the mass of 1 atomic mass unit (1 u = kg),
- is the speed of light ( m/s).
First, calculating the energy in joules: Now, converting this energy to MeV (1 J = MeV): Thus, the energy equivalent of one atomic mass unit is approximately 1.494 \times joules or 9.33 MeV.

The energy of an electron in the th orbit of hydrogen atom is given by: where is the principal quantum number. The electron can transition between different energy levels, emitting or absorbing energy in the form of electromagnetic radiation.
The energy levels are represented by the following diagram: The transitions for the Balmer series (visible spectrum) occur when the electron moves to the level. The wavelengths for these transitions can be calculated using: Similarly, the transitions for the Paschen series (infrared spectrum) occur when the electron moves to the level.

The major drawbacks of Rutherford’s atomic model are:
According to classical electromagnetic theory, electrons revolving in circular orbits should radiate energy continuously because they are accelerated charges.
Due to continuous energy loss, electrons should spiral into the nucleus, leading to the collapse of the atom. This implies that atoms should be unstable, which contradicts experimental evidence of stable atoms.
The model could not explain the line spectrum of hydrogen and other elements. Instead of discrete lines, it predicted a continuous spectrum.
Final Answer: Rutherford’s model failed to explain atomic stability and discrete line spectra.