Bohr S Model Of Hydrogen Atom

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.

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 .

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) .

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.

Step 1: Understanding the Concept:
This problem combines two key principles of the Bohr model for the hydrogen atom: the formula for quantized energy levels and the postulate for quantized angular momentum. First, we must identify the principal quantum number ( ) of the electron's orbit from its given energy. Then, we can use this value of to calculate the angular momentum.

Step 2: Key Formula or Approach:
1. The energy ( ) of an electron in the orbit of a hydrogen atom is given by: 2. The angular momentum ( ) of an electron in the orbit is quantized according to Bohr's second postulate: where is Planck's constant ( J·s).

Step 3: Detailed Explanation:
Part 1: Find the Principal Quantum Number (n)
We are given the energy of the electron, eV. We can use the energy formula to find : Taking the square root, we get: This means the electron is in the first excited state ( ).
Part 2: Calculate the Angular Momentum (L)
Now that we have , we can calculate the angular momentum using Bohr's quantization rule: Substituting the value of Planck's constant:

Step 4: Final Answer:
The angular momentum of the electron is approximately J·s.

Step 1: Understanding the Concept:
This question relates to Bohr's model of the hydrogen atom, specifically his second postulate, which deals with the quantization of angular momentum. According to Bohr, an electron can only revolve in certain stable orbits where its angular momentum is an integral multiple of a fundamental unit.

Step 2: Key Formula or Approach:
Bohr's quantization condition for the angular momentum ( ) of an electron in the orbit is given by: where is the principal quantum number ( ) and is Planck's constant.

Step 3: Detailed Explanation:
We need to find the difference in angular momentum between two successive orbits. Let's consider two consecutive orbits with principal quantum numbers and .
The angular momentum in the orbit is: The angular momentum in the orbit is: The difference ( ) between them is: The difference is constant and independent of .

Step 4: Final Answer:
The difference in angular momentum between any two successive orbits is . This is also often written as (h-bar). Therefore, option (A) is correct.