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Q12.6(b):
(b) Calculate the orbital period in each of these levels.
Solution :
Step 1: Theoretical Foundation for Orbital Period
According to the Bohr model of the atom, the angular velocity $\omega$ of an electron in a stable orbit is related to the principal quantum number $n$. The orbital period $T$ is defined as the time taken for one complete revolution: $T = \frac{2\pi}{\omega}$.
From the derivation of the Bohr radius ($r_n$) and the velocity ($v_n$), we know that $v_n = \frac{v_1}{n}$ and $r_n = n^2 r_1$. Since $\omega = \frac{v_n}{r_n}$, we substitute the expressions:
$\omega_n = \frac{v_1/n}{n^2 r_1} = \frac{v_1}{n^3 r_1} = \frac{\omega_1}{n^3}$
Thus, the orbital period $T_n$ is given by: $T_n = \frac{2\pi}{\omega_1/n^3} = T_1 \cdot n^3$, where $T_1$ is the period of the ground state ($n=1$). [Per Bohr’s Postulates of Quantization]
Step 2: Calculating Ground State Period ($T_1$)
For the hydrogen atom, the velocity in the first orbit is $v_1 \approx 2.18 \times 10^6 \text{ m/s}$ and the Bohr radius is $r_1 \approx 0.529 \times 10^{-10} \text{ m}$.
The ground state angular frequency $\omega_1$ is: $\omega_1 = \frac{v_1}{r_1} = \frac{2.18 \times 10^6}{0.529 \times 10^{-10}} \approx 4.12 \times 10^{16} \text{ rad/s}$
Therefore, the ground state period $T_1$ is: $T_1 = \frac{2\pi}{\omega_1} = \frac{2 \times 3.14159}{4.12 \times 10^{16}} \approx 1.52 \times 10^{-16} \text{ s}$
Step 3: Calculating Periods for Higher Energy Levels
Using the relation $T_n = T_1 \cdot n^3$, we calculate the periods for $n=1, 2, 3$:
| Quantum Number ($n$) | Calculation ($1.52 \times 10^{-16} \times n^3$) | Orbital Period ($T_n$) |
|---|---|---|
| $n=1$ | $1.52 \times 10^{-16} \times 1$ | $1.52 \times 10^{-16} \text{ s}$ |
| $n=2$ | $1.52 \times 10^{-16} \times 8$ | $1.22 \times 10^{-15} \text{ s}$ |
| $n=3$ | $1.52 \times 10^{-16} \times 27$ | $4.10 \times 10^{-15} \text{ s}$ |
Visualization of Orbital Expansion
Final Solution: The orbital periods for $n=1, 2,$ and $3$ are $1.52 \times 10^{-16} \text{ s}$, $1.22 \times 10^{-15} \text{ s}$, and $4.10 \times 10^{-15} \text{ s}$ respectively.
More Questions from Class 12 Physics Atoms EXERCISES
- Q12.1(a): Choose the correct alternative from the clues given at the end of the each statement: (a) The size of the atom in Thomson’s model is .......... the atomic size in Rutherford’s model.
- Q12.1(a): Choose the correct alternative from the clues given at the end of the each statement: The size of the atom in Thomson’s model is .......... the atomic size in Rutherford’s model.
- Q12.1(b): Choose the correct alternative from the clues given at the end of the each statement: (b) In the ground state of .......... electrons are in stable equilibrium, while in .......... electrons always experience a net force.
- Q12.1(b): Choose the correct alternative from the clues given at the end of the each statement: In the ground state of .......... electrons are in stable equilibrium, while in .......... electrons always experience a net force.
- Q12.1(c): Choose the correct alternative from the clues given at the end of the each statement: (c) A classical atom based on .......... is doomed to collapse.
- Q12.1(c): Choose the correct alternative from the clues given at the end of the each statement: A classical atom based on .......... is doomed to collapse.
- Q12.1(d): Choose the correct alternative from the clues given at the end of the each statement: (d) An atom has a nearly continuous mass distribution in a .......... but has a highly non-uniform mass distribution in ..........
- Q12.1(d): Choose the correct alternative from the clues given at the end of the each statement: An atom has a nearly continuous mass distribution in a .......... but has a highly non-uniform mass distribution in ..........
- Q12.1(e): Choose the correct alternative from the clues given at the end of the each statement: (e) The positively charged part of the atom possesses most of the mass in ..........
- Q12.1(e): Choose the correct alternative from the clues given at the end of the each statement: The positively charged part of the atom possesses most of the mass in ..........
- Q12.2: Suppose you are given a chance to repeat the alpha-particle scattering experiment using a thin sheet of solid hydrogen in place of the gold foil. (Hydrogen is a solid at temperatures below 14 K.) What results do you expect?
- Q12.2: Suppose you are given a chance to repeat the alpha-particle scattering experiment using a thin sheet of solid hydrogen in place of the gold foil. (Hydrogen is a solid at temperatures below 14 K.) What results do you expect?
- Q12.3: A difference of 2.3 eV separates two energy levels in an atom. What is the frequency of radiation emitted when the atom make a transition from the upper level to the lower level?
- Q12.3: A difference of 2.3 eV separates two energy levels in an atom. What is the frequency of radiation emitted when the atom make a transition from the upper level to the lower level?
- Q12.4: The ground state energy of hydrogen atom is –13.6 eV. What are the kinetic and potential energies of the electron in this state?
- Q12.4: The ground state energy of hydrogen atom is –13.6 eV. What are the kinetic and potential energies of the electron in this state?
- Q12.5: A hydrogen atom initially in the ground level absorbs a photon, which excites it to the $n = 4$ level. Determine the wavelength and frequency of photon.
- Q12.5: A hydrogen atom initially in the ground level absorbs a photon, which excites it to the ((n)) = 4 level. Determine the wavelength and frequency of photon.
- Q12.6(a): (a) Using the Bohr’s model calculate the speed of the electron in a hydrogen atom in the $n = 1, 2$, and $3$ levels.
- Q12.6(a): (a) Using the Bohr’s model calculate the speed of the electron in a hydrogen atom in the ((n)) = 1, 2, and 3 levels.
- Q12.6(b): (b) Calculate the orbital period in each of these levels.
- Q12.7: The radius of the innermost electron orbit of a hydrogen atom is $5.3 \times 10^{-11}$ m. What are the radii of the $n = 2$ and $n = 3$ orbits?
- Q12.7: The radius of the innermost electron orbit of a hydrogen atom is ((5.3 \times 10^{-11})) m. What are the radii of the ((n)) = 2 and ((n)) = 3 orbits?
- Q12.8: A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?
- Q12.8: A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?
- Q12.9: In accordance with the Bohr’s model, find the quantum number that characterises the earth’s revolution around the sun in an orbit of radius $1.5 \times 10^{11}$ m with orbital speed $3 \times 10^{4}$ m/s. (Mass of earth = $6.0 \times 10^{24}$ kg.)
- Q12.9: In accordance with the Bohr’s model, find the quantum number that characterises the earth’s revolution around the sun in an orbit of radius ((1.5 \times 10^{11})) m with orbital speed ((3 \times 10^{4})) m/s. (Mass of earth = ((6.0 \times 10^{24})) kg.)
CBSE Solutions for Class 12 Physics Atoms
Chapters in CBSE - Class 12 Physics
- Electric Charges and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Waves
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Nuclei
- Semiconductor Electronics
- Communication Systems
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