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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.
Solution :
Step 1: Establishing the Theoretical Framework
According to Bohr’s postulate of quantization of angular momentum, an electron in a hydrogen atom moves in a circular orbit such that its angular momentum is an integral multiple of $\frac{h}{2\pi}$. The velocity $v_n$ of an electron in the $n^{th}$ orbit is derived from the balance between the electrostatic force of attraction and the necessary centripetal force:
$\frac{mv_n^2}{r_n} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Ze^2}{r_n^2}$
By applying the quantization condition $mvr = \frac{nh}{2\pi}$, the expression for velocity in the $n^{th}$ orbit for a hydrogen atom ($Z=1$) is given by:
$v_n = \frac{e^2}{2\epsilon_0 nh}$
Substituting the universal constants:
- Charge of electron ($e$) = $1.6 \times 10^{-19}\ C$
- Permittivity of free space ($\epsilon_0$) = $8.854 \times 10^{-12}\ C^2N^{-1}m^{-2}$
- Planck’s constant ($h$) = $6.626 \times 10^{-34}\ Js$
Step 2: Calculating the Velocity Constant
To simplify, we utilize the relationship between the velocity of the first orbit ($v_1$) and the fine-structure constant ($\alpha$): $v_1 = \frac{e^2}{2\epsilon_0 h} \approx 2.188 \times 10^6\ m/s$
The velocity for any level $n$ is represented as: $v_n = \frac{v_1}{n}$
Step 3: Calculating Velocity for Specific Levels
| Energy Level ($n$) | Calculation Formula | Calculated Speed ($m/s$) |
|---|---|---|
| $n = 1$ | $v_1 = \frac{2.188 \times 10^6}{1}$ | $2.188 \times 10^6\ m/s$ |
| $n = 2$ | $v_2 = \frac{2.188 \times 10^6}{2}$ | $1.094 \times 10^6\ m/s$ |
| $n = 3$ | $v_3 = \frac{2.188 \times 10^6}{3}$ | $7.293 \times 10^5\ m/s$ |
Step 4: Visualizing the Orbital Velocity Decay
Final Solution: The speeds of the electron in the $n=1, 2,$ and $3$ levels are $2.188 \times 10^6\ m/s$, $1.094 \times 10^6\ m/s$, and $7.293 \times 10^5\ m/s$ 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(b): (b) Calculate the orbital period in each of these 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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