Physics · Unit 3 · 8 hrs
Fundamentals of Atomic Theory
Exam-focused notes for Fundamentals of Atomic Theory (Physics, PHY118): what the TU syllabus asks and how it has actually been tested, with 13 solved past questions from this unit.
What this unit covers
- Blackbody radiation
- Bohr atom
- Spectrum of Hydrogen
- Franck-Hertz experiment
- de Broglie's hypothesis and its experimental verification
- Uncertainty principle and its origin
- matter waves and the uncertainty principle
- group velocity
group velocity
Explain group velocity. [5]
The velocity with which the wave packet obtained due to superposition of waves travelling in a group is called group velocity. It is denoted by vg. In other words, when a number of waves of slightly different frequencies and wavelengths are superimposed, th...
Full solved answer →Uncertainty principle and its origin
A small particle of mass $10^{-6}$ g moves along the x axis; its speed is uncertain by $10^{-6}$ m/sec. (a) What is the uncertainty in the x coordinate of the particle? (b) Repeat the calculation for an electron assuming that the uncertainty in its velocity is also $10^{-6}$ m/sec. [5]
- Mass of small particle: $m = 10^{-6}$ g $= 10^{-6} \times 10^{-3}$ kg $= 10^{-9}$ kg - Uncertainty in speed: $\Delta vx = 10^{-6}$ m/s - Mass of electron: $me = 9.11 \times 10^{-31}$ kg - Planck's constant: $h = 6.626 \times 10^{-34}$ J·s The uncertainty ...
Full solved answer →Calculate uncertainty in the momentum of electron if uncertainty in its position is $10^{-10}$ m. [5]
- Uncertainty in position: $\Delta x = 10^{-10}$ m - Planck's constant: $h = 6.626 \times 10^{-34}$ J·s - (Mass of electron $me = 9.11 \times 10^{-31}$ kg is available but not required for momentum uncertainty) --- $$\Delta x \cdot \Delta p \geq \frac{h}{4\...
Full solved answer →A small particle of mass $10^{-6}$ gm moves along the x axis; its speed is uncertain by $10^5$ m/sec. (a) what is the uncertainty in the x coordinate of the particle? (b) Repeat the calculation for an electron assuming that the uncertainty in its velocity is also $10^5$ m/sec. use the known values for electrons and Planck's constant. [5]
- Mass of small particle: $m = 10^{-6}$ g $= 10^{-9}$ kg - Uncertainty in speed: $\Delta vx = 10^5$ m/s - Mass of electron: $me = 9.11 \times 10^{-31}$ kg - Planck's constant: $h = 6.626 \times 10^{-34}$ J·s The uncertainty principle: $$\Delta x \cdot \Delt...
Full solved answer →Spectrum of Hydrogen
What are (a) the energy, (b) the momentum, and (c) the wavelength of the photon that is emitted when a hydrogen atom undergoes a transition from the state n = 4 to n = 2? [5]
- Initial state: $ni = 4$ - Final state: $nf = 2$ - Hydrogen ground-state constant: $En = -\dfrac{13.6}{n^2}$ eV - Constants: $h = 6.626\times10^{-34}$ J·s, $c = 3\times10^8$ m/s, $1\text{ eV} = 1.6\times10^{-19}$ J $$E4 = -\frac{13.6}{16} = -0.85 \text{ eV...
Full solved answer →Give spectrum of Hydrogen atom and discuss its lines. [5]
When an electron jumps from a higher energy level (n₂) to a lower energy level (n₁), energy is emitted in the form of radiation equal to the difference in energies: $$E2 - E1 = h\nu$$ From Bohr's theory, the total energy of an electron in the n-th orbit is:...
Full solved answer →de Broglie's hypothesis and its experimental verification
An α particle is emitted from a radioactive nuclei with an energy of 6.8 MeV. Calculate its wavelength and compare it with the size of the emitting nucleus that has a radius of $8 \times 10^{-15}$ m. [5]
- Kinetic energy of α particle: $E = 6.8$ MeV - Radius of emitting nucleus: $r = 8 \times 10^{-15}$ m Constants used: - Mass of α particle: $m = 4u = 4 \times 1.66 \times 10^{-27} = 6.64 \times 10^{-27}$ kg - Planck's constant: $h = 6.626 \times 10^{-34}$ J...
Full solved answer →In neutron spectroscopy a beam of mono-energetic neutrons is obtained by reflecting reactor neutrons from a beryllium crystal. If the separation between the atomic planes of the beryllium crystal is $0.732 \text{ Å}$, what is the angle between the incident neutron beam and the atomic planes that will yield a monochromatic beam of neutrons of wavelength $0.1 \text{ Å}$? [5]
- Interplanar spacing: $d = 0.732\ \text{Å}$ - Neutron wavelength: $\lambda = 0.1\ \text{Å}$ - Order: $n = 1$ (first order) Bragg's Law: $$2d\sin\theta = n\lambda$$ Solve for $\theta$: $$\sin\theta = \frac{n\lambda}{2d} = \frac{1 \times 0.1}{2 \times 0.732}...
Full solved answer →In neutron spectroscopy a beam of monoenergetic neutrons is obtained by reflecting reactor neutrons from a beryllium crystal. If the separation between the atomic planes of the beryllium crystal is $0.732 \text{ Å}$, what is the angle between the incident neutron beam and the atomic planes that will yield a monochromatic beam of neutrons of wavelength $0.1 \text{ Å}$? [5]
- Plane separation: $d = 0.732\ \text{Å} = 0.732 \times 10^{-10}\ \text{m}$ - Required wavelength: $\lambda = 0.1\ \text{Å} = 0.1 \times 10^{-10}\ \text{m}$ - Order of diffraction: $n = 1$ (first order, assumed) Bragg's Law: $$n\lambda = 2d\sin\theta$$ For ...
Full solved answer →The uncertainty in the position of a particle is equal to the de Broglie wavelength of particle. Calculate the uncertainty in the velocity of the particle in terms of the velocity of the de Broglie wave associated with the particle. [5]
- Uncertainty in position: $\Delta x = \lambda$ (the de Broglie wavelength) - de Broglie wavelength: $\lambda = \dfrac{h}{mv}$, where $m$ = mass, $v$ = particle velocity - Required: $\Delta v$ expressed in terms of the velocity of the de Broglie wave associ...
Full solved answer →Franck-Hertz experiment
Describe Frank-Hertz experiment. Interpret how the results of this experiment advocate atomic model proposed by Bohr?[10]
The Frank-Hertz experiment, performed by James Franck and Gustav Hertz in 1914, provided direct experimental evidence for the existence of discrete (quantized) energy levels in atoms. This experiment strongly supported the atomic model proposed by Niels Boh...
Full solved answer →Describe Frank Hertz experiment. Discuss its result and outline limitations.[10]
The Frank-Hertz experiment, performed by James Franck and Gustav Hertz in 1914, provided direct experimental evidence for the existence of discrete (quantized) energy levels in atoms. It confirmed Bohr's atomic model by showing that atoms can only absorb en...
Full solved answer →Blackbody radiation
Explain the theory of black body radiation. Why this theory needs quantum mechanical interpretation? How this interpretation became experimentally successful? Explain.[10]
A black body is an ideal body that absorbs all incident radiation falling on it, regardless of wavelength or angle of incidence. When such a body is heated, it emits radiation called black body radiation (also called thermal radiation). Key characteristics:...
Full solved answer →Make Unit 3 stick
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