Atomic Models

Step 1: Understanding photons. Photons are particles of light, which have zero rest mass and travel at the speed of light in a vacuum. However, they do carry energy and momentum, as per .
Step 2: Conclusion. Thus, option (B) is incorrect because photons do carry momentum despite having zero rest mass.

Step 1: Angular momentum of the electron. In the Bohr model of the hydrogen atom, the angular momentum is quantized and is given by: Where is the principal quantum number and is the reduced Planck’s constant.
Step 2: Conclusion. Thus, for the hydrogen atom, the principal quantum number for the angular momentum is 2.

Step 1: Understanding transition series. The transition of an electron from a higher to a lower energy level in hydrogen emits radiation in specific series. The transition from to corresponds to the Lyman series.
Step 2: Conclusion. Thus, the correct series is the Lyman series.

Step 1: Use the relation between wavelengths in different orders. For the hydrogen spectrum, the wavelength in the -th order is related to the wavelength in the first order by: Where is the wavelength in the first order and is the order of the spectrum.
Step 2: Solve for . Given for the second order, we have:

Step 1: Using relativistic velocity addition formula.
When two particles are moving in opposite directions, the relative speed is given by the relativistic velocity addition formula: where and .
Step 2: Substituting the values.

Step 3: Conclusion.
Hence, the correct answer is option (3).

Final Answer:

Step 1: Understanding Bragg's Law.
Bragg's law explains the condition for constructive interference of X-rays diffracted by a crystal lattice: where is the wavelength, is the interplanar distance, and is the angle of diffraction.
Step 2: Analysis of the options.
- (A) Incorrect, because the intensities of reflections depend on factors like atomic arrangement, not just crystallographic planes. - (B) Correct statement, higher order reflections have higher -values for a given wavelength. - (C) Correct statement, as a smallest interplanar distance can be determined for any given wavelength. - (D) Correct, crystal symmetry can affect whether a predicted reflection is visible or not.
Step 3: Conclusion.
The incorrect statement is (A). Hence, the correct answer is option (A).

Final Answer:

Step 1: Understanding Moseley's law.
Moseley's law states that the square root of the frequency of the X-ray line is proportional to the atomic number of the element. Mathematically:
Step 2: Analyzing the options.
The graph should plot the square of the frequency ( ) against the atomic number . The correct representation is option (B) vs .
Step 3: Conclusion.
Therefore, the correct graph is option (B).

Final Answer:

Step 1: Frequency of Balmer series lines.
The Balmer series of hydrogen has frequencies that follow a specific ratio. The second line of the series is times the first line.
Step 2: Conclusion.
Hence, the frequency of the second line is . Therefore, the correct answer is option (B).

Final Answer:

Step 1: Relating kinetic and rest energy.
At the velocity , the kinetic energy is equal to the rest energy. This gives us the relation , leading to .
Step 2: Conclusion.
Thus, the velocity at which the kinetic energy equals the rest energy is , so the correct answer is option (D).

Final Answer:

Step 1: Using deBroglie relation.
The deBroglie wavelength of a particle is given by where is the mass and is the kinetic energy.
Step 2: Analyzing the options.
Since the kinetic energy is the same for both, the ratio of wavelengths depends on the masses of the particles. Hence,
Step 3: Conclusion.
Therefore, the correct answer is option (D).

Final Answer:

Step 1: Identify the ground state of the atom.
The magnetic moment of an atom in a ground state is given by where is the number of electrons in the unpaired orbital. In this case, the number of unpaired electrons is 5. Thus, we apply the formula:
Step 2: Conclusion.
Hence, the correct answer is option (D).

Final Answer:

Step 1: Use de Broglie wavelength for electrons.

Step 2: Effect of increasing speed.
If increases, momentum increases, so de Broglie wavelength decreases:

Step 3: Diffraction angular width relation.
For single slit diffraction:

So if decreases, decreases.
Step 4: Conclusion.
Thus, the angular width of central maximum decreases when electron speed increases.
Final Answer:

Step 1: Understand K-series X-rays.
K-series X-rays are produced when an electron falls into the K-shell ( ) from a higher level.
Step 2: Identify .
line corresponds to transition from L-shell to K-shell:

Step 3: Identify .
line corresponds to transition from M-shell to K-shell:

Step 4: Match with correct option.
So correct answer is option (A).
Final Answer:

Step 1: Identify what proves wave nature.
Wave nature is directly confirmed when a particle shows phenomena like diffraction or interference, which are purely wave effects.
Step 2: Apply to matter particles.
Electrons are matter particles. When electrons show diffraction (as in Davisson--Germer experiment), it proves electrons behave like waves.
Step 3: Eliminate other options.
Momentum is a particle property, not proof of wave nature. Photon diffraction proves wave nature of light, not matter.
Step 4: Final conclusion.
Hence, electron diffraction best supports wave nature of matter.
Final Answer:

Step 1: Current due to revolving electron.
Electron revolving in orbit forms a current:
where is revolutions per second.
Given .
Step 2: Magnetic field at centre of circular loop.
Given .
Step 3: Substitute values.
Final Answer:

Step 1: Write centripetal force equation.
Centripetal force required:
Coulomb force:
Step 2: Equate centripetal and coulomb force.
Step 3: Substitute .
Step 4: Solve for speed .
Final Answer:

Step 1: Bohr radius relation.
Radius of nth orbit:
Step 2: Velocity relation.
Electron velocity in nth orbit:
Step 3: Energy relation.
Total energy of electron:
Magnitude varies as .
Step 4: Match option.
Thus variation is:
Final Answer:

Step 1: Current due to revolving electron.
An electron moving in circular orbit forms a current:
where is time period.
Step 2: Time period of revolution.
So,
Step 3: Substitute given values of and .
Step 4: Simplify.
Now converting into form gives:
which matches option (B).
Final Answer:

Step 1: Momentum of the particle.
The momentum of a particle is given by: where is the mass, is the charge, and is the potential difference.
Step 2: Ratio of momentum.
Since the momentum depends on mass, the ratio of momentum between the electron and -particle is:
Final Answer:

Step 1: Formula for number of excess electrons.
The number of excess electrons is given by: where is the total charge and is the charge of an electron.
Step 2: Calculation.
Substituting the given charge of the drop and the charge of an electron:
Final Answer:

Step 1: Understanding spin quantum number.
The spin quantum number indicates the direction of the electron's spin. corresponds to a clockwise spin, and corresponds to an anti-clockwise spin.
Step 2: Explanation.
Thus, and represent the clockwise and anti-clockwise directions of the electron's spin, respectively.
Final Answer:

When a hydrogen atom absorbs energy, it jumps to a higher energy level. The orbital angular momentum changes accordingly. Using the Bohr model, we calculate the increase in angular momentum.

Step 2: Conclusion.
The increase in orbital angular momentum is , corresponding to option (b).

The relationship between the frequency of the X-ray spectrum and the atomic number is given by Moseley's law, which is a linear relationship. The graph will have a negative square root dependence.

Step 2: Conclusion.
The graph follows the form , corresponding to option (b).

Step 1: Understanding Bohr’s model.
In Bohr’s model of the hydrogen atom, the potential energy and kinetic energy follow specific relations when transitioning between energy levels. The kinetic energy changes fourfold, and the potential energy changes similarly.

Step 2: Conclusion.
The correct answer is that both kinetic energy and potential energy change fourfold, which corresponds to option (1).

Step 1: Understanding the relationship.
The wavelength of -X-ray emitted by an element is inversely proportional to the square of its atomic number. Using the relationship , we can solve for the atomic number emitting a wavelength of .

Step 2: Calculations.
Since and the wavelength changes by a factor of 4, we find the new atomic number is 6.

The wavelength of the emitted radiation is given by the Rydberg formula. By applying the formula for the transition between the second and third orbits, the wavelength of the radiation can be calculated.

In this case, when a hydrogen atom is raised to an excited state, it gains potential energy, but the kinetic energy decreases because the atom is less tightly bound in the higher energy state. The potential energy increases as the electron moves farther from the nucleus.

For the electron in Bohr’s model, the angular momentum is quantized and given by , where for the ground state. The angular momentum for a -3.4 eV energy corresponds to .

Step 1: Relation between Wavelength and Energy.
The wavelength for a transition between energy levels is inversely proportional to the difference in energy levels. Using the given information, the wavelength for the reverse transition would be directly related to the energy difference, resulting in .
Step 2: Conclusion.
The correct answer is (C), .

Step 1: Using the Rydberg Formula.
The Rydberg formula for the wavelength of the emitted photon is given by: where is the Rydberg constant and is the principal quantum number. Solving for using this equation, we get . Step 2: Conclusion.
The correct answer is (A), .

Step 1: The wavelengths of emitted radiation during electron transitions in hydrogen atoms can be calculated using the Rydberg formula for the emission spectrum.
Step 2: The wavelengths and correspond to the transitions from the third to the second excited state and from the second to the first excited state, respectively. The ratio of these wavelengths comes out to be .

Final Answer:

Step 1: According to the Bohr model of the atom, the radius of the -th orbit is given by: where is the radius of the first orbit.
Step 2: The ratio of the radii of the first three orbits is:
Final Answer:

Step 1: de-Broglie wavelength formula.
The de-Broglie wavelength is given by: where is Planck’s constant and is the momentum of the electron. Step 2: Relationship between kinetic energy and momentum.
The kinetic energy of the electron is related to its momentum by: where is the mass of the electron. If the kinetic energy doubles, the momentum increases by a factor of . Therefore, the wavelength decreases by a factor of .
Final Answer:

The energy levels of the hydrogen atom are given by: where is the principal quantum number. The energy ratio for the first and second excited states is:
Final Answer: