de Broglies Explanation of Bohrs Quantisation - UNSOLVED PRACTICE SET
Chapter: Atoms | Topic: de Broglies Explanation of Bohrs Quantisation
DE BROGLIES EXPLANATION OF BOHRS QUANTISATION - UNSOLVED PRACTICE SET
Topic: de Broglies Explanation of Bohrs Quantisation
Multiple Choice Questions
Q1. According to de Broglie, the electron in a Bohr orbit behaves as a:
- Particle only
- Wave only
- Standing wave
- Travelling wave
Q2. De Broglie's explanation of Bohr's quantization condition is based on the idea that:
- The electron's orbit must contain an integer number of electron wavelengths
- The electron must travel at the speed of light
- The electron's mass must be quantized
- The electron's charge must be quantized
Q3. The circumference of the nth Bohr orbit is equal to:
- n times the de Broglie wavelength of the electron
- nΒ² times the de Broglie wavelength
- The de Broglie wavelength divided by n
- A constant independent of n
Q4. De Broglie's explanation implies that the electron in a stable orbit forms:
- A progressive wave
- A standing wave with nodes and antinodes
- A shock wave
- No wave pattern
Q5. The de Broglie wavelength of the electron in the first Bohr orbit of hydrogen is:
- Equal to the circumference of the orbit
- Twice the circumference of the orbit
- Half the circumference of the orbit
- Equal to the radius of the orbit
Q6. De Broglie's explanation provides:
- A proof of Bohr's quantization condition from wave mechanics
- A rejection of Bohr's model
- An alternative to quantum mechanics
- A classical explanation of atomic spectra
Short Answer Questions
Q7. Explain how de Broglie's hypothesis accounts for Bohr's quantization condition of angular momentum.
Q8. What is the physical significance of the electron forming a standing wave in a Bohr orbit?
Q9. Show that the de Broglie wavelength of the electron in the nth orbit is Ξ» = 2Οr/n, where r is the radius of the orbit.
Q10. Why does the electron not radiate energy when it forms a standing wave in a stationary orbit?
Q11. Calculate the de Broglie wavelength of the electron in the first Bohr orbit of hydrogen.
Q12. How does de Broglie's explanation connect the particle and wave nature of the electron in the atom?
Long Answer Questions
Q13. Explain de Broglie's explanation of Bohr's quantization condition. Show that the condition for standing waves in an orbit leads directly to Bohr's quantization of angular momentum.
Q14. Derive the expression for the de Broglie wavelength of an electron in the nth Bohr orbit. Show that the circumference of the orbit contains exactly n de Broglie wavelengths.
Q15. Discuss the significance of de Broglie's explanation in the development of quantum mechanics. How does it bridge the gap between Bohr's ad hoc postulate and the wave mechanical model of the atom?
Numerical & Application-based Problems
Q16. For a hydrogen atom:
(a) Calculate the de Broglie wavelength of the electron in the first Bohr orbit.
(b) Verify that the circumference of the first orbit contains exactly one de Broglie wavelength.
(c) Calculate the de Broglie wavelength of the electron in the third Bohr orbit.
(d) Verify that the circumference of the third orbit contains exactly three de Broglie wavelengths.
Q17. An electron in a hydrogen atom is in the n = 4 state.
(a) Calculate the radius of this orbit.
(b) Calculate the speed of the electron in this orbit.
(c) Calculate the de Broglie wavelength of the electron.
(d) Verify that the circumference of the orbit contains exactly four de Broglie wavelengths.
Q18. In your school's physics exhibition, a student creates a demonstration of de Broglie's explanation using a string model.
(a) She stretches a string of length 1 m between two fixed points and vibrates it to form standing waves. She observes that only certain frequencies produce stable standing waves. Calculate the wavelengths of the first three standing wave modes and compare them with the de Broglie wavelengths in the first three Bohr orbits.
(b) The student then explains that if the string were a circular loop, the standing wave condition would be similar to Bohr's orbits. Calculate the allowed wavelengths for a circular loop of circumference 2Οr and show that they correspond to the de Broglie wavelengths in Bohr's model.
(c) A classmate asks why we don't observe the wave nature of electrons in everyday objects like cricket balls. Calculate the de Broglie wavelength of a 0.1 kg ball moving at 10 m/s and explain why its wave nature is not observable.
(d) The student learns that the wave nature of electrons is used in electron microscopes. Explain why electrons with de Broglie wavelengths of about 0.01 nm can resolve much smaller details than visible light microscopes (Ξ» β 500 nm).
(e) In the context of modern physics, explain how de Broglie's standing wave picture, while insightful, was eventually superseded by SchrΓΆdinger's wave equation, and what additional information the modern theory provides.