Problem 12 - College Physics

Question

According to the Bohr theory of the hydrogen atom, the speed of the electron in the first Bohr orbit is \(2.19 \times 10^{6} \mathrm{~m} / \mathrm{s} .\) (a) What is the wavelength of the matter wave associated with the electron? (b) How does this wavelength compare with the circumference of the first Bohr orbit?

Step-by-Step Solution

Answer

The wavelength of the electron is approximately equal to the circumference of the first Bohr orbit.

Step 1: Determine Wavelength Using de Broglie Equation

To find the wavelength \( \lambda \) of the matter wave associated with the electron, we use the de Broglie wavelength formula: \[ \lambda = \frac{h}{mv} \]where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ Js}\), \( m \) is the mass of the electron \( 9.109 \times 10^{-31} \text{ kg}\), and \( v \) is the speed of the electron \( 2.19 \times 10^6 \text{ m/s}\). Substituting the values, we calculate:\[ \lambda = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 2.19 \times 10^6} \approx 3.32 \times 10^{-10} \text{ m} \].

Step 2: Find Circumference of First Bohr Orbit

The circumference of the first Bohr orbit \( C \) can be calculated using the formula for the circumference of a circle \( C = 2\pi r \), where the radius \( r \) is the Bohr radius \( 5.29 \times 10^{-11} \text{ m}\). Thus:\[ C = 2 \pi \times 5.29 \times 10^{-11} \approx 3.32 \times 10^{-10} \text{ m}\].

Step 3: Compare Wavelength with Circumference

Comparing the calculated wavelength \( \lambda \approx 3.32 \times 10^{-10} \text{ m}\) to the circumference of the first Bohr orbit \( C = 3.32 \times 10^{-10} \text{ m}\), we see that the wavelength is approximately equal to the circumference of the orbit.

Key Concepts

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that reveals the wave-like nature of particles. Louis de Broglie proposed that all particles could exhibit wave properties, a revolutionary idea linking particle physics with wave theory.

• de Broglie Equation: The equation used to calculate the wavelength (\( \lambda \)) of a particle's wave is \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.
• Planck's Constant: This constant \( (6.626 \times 10^{-34} \text{ Js}) \) is crucial in both the de Broglie equation and the broader field of quantum mechanics.
• Application: We used this formula to find the matter wave wavelength of the electron moving in a hydrogen atom's Bohr orbit.

This idea explains how small particles, like electrons, behave similarly to waves, providing deeper insight into the mysteries of atomic and subatomic behaviors.

Electrons

Electrons are subatomic particles that play a critical role in chemistry and physics. They are negatively charged and orbit the nucleus of an atom in distinct energy levels.

• Charge and Mass: Electrons carry a negative charge and have a mass of \(9.109 \times 10^{-31} \text{ kg}\). Though incredibly small, their interactions are vital for chemical bonds and electricity.
• Bohr Model: Niels Bohr’s model places electrons in fixed orbits around an atom's nucleus. The de Broglie wavelength provides a way to view these orbits as wave-like paths.
• Wave-Particle Duality: Electrons exhibit wave-particle duality, meaning they have properties of both particles and waves. This duality is a cornerstone of quantum theory.

Understanding these properties helps explain complex phenomena, from how atoms form to the operations of electrical circuits.

Atomic Physics

Atomic physics is the field of physics that examines atoms as a system of electrons and a nucleus. This field has led to understanding fundamental interactions within an atom.

• Atomic Structure: Atoms consist of a dense nucleus surrounded by electrons. The arrangement and behavior of electrons influence the atom's chemical properties.
• Bohr's Contributions: The Bohr model introduced quantized orbits for electrons, explaining why atoms emit light at specific wavelengths, a phenomenon known as atomic spectra.
• Experimental Techniques: Tools like spectroscopy allow scientists to study atomic structures and behaviors insightfully.

In essence, atomic physics helps us comprehend the building blocks of matter and the forces at play within atoms, driving modern technological advances.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with phenomena at the smallest scales, such as atoms and subatomic particles. It diverges from classical mechanics and provides unique insights into the behavior of matter.

• Uncertainty Principle: Proposed by Heisenberg, it states that certain pairs of physical properties, like position and momentum, cannot be simultaneously known to arbitrary precision.
• Quantization: Quantum mechanics introduced the concept that energy levels are quantized, meaning electrons can only occupy certain energy levels around an atom's nucleus.
• Quantum State: Particles are described by a wave function, representing the probabilities of finding a particle in various states.

These concepts often defy common sense but are essential for explaining many natural phenomena and enabling technological developments, such as semiconductors and lasers.