Problem 43 - College Physics

Question

\(\bullet\) In the Bohr model of the hydrogen atom, what is the de Broglie wavelength for the electron when it is in (a) the \(n=1\) level and (b) the \(n=4\) level? In each case, compare the de Broglie wavelength to the circumference 2\(\pi r_{n}\) of the orbit.

Step-by-Step Solution

Answer

For both levels, the de Broglie wavelength equals the orbit circumference, showing quantized orbits.

Step 1: Formula for de Broglie Wavelength

The de Broglie wavelength \(\lambda\) of a particle is given by the formula \(\lambda = \frac{h}{p}\), where \(h\) is the Planck constant and \(p\) is the momentum of the particle. For an electron orbiting in the Bohr model, the momentum \(p\) is \(mv\), where \(m\) is the mass of the electron and \(v\) is the velocity.

Step 2: Velocity of Electron in Bohr Model

In the Bohr model, the velocity of an electron in the nth level is given by \(v = \frac{e^2}{2\epsilon_0 h} \cdot \frac{1}{n}\), where \(e\) is the charge of the electron, and \(\epsilon_0\) is the vacuum permittivity.

Step 3: Calculate de Broglie Wavelength for n=1

For \(n = 1\), substitute the expression for velocity into the de Broglie wavelength formula: \(\lambda = \frac{h}{mv}\). Using the velocity \(v = \frac{e^2}{2\epsilon_0 h}\), calculate the de Broglie wavelength.

Step 4: Calculate de Broglie Wavelength for n=4

For \(n = 4\), similarly substitute the corresponding velocity into \(\lambda = \frac{h}{mv}\). The velocity for \(n = 4\) is \(\frac{1}{4}\) times the velocity for \(n = 1\). Calculate the de Broglie wavelength.

Step 5: Compare Wavelength to Orbit Circumference

In the Bohr model, the circumference of the nth orbit is \(2\pi r_n = 2\pi\left(\frac{n^2h^2}{4\pi^2 m e^2}\right)\). Calculate and compare this circumference to the corresponding de Broglie wavelength \(\lambda\) for each level.

Key Concepts

de Broglie Wavelength

In atomic physics, the de Broglie wavelength is an important concept that helps explain the wave-particle duality of electrons. De Broglie proposed that particles like electrons also exhibit wave-like properties. The de Broglie wavelength \[\lambda = \frac{h}{p} \]where:

• \(\lambda\) is the de Broglie wavelength,
• \(h\) is Planck constant, and
• \(p\) is the momentum of the particle.

For an electron in a hydrogen atom, its momentum is described as \(p = mv\)where \(m\) is the mass of the electron and \(v\) is its velocity. The de Broglie wavelength helps illustrate the electron's wave-like nature as it orbits the nucleus, demonstrating a balance between particle-like and wave-like behavior.

Electron Orbit

In the Bohr model, electrons are considered to move around the nucleus in fixed orbits, similar to planets around the sun. These orbits or energy levels have certain characteristics. The electron's velocity affects its orbit and is given by:\[v = \frac{e^2}{2\epsilon_0 h} \cdot \frac{1}{n}\]Where:

• \(e\) is the electron charge,
• \(\epsilon_0\) is the permittivity of free space,
• \(n\) is the principal quantum number reflecting the orbit level.

These orbits are not arbitrary; they have specific radii and circumferences, calculated by:\[2\pi r_n = 2\pi\left(\frac{n^2h^2}{4\pi^2 m e^2}\right)\]The calculated circumference is crucial in comparing with the de Broglie wavelength, serving as a bridge between electron standing waves and classical orbits.

Quantum Levels

Quantum levels represent the discrete energy stages available to electrons in an atom. An electron in a higher quantum level has more energy compared to one in a lower level.Each level can be identified by a unique principal quantum number \(n\), determining:

• Size of the electron orbit,
• Energy associated with the electron, and
• Velocity of the electron.

As \(n\) increases, the orbit size becomes larger and the energy increases. This quantization is essential for understanding electron transitions, where electrons move between levels absorbing or emitting specific energy quanta. Quantum levels thus explain why atoms absorb or emit light at specific wavelengths.

Hydrogen Atom

The hydrogen atom is the simplest atom, containing one proton and one electron. Its simplicity makes it an ideal model for studying fundamentals of atomic physics like quantized orbits and energy levels. Bohr's model, which applies well to hydrogen, is based on:

• Quantized electron orbits,
• Specific energy levels, and
• Absorption/emission of energy as electrons transition between levels.

For the hydrogen atom, the wavelength of the absorbed or emitted light corresponds to the difference in energy levels of the electron. Studying these transitions helps elucidate the structural and electronic properties of atoms, as well as the principles behind atomic spectra.

Atomic Physics

Atomic physics focuses on the study of atoms as isolated systems of electrons and an atomic nucleus. Central topics include electron configuration, atomic spectra, and electron interactions.
Bohr's model is pivotal in atomic physics, providing insights into:

• Electron orbit structure,
• Quantization of energy levels, and
• Wave-particle duality through de Broglie's theory.

Understanding atomic physics lays the groundwork for many modern technologies such as lasers, quantum computing, and nuclear energy. At its heart, atomic physics delves into fundamental questions about matter and energy, shaping our understanding of the universe on the smallest scales.