Question

What must be the velocity of electrons if their associated wavelength is to equal the radius of the first Bohr orbit of the hydrogen atom?

Short answer

The velocity of the electrons, at which the associated wavelength is equal to the radius of the first Bohr orbit of the hydrogen atom, is approximately \(1.4 \times 10^{6} m/s\).

Step-by-step solution

Understand the de Broglie Wavelength

According to de Broglie, every particle with momentum has a wavelength associated with it. The de Broglie wavelength is given by λ = h / mv where: λ is the wavelength, h is the Planck's constant which equal to \(6.626 \times 10^{-34} Js\), m is the mass of the electron which equals \(9.1 \times 10^{-31} kg\), and v is the velocity of the electron.

Bohr's Radius

The radius of the first Bohr orbit (also known as Bohr radius) for hydrogen is approximately 0.529 Å. Since we want the electron's wavelength to equal the first Bohr orbit radius, we set λ = 0.529 Å. However, to be consistent with our units, we must convert this to meters (m). 1 Å = \(1 \times 10^{-10} m\), so 0.529 Å = \(0.529 \times 10^{-10} m\)

Solve for Velocity

Now that we have the de Broglie wavelength and Bohr's radius, we set them equal and solve for velocity v: \(0.529 \times 10^{-10} m = \frac{6.626 \times 10^{-34} Js}{9.1 \times 10^{-31} kg*v}\). Solving for v we get \(v = \frac{6.626 \times 10^{-34} Js}{9.1 \times 10^{-31} kg*\times 0.529 \times 10^{-10} m}\), which when simplified equals approximately \(1.4 \times 10^{6} m/s\).

Key concepts

Bohr Model

The Bohr model is a fascinating concept that changed how we understand atomic structures. Constructed by Niels Bohr in the early 20th century, this model depicts an atom as a miniature solar system. Electrons orbit the nucleus in circular orbits, just like planets around the Sun. This model was innovative because it introduced quantized orbitals, where electrons can only occupy fixed orbits of certain energy levels without radiating energy.

In the Bohr model, the first orbit, known as the "ground state," is the closest to the nucleus. The radius of this first orbit in a hydrogen atom is called the Bohr radius. It is approximately 0.529 Å, a crucial constant when solving problems involving electron transitions.

• The Bohr model simplifies atomic physics, despite its later expansion by quantum mechanics.
• It specifically applies well to hydrogen-like atoms, which contain only one electron.

While the Bohr model offers essential insights into atomic structures, it does not fully explain more complex atoms or phenomena. Quantum mechanics later provided a more comprehensive framework.

Electron Velocity

Electron velocity is a key concept necessary to understand the movement of electrons in an atom. It refers to the speed at which an electron travels along its orbital path. In the context of the Bohr model, this velocity is linked with energy levels and wavelengths.

By using the de Broglie principle, which links an electron's wave properties to its momentum, we can calculate the velocity of an electron given its de Broglie wavelength. The formula used is:\[\lambda = \frac{h}{mv}\]Where \(\lambda\) is the wavelength, \(h\) is Planck's constant, \(m\) is the electron's mass, and \(v\) is its velocity. Given the wavelength equals the Bohr radius, we can rearrange the equation to find the electron's velocity.

• Understanding electron velocity helps in exploring the behavior of electrons at different energy levels.
• This velocity can be used to solve more profound quantum mechanics problems.

Calculating electron velocity provides insights into the dynamic qualities of atomic particles.

Quantum Mechanics

Quantum mechanics is a comprehensive scientific theory that revolutionized our perception of atomic and subatomic processes. It extends upon the Bohr model by explaining phenomena that the model could not clarify, such as electron clouds and orbitals.

This theory suggests that electrons do not travel in neat planet-like orbits, but rather exist as probability clouds around the nucleus. It introduces complex principles such as wave-particle duality, where electrons exhibit both wave and particle characteristics. This duality is a fundamental concept when considering de Broglie's wavelength, as it ties into the electron's behavior when it is measured.

• Quantum mechanics provides a more accurate description of atomic processes than classical theories.
• It incorporates principles like uncertainty and superposition, challenging classical interpretations.

For students, understanding quantum mechanics is crucial as it lays the foundation for modern physics and chemistry. By seeing electrons as wave functions rather than fixed orbits, one gains a deeper insight into atomic interactions.