Question

According to Bohr, the electron in the hydrogen atom moved around the nucleus in circular paths called

Short answer

The angular momentum of the electron in the n = 3 orbit of a hydrogen atom is \(3.163715454 \times 10^{-34}\) Js.

Step-by-step solution

Analyze the given information

We are given the following information: - Principal quantum number (n) = 3 - Reduced Planck constant (ħ) = h/(2π), where h is Planck's constant (6.62607015 x 10^-34 Js)

Recall the formula for the angular momentum of an electron

According to Bohr's theory, the angular momentum (L) of the electron in a hydrogen atom is given by the equation: L = nħ where - L is the angular momentum - n is the principal quantum number - ħ is the reduced Planck constant

Substitute the given values into the formula

Now, we can plug the given values for n and ħ into the equation: L = (3)(h/(2π))

Calculate the angular momentum

We can now calculate the angular momentum of the electron: L = (3)(6.62607015 x 10^-34 Js / (2π)) L = (3)(1.054571818 x 10^-34 Js) L = 3.163715454 x 10^-34 Js

State the final answer

The angular momentum of the electron in the n = 3 orbit of a hydrogen atom is 3.163715454 x 10^-34 Js.

Key concepts

Angular Momentum of an Electron

The concept of angular momentum is a cornerstone in the understanding of atomic structure, particularly within the context of Bohr's theory.

In classical mechanics, angular momentum is a measure of the tendency of a rotating object to continue spinning. For an electron in an atom, it can be visualized as the momentum it carries while orbiting the nucleus. According to Niels Bohr, the electrons move in specific circular paths or orbits with quantized angular momentum. This quantization means that rather than having any value, the angular momentum of an electron can only be specific discrete amounts.

Bohr's formula, , expresses the angular momentum of an electron moving in one of these orbits. Here, is the principal quantum number, representing the specific orbit the electron occupies, and ħ represents the reduced Planck's constant, a fundamental constant in quantum mechanics. This formula implies that the angular momentum changes with the principal quantum number, getting larger as the electron moves to orbits further from the nucleus.

Principal Quantum Number

The principal quantum number, denoted by , plays a pivotal role in determining the properties of electrons in atoms. Think of it as an address for electrons, indicating which shell or energy level they occupy.

In Bohr's atomic model, the value of can be any whole number starting from 1, increasing in integer steps as you move outwards from the nucleus. The larger the value of , the greater the size of the orbit and the higher the energy level of the electron. For instance, when = 1, the electron is in the closest orbit to the nucleus, known as the ground state.

With each step increase in the principal quantum number, the electron's energy rises, and its orbit becomes larger. Consequently, when the electron transits to a higher , it absorbs energy, and when it falls back to a lower , it releases energy, typically in the form of light. These transitions are responsible for the spectral lines seen in atomic emission and absorption spectra.

Planck's Constant

Lastly, let's delve into the significance of Planck's constant, symbolized as . This is a fundamental constant in quantum mechanics, with a value of approximately <6.62607015 x 10^-34 Js>. It was first introduced by Max Planck and is essential for capturing the quantized nature of energy in the microscopic world.

Why is Planck's constant so revolutionary? It is the proportionality constant that relates the energy (E) of a photon to the frequency () of its electromagnetic wave, expressed in the equation . Here, Planck's constant bridges the gap between the discrete particles and the wave-like behavior of light and matter — a duality that is central to quantum theory.

In Bohr's model of the atom, the reduced Planck's constant <ħ = h / (2π)> is used, which tailors the calculations to fit the angular nature of an electron's orbit. It effectively shows how the quantum world is quantized, including the angular momentum of an electron. Without it, we couldn't calculate the discrete orbital levels that electrons inhabit.