Problem 23 - College Physics

Question

\(\bullet\) Use the Bohr model for the following calculations: (a) What is the speed of the electron in a hydrogen atom in the \(n=1,2\) and 3 levels? (b) Calculate the radii of each of these levels. (c) Find the total energy (in eV) of the atom in each of these levels.

Step-by-Step Solution

Answer

Speed at \(n=1,2,3\): \(2.18, 1.09, 0.727\times 10^6\text{ m/s}\). Radii: \(0.529, 2.116, 4.761\times 10^{-10}\text{ m}\). Energies: \(-13.6, -3.4, -1.51\text{ eV}\).

Step 1: Understand Bohr's Model

Bohr's model postulates that electrons orbit the nucleus at discrete energy levels without radiating energy. Each energy level, characterized by the principal quantum number \(n\), has a specific radius and energy associated with it. The electron's speed also changes with the energy level.

Step 2: Use Bohr's Formula for Speed Calculation

The speed of an electron in Bohr's model is given by \(v_n = \dfrac{e^2}{2\epsilon_0 h} \cdot \dfrac{1}{n}\). We use this formula to find the speed for \(n = 1, 2,\) and \(3\). For hydrogen, \(v_1 = 2.18 \times 10^6 m/s\), \(v_2 = \dfrac{2.18 \times 10^6}{2} = 1.09 \times 10^6 m/s\), and \(v_3 = \dfrac{2.18 \times 10^6}{3} = 7.27 \times 10^5 m/s\).

Step 3: Determine the Radius using Bohr's Radius Formula

The radius of the electron's orbit is given by \(r_n = n^2 a_0\), where \(a_0 = 0.529 \times 10^{-10} m\) (Bohr radius). Therefore, \(r_1 = 1^2 \times 0.529 \times 10^{-10} m = 0.529 \times 10^{-10} m\), \(r_2 = 4 \times 0.529 \times 10^{-10} m = 2.116 \times 10^{-10} m\), and \(r_3 = 9 \times 0.529 \times 10^{-10} m = 4.761 \times 10^{-10} m\).

Step 4: Calculate Total Energy in Each Level

The total energy for an electron in Bohr's model is given by \(E_n = -rac{13.6}{n^2} \, ext{eV}\). For \(n = 1\), \(E_1 = -13.6 \, ext{eV}\). For \(n = 2\), \(E_2 = -rac{13.6}{4} = -3.4 \, ext{eV}\). For \(n = 3\), \(E_3 = -rac{13.6}{9} = -1.51 \, ext{eV}\).

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest atom, consisting of only one proton in its nucleus and a single electron orbiting around it. This simple structure makes it an ideal candidate for studying atomic theory. In the Bohr model, this electron moves in defined circular orbits, which are quantized, meaning they have specific allowable radii and energy values. The Bohr model simplifies the complex nature of atoms by assuming electrons move in precise paths instead of the more unpredictable distributions described by quantum mechanics.
Understanding how electrons behave in a hydrogen atom allows us to predict several important properties such as electron speed, orbit radius, and energy levels. These properties then help us understand how atoms interact with light and other particles, which is crucial in fields such as chemistry and materials science.
The Bohr model, though limited to hydrogen-like atoms, provides a crucial framework to describe atomic structure and phenomena like spectral lines in a relatively straightforward manner.

Energy Levels

In the Bohr model, energy levels are crucial for understanding an atom's structure. Each level is denoted by a principal quantum number, \( n \). These numbers represent different energy states an electron can inhabit. The lowest energy level, \( n = 1 \), is the closest to the nucleus and is called the "ground state." Higher energy levels (\( n = 2, 3, \ldots \)) are termed as "excited states."

• The energy of an electron in the hydrogen atom is negative, indicating it is bound to the atom. The energy becomes less negative (i.e., increases) as \( n \) increases, showing that the electron is less tightly bound with the atom.
• For example, the energy for \( n = 1 \) is \(-13.6 \text{ eV}\), \( n = 2 \) is \(-3.4 \text{ eV}\), and \( n = 3 \) is \(-1.51 \text{ eV}\).

The change in energy levels allows the emission or absorption of photons, which contributes to an atom's spectral lines, thereby giving rise to its unique spectral fingerprint.

Electron Speed

Electron speed in the Bohr model refers to how fast an electron moves along its orbit at a certain energy level. In hydrogen, the speed is not constant but depends on the principal quantum number, \( n \).
The formula for the electron speed in a specific orbit is \( v_n = \dfrac{e^2}{2\epsilon_0 h} \cdot \dfrac{1}{n} \). Here, \( e \) is the electron charge, \( \epsilon_0 \) is the vacuum permittivity, and \( h \) is Planck's constant.

• For \( n = 1 \), the electron speed is approximately \( 2.18 \times 10^6 \text{ m/s} \).
• For \( n = 2 \), it decreases to \( 1.09 \times 10^6 \text{ m/s} \).
• For \( n = 3 \), it further reduces to \( 7.27 \times 10^5 \text{ m/s} \).

This means as the electron moves to higher orbits, it slows down. This inverse relationship between speed and quantum number reflects how electrons transition between orbits, involving the absorption or emission of energy.

Orbit Radius

The orbit radius in the Bohr model determines the size of an electron's path around the nucleus. It's given by the formula \( r_n = n^2 a_0 \), where \( a_0 \) is the Bohr radius about \( 0.529 \times 10^{-10} \text{ m} \).
Here’s how the formula plays out for different energy levels:

• At \( n = 1 \), the orbit radius is \( 0.529 \times 10^{-10} \text{ m} \), considered the smallest orbit in the hydrogen atom.
• At \( n = 2 \), the radius increases to \( 2.116 \times 10^{-10} \text{ m} \), demonstrating the larger path required for higher energy states.
• For \( n = 3 \), the radius further grows to \( 4.761 \times 10^{-10} \text{ m} \).

This increase in radius with higher energy levels reflects the electron's lesser binding to the nucleus and allows larger orbits for higher energy states. The larger the orbit radius, the less energy is required to remove the electron from the atom, which is crucial for understanding atomic reactivity and bonding.