Problem 79 - College Physics

Question

In the Bohr model of the hydrogen atom, a single electron revolves around a single proton in a circle of radius \(r .\) Assume that the proton remains at rest. (a) By equating the electric force to the electron mass times its acceleration, derive an expression for the electron's speed. (b) Obtain an expression for the electron's kinetic energy, and show that its magnitude is just half that of the electric potential energy. (c) Obtain an expression for the total energy, and evaluate it using \(r=\) \(5.29 \times 10^{-11} \mathrm{m} .\) Give your numerical result in joules and in electron volts.

Step-by-Step Solution

Answer

1.\( v = \sqrt{\frac{k e^2}{mr}} \). 2.K = \( \frac{k e^2}{2r} \). 3.E = -\( \frac{k e^2}{2r} \), E ≈ -13.6 eV.

Step 1: Understand Electric Force and Electron Motion

In the Bohr model, the centripetal force that keeps the electron in a circular orbit is provided by the electric force. The electric force is given by Coulomb's Law: \[ F = \frac{k \cdot e^2}{r^2} \] where \( k \) is Coulomb's constant \( 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \), and \( e \) is the elementary charge \( 1.6 \times 10^{-19} \text{ C} \). This force equals the centripetal force required to keep the electron in circular motion, given by \( m v^2/r \), where \( m \) is the electron mass.

Step 2: Derive Expression for Electron's Speed

Equate the electric force to the centripetal force:\[ \frac{k \cdot e^2}{r^2} = \frac{m v^2}{r} \] Solve for velocity \( v \):\[ v = \sqrt{\frac{k \cdot e^2}{m \cdot r}} \]Here, use \( m = 9.11 \times 10^{-31} \text{ kg} \) for the electron's mass.

Step 3: Determine Expression for Kinetic Energy

The kinetic energy \( K \) of the electron is given by:\[ K = \frac{1}{2} m v^2 \] Substitute \( v^2 \) from the expression in the previous step:\[ K = \frac{1}{2} m \left( \frac{k \cdot e^2}{m \cdot r} \right) = \frac{k \cdot e^2}{2r} \] This shows it is half the electric potential energy magnitude.

Step 4: Relate Kinetic and Potential Energy

The electric potential energy \( U \) is given by:\[ U = -\frac{k \cdot e^2}{r} \] This confirms that \( |K| = \frac{1}{2} |U| \), consistent with the Bohr model.

Step 5: Find the Total Energy

The total energy \( E \) is the sum of kinetic and potential energies:\[ E = K + U = \frac{k \cdot e^2}{2r} - \frac{k \cdot e^2}{r} = -\frac{k \cdot e^2}{2r} \]

Step 6: Evaluate Total Energy Numerically

Given \( r = 5.29 \times 10^{-11} \text{ m} \), substitute in the expression:\[ E = -\frac{(8.99 \times 10^9)(1.6 \times 10^{-19})^2}{2 \times 5.29 \times 10^{-11}} \]This calculates to \( E = -2.18 \times 10^{-18} \text{ Joules} \).In electron volts (1 eV = \( 1.6 \times 10^{-19} \text{ J} \)), the energy is \[ E = \frac{-2.18 \times 10^{-18}}{1.6 \times 10^{-19}} \approx -13.6 \text{ eV} \].

Key Concepts

Electric Force

Electric force plays a crucial role in the Bohr model of the hydrogen atom. This model considers a single electron revolving around a stationary proton. The force keeping the electron in its circular orbit is the electric force, governed by Coulomb's Law. This law states that the electric force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. It is expressed as:\[ F = \frac{k \cdot e^2}{r^2} \]where:

• \( k \) is Coulomb's constant, valued at \( 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \),
• \( e \) is the elementary charge, \( 1.6 \times 10^{-19} \text{ C} \),
• \( r \) is the radius of the circle.

This electric force acts as the centripetal force, needed to keep the electron in motion, and it can be expressed as \( m v^2/r \), where \( m \) is the electron's mass and \( v \) is its velocity. By equating the electric and centripetal forces, one can derive the speed of the electron.

Kinetic Energy

Kinetic energy, often abbreviated as \( K \), is the energy that an object possesses due to its motion. In the context of the Bohr model, the electron spinning around the proton has kinetic energy given by the equation:\[ K = \frac{1}{2} m v^2 \]Since we have already derived an expression for the electron's speed \( v \), we substitute it into the kinetic energy formula to get:\[ K = \frac{1}{2} m \left( \frac{k \cdot e^2}{m \cdot r} \right) = \frac{k \cdot e^2}{2r} \]This expression shows that the electron's kinetic energy is half of the magnitude of its electric potential energy. This relationship is fundamental in understanding the dynamics of the atom's energy states in the Bohr model.

Potential Energy

Potential energy in the Bohr model arises due to the position of the electron relative to the proton. For an electron orbiting a proton, the electric potential energy \( U \) is negative, indicating an attractive interaction. It is calculated as:\[ U = -\frac{k \cdot e^2}{r} \]The negative sign indicates that work would be required to separate the electron from the proton completely. A key insight from this formula is that the magnitude of the potential energy is twice that of the kinetic energy, as seen by substituting the expression for kinetic energy. This relationship helps explain the stability of the atom, with electrons in stable orbits having a very specific energy configuration.

Total Energy

The total energy \( E \) of the electron-proton system in the Bohr model is a vital concept that helps us understand how energy levels work. It is the sum of kinetic and potential energies and is given by:\[ E = K + U \]Substituting the known expressions:\[ E = \frac{k \cdot e^2}{2r} - \frac{k \cdot e^2}{r} = -\frac{k \cdot e^2}{2r} \]This result shows that the total energy is negative, a typical trait for bound systems, indicating that the electron is bound to the proton. Using the specific radius \( r = 5.29 \times 10^{-11} \text{ m} \), the total energy evaluates to \( -2.18 \times 10^{-18} \text{ Joules} \). In electron volts, another common unit for atomic energies, this corresponds to approximately \( -13.6 \text{ eV} \). This particular energy value is significant in physics because it reflects the most stable energy state (ground state) of the hydrogen atom in the Bohr model.