Problem 56 - University Physics with Modern 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

The electron's speed is \( v = \sqrt{\frac{k e^2}{m r}} \), kinetic energy \( K = \frac{k e^2}{2 r} \), and total energy \( E = -\frac{k e^2}{2 r} = -2.18 \times 10^{-18} \) J or -13.6 eV.

Step 1: Understanding the Electric Force

The electric force between the proton and the electron in the Bohr model is given by Coulomb's law. This force can be expressed as: \[ F = \frac{k \, e^2}{r^2} \]where \( k \) is Coulomb's constant \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \), and \( e \) is the charge of the electron \( 1.6 \times 10^{-19} \, \text{C} \).

Step 2: Equating Forces to Find Speed

The acceleration of the electron moving in a circular orbit is given by \( a = \frac{v^2}{r} \). Using Newton's second law, the force is also defined as mass times acceleration: \[ F = m \cdot a = m \cdot \frac{v^2}{r} \]Equating the expressions from Coulomb's force and Newton's second law, we have:\[ \frac{k \, e^2}{r^2} = m \cdot \frac{v^2}{r} \]Solving for \( v \), the speed of the electron, we obtain:\[ v = \sqrt{\frac{k \, e^2}{m \, r}} \]where \( m \) is the mass of the electron \( 9.11 \times 10^{-31} \, \text{kg} \).

Step 3: Calculating Electron's Kinetic Energy

The kinetic energy (KE) of the electron can be expressed using the formula:\[ \text{KE} = \frac{1}{2} m v^2 \]Substitute the expression we obtained for \( v \) into the kinetic energy formula:\[ \text{KE} = \frac{1}{2} m \left( \frac{k \, e^2}{m \, r} \right) = \frac{k \, e^2}{2 \, r} \]

Step 4: Finding the Electric Potential Energy

The electric potential energy (U) of the system is given by:\[ U = -\frac{k \, e^2}{r} \]The negative sign indicates the binding nature of the energy.

Step 5: Verify Kinetic Energy is Half of Potential

Compare the magnitudes of KE and U:\[ \text{KE} = \frac{k \, e^2}{2 \, r}, \quad |U| = \frac{k \, e^2}{r} \]Thus, the magnitude of the kinetic energy is indeed half that of the electric potential energy.

Step 6: Calculate Total Energy

The total energy (E) of the electron in the Bohr model is the sum of its kinetic and potential energies:\[ E = \text{KE} + U = \frac{k \, e^2}{2 \, r} - \frac{k \, e^2}{r} \]By simplifying, the total energy becomes:\[ E = -\frac{k \, e^2}{2 \, r} \]

Step 7: Evaluate Total Energy for Given Radius

Substitute \( r = 5.29 \times 10^{-11} \, \text{m} \) into the expression for \( E \):\[ E = -\frac{8.99 \times 10^9 \, \times (1.6 \times 10^{-19})^2}{2 \times 5.29 \times 10^{-11}} \approx -2.18 \times 10^{-18} \, \text{J} \]

Step 8: Convert Total Energy to Electron Volts

To convert the energy from joules to electron volts, use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):\[ E \approx \frac{-2.18 \times 10^{-18}}{1.6 \times 10^{-19}} \approx -13.6 \, \text{eV} \]

Key Concepts

Electric Force

When examining the Bohr model of the hydrogen atom, the electric force plays a crucial role. This force is the attraction between the positively charged proton and the negatively charged electron. It is governed by Coulomb's law, which mathematically describes the electric force as: \[ F = \frac{k \, e^2}{r^2} \] where:

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

The force is directed along the line joining the electron and proton and provides the necessary centripetal force for the electron's circular motion around the proton.
By substituting these values into the formula, we quantify the strength of the attraction necessary for the electron's stable orbit in the Bohr model.

Electron Kinetic Energy

In the Bohr model, the electron moves in a circular path with constant speed. The kinetic energy (KE) of the electron is calculated using the formula:\[ \text{KE} = \frac{1}{2}m v^2 \]where:

• \( m \) is the mass of the electron, approximately \( 9.11 \times 10^{-31} \, \text{kg} \).
• \( v \) is the speed of the electron.

Using the expression derived for electron speed \( v = \sqrt{\frac{k \, e^2}{m \, r}} \), we substitute into the kinetic energy formula and obtain:\[ \text{KE} = \frac{k \, e^2}{2 \, r} \]One vital observation in the Bohr model is that the magnitude of kinetic energy is half that of the magnitude of electric potential energy. This relationship highlights the balance of the forces that maintain the stability of the electron's orbit.

Electric Potential Energy

Electric potential energy (U) in the Bohr model arises from the interaction between the electron and the proton. This energy is given by the formula:\[ U = -\frac{k \, e^2}{r} \]The negative sign is particularly important here, indicating that the energy is binding. This means that energy is released as the electron comes near the proton, effectively binding them together. The potential energy reflects the considerable force of attraction, keeping the electron in its orbit.
The magnitude of the electric potential energy illustrates the energy required to move the electron infinitely far from the proton, thus requiring input to overcome the attractive force.

Total Energy in Bohr Model

The total energy (E) of an electron in the Bohr model is the sum of its kinetic energy (KE) and its electric potential energy (U). Mathematically, it is expressed as:\[ E = \text{KE} + U = \frac{k \, e^2}{2 \, r} - \frac{k \, e^2}{r} \]Upon simplifying, this reduces to:\[ E = -\frac{k \, e^2}{2 \, r} \]This result indicates that the total energy in the Bohr model is negative. The negative total energy signifies that the electron is bound to the nucleus, meaning energy must be supplied to free the electron from the atom.
For example, inserting a specific orbital radius \( r = 5.29 \times 10^{-11} \, \text{m} \), we find the total energy \( E \approx -2.18 \times 10^{-18} \, \text{J} \). Converting this to electron volts using the conversion \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \), the energy is approximately \(-13.6 \, \text{eV}\), matching the observed energies for hydrogen atom transitions.