Problem 54 - University Physics with Modern Physics

Question

In the Rohr 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

(a) \( v = \sqrt{\frac{k_e e^2}{m_e r}} \). (b) \( KE = \frac{1}{2} |U| \). (c) Total energy is -2.18x10^{-18} J or -13.6 eV.

Step 1: Understanding Electric Force

The electric force between a proton and an electron is given by Coulomb's law: \( F = \frac{k_e e^2}{r^2} \), where \( k_e \) is the Coulomb's constant and \( e \) is the elementary charge.

Step 2: Equate Electric Force to Centripetal Force

For the electron in circular motion, the centripetal force \( F_c \) is given by \( F_c = \frac{m_ev^2}{r} \). Set this equal to the electric force: \[ \frac{m_ev^2}{r} = \frac{k_e e^2}{r^2} \].

Step 3: Solve for the Electron's Speed

Solve the equation from Step 2 for \( v \):\[ m_ev^2 = \frac{k_e e^2}{r} \]. Thus, \( v = \sqrt{\frac{k_e e^2}{m_e r}} \).

Step 4: Calculate the Electron's Kinetic Energy

The kinetic energy \( KE \) of the electron is given by \( KE = \frac{1}{2}m_ev^2 \). Substitute \( v^2 = \frac{k_e e^2}{m_e r} \) from Step 3: \[ KE = \frac{1}{2} \left( \frac{k_e e^2}{r} \right) \].

Step 5: Calculate the Electric Potential Energy

The electric potential energy \( U \) between the proton and electron is \( U = -\frac{k_e e^2}{r} \).

Step 6: Show KE is Half of the Magnitude of U

Given \( KE = \frac{1}{2} \left( \frac{k_e e^2}{r} \right) \) and \( U = -\frac{k_e e^2}{r} \), the magnitude of the potential energy is \( \frac{k_e e^2}{r} \), hence \( KE = \frac{1}{2} |U| \).

Step 7: Calculate the Total Energy

The total energy \( E \) is the sum of the kinetic energy and the potential energy: \( E = KE + U = \frac{1}{2} \left( \frac{k_e e^2}{r} \right) - \frac{k_e e^2}{r} = -\frac{1}{2} \left( \frac{k_e e^2}{r} \right) \).

Step 8: Evaluate Total Energy Numerically

Substitute \( r = 5.29 \times 10^{-11} \text{m} \), \( e = 1.6 \times 10^{-19} \text{C} \), and \( k_e = 8.988 \times 10^9 \text{N m}^2/\text{C}^2 \) into the formula for total energy: \[ E = -\frac{1}{2} \left( \frac{8.988 \times 10^9\cdot(1.6 \times 10^{-19})^2}{5.29 \times 10^{-11}} \right) = -2.18 \times 10^{-18} \text{J} \].

Step 9: Convert Total Energy to Electron Volts

Use the conversion \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{J} \). Thus, \[ E = \frac{-2.18 \times 10^{-18}}{1.6 \times 10^{-19}} = -13.6 \text{ eV} \].

Key Concepts

Coulomb's Law

Coulomb's Law is a foundational principle that describes the force between two charged particles. It states that the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Mathematically, for a proton and an electron, this force is given by the equation:

• \( F = \frac{k_e e^2}{r^2} \)

The constant \( k_e \) is known as Coulomb's constant, valued at approximately \( 8.988 \times 10^9 \ \text{N m}^2/\text{C}^2 \).
The charges \( e \) of the proton and electron are equal and opposite, each having a magnitude of approximately \( 1.6 \times 10^{-19} \ \text{C} \).
Understanding Coulomb’s Law is crucial in explaining how a singular electron orbits a proton in an atom, such as in the Hydrogen atom. This force acts as the centripetal force needed for the electron's circular motion.

Kinetic Energy

Kinetic Energy (KE) is the energy that a body possesses by virtue of being in motion. For the electron moving around the proton in a circular path, we apply the kinetic energy formula:\( KE = \frac{1}{2}m_ev^2 \).
Here, \( m_e \) is the mass of the electron, and \( v \) is its speed.
In this scenario, the kinetic energy stems from the electric force which acts as the centripetal force, keeping the electron in a circular orbit.
After finding the speed \( v \) using Coulomb's law (as derived in the solution), this expression for the electron's kinetic energy becomes :

• \( KE = \frac{1}{2} \left( \frac{k_e e^2}{r} \right) \).

This shows the dependency of kinetic energy on the radius \( r \) of the electron’s orbit around the nucleus.

Electric Potential Energy

Electric Potential Energy (U) refers to the energy stored within a system due to the positioning of charged particles. In our hydrogen atom model, this energy comes from the interaction between the proton and the electron. The formula for this energy is:

• \( U = -\frac{k_e e^2}{r} \)

The negative sign signifies that the electron is bound to the proton — a lower energy state compared to when they are far apart.
As the electron gets closer to the proton, its electric potential energy becomes more negative.
The potential energy helps us understand the balance of energies in the atom, as it directly relates to kinetic energy to determine the total energy of the electron.

Total Energy

The Total Energy (E) of the electron in an atom results from the sum of its kinetic energy and electric potential energy. This is a critical concept in atomic physics, showing the balance of energies within the atom.
The equation for total energy is:

• \( E = KE + U = \frac{1}{2} \left( \frac{k_e e^2}{r} \right) - \frac{k_e e^2}{r} \)

Which simplifies to:

• \( E = -\frac{1}{2} \left( \frac{k_e e^2}{r} \right) \)

This expression reveals that the total energy is negative, indicating that the electron is in a bound state.
Even when absolute values aren't favorable, the negative sign shows stability in the atom.
It's essential to understand that the total energy gives insight into the electron's binding level, depth, and stability in its orbit.

Hydrogen Atom

A Hydrogen Atom is the simplest and most fundamental atom, consisting of one proton and one electron. This simplicity makes it an ideal model for studying atomic structures and energy interactions.
In a Hydrogen atom, the electron forms a circular path around the proton, helped by electric forces, as described in Coulomb's law.
This simple structure is crucial for understanding more complex atoms and molecules since many principles of quantum mechanics and atomic interaction can be deduced from it. Further, the Bohr Model of the hydrogen atom provides foundational insight into how electrons orbit and how energy levels are quantized.
Through this, we see that higher energies correspond to electrons at greater distances from the nucleus, while lower energies signify tighter, more stable orbits.

Electron Speed

The speed of an electron in its orbit around a nucleus is a function of the forces acting upon it. For the hydrogen atom, using Coulomb's Law and the concept of centripetal force, we derive the expression for electron speed as:

• \( v = \sqrt{\frac{k_e e^2}{m_e r}} \)

This formula highlights how the electron's speed depends on the constants of nature and the radius of orbit.
As the radius \( r \) changes, the speed of the electron will adjust accordingly, demonstrating a dynamic balance between electric force and the resulting motion.
Understanding electron speed is important for interpreting the energy levels within an atom and how electrons transition between these levels.