Problem 61 - University Physics with Modern Physics

Question

The \(\mathrm{H}_{2}^{+}\) Ion. The \(\mathrm{H}_{2}^{+}\) ion is composed of two protons, each of charge \(+e=1.60 \times 10^{-19} \mathrm{C},\) and an electron of charge \(-e\) and mass \(9.11 \times 10^{-31} \mathrm{kg} .\) The separation between the protons is \(1.07 \times 10^{-10} \mathrm{m} .\) The protons and the electron may be treated as point charges. (a) Suppose the electron is located at the point midway between the two protons. What is the potential energy of the interaction between the electron and the two protons? (Do not include the potential energy due to the interaction between the two protons.) (b) Suppose the electron in part (a) has a velocity of magnitude \(1.50 \times 10^{6} \mathrm{m} / \mathrm{s}\) in a direction along the perpendicular bisector of the line connecting the two protons. How far from the point midway between the two protons can the electron move? Because the masses of the protons are much greater than the electron mass, the motions of the protons are very slow and can be ignored. (Note: A realistic description of the electron motion requires the use of quantum mechanics, not Newtonian mechanics.)

Step-by-Step Solution

Answer

Potential energy: \(-4.28 \times 10^{-18} \, \mathrm{J}\). Max distance: \(1.07 \times 10^{-10} \, \mathrm{m}\).

Step 1: Understand the Configuration

The configuration consists of two protons and an electron placed along a line. The protons are at positions separated by a distance of \(1.07 \times 10^{-10} \mathrm{m}\). The electron is initially placed equidistant from both protons.

Step 2: Compute Potential Energy of Electron between Protons

The electric potential energy of a charge \(q\) in an electric field due to a charge \(Q\) separated by distance \(r\) is given by \(U = \frac{k_e Q q}{r}\), where \(k_e = 8.9875 \times 10^9 \, \mathrm{N\,m^2/C^2}\). Calculate the contribution to the potential energy from one proton: \[ U_1 = \frac{k_e (+e)(-e)}{0.535 \times 10^{-10}} \]\[ e = 1.60 \times 10^{-19} \, \mathrm{C} \text{ and } \frac{1.07}{2} = 0.535 \, \times 10^{-10} \, \mathrm{m} \].Calculate the total potential energy by adding up contributions from both protons since the electron is midway:\[ U_{total} = 2 \times U_1 \].

Step 3: Solve for Potential Energy Value

Substitute the values into the expression:\[ U_1 = \frac{(8.9875 \times 10^9) (1.60 \times 10^{-19})^2}{0.535 \times 10^{-10}} \]Calculate the total potential energy:\[ U_{total} = 2 \times U_1 \approx -4.28 \times 10^{-18} \, \mathrm{J} \].

Step 4: Calculate Kinetic Energy of the Electron

The electron has initial kinetic energy due to its velocity:\[ K = \frac{1}{2} m v^2 \]Substitute the given values:\[ m = 9.11 \times 10^{-31} \, \mathrm{kg} \text{ and } v = 1.50 \times 10^{6} \, \mathrm{m/s} \]\[ K = \frac{1}{2} (9.11 \times 10^{-31}) (1.50 \times 10^{6})^2 \approx 1.02 \times 10^{-18} \, \mathrm{J} \].

Step 5: Equate Total Energy at Start and Far Point

The total energy (kinetic plus potential) must be conserved as the electron moves:Initially: \[ E_{initial} = U_{total} + K \].At furthest point, kinetic energy is zero, so:\[ E_{final} = U_{final} \].Set \(E_{initial} = E_{final}\) since the energy is conserved:\[ -4.28 \times 10^{-18} + 1.02 \times 10^{-18} = U_{final} \].Find \(U_{final}\).

Step 6: Calculate Maximum Distance Using Potential Energy

The potential energy when at the max distance \(r'\) is:\[ U_{final} = U_1' + U_1' \] (due to symmetry and similar configuration as initial but different distances)\[ 2\frac{k_e (-e)(+e)}{r'} = -3.26 \times 10^{-18} \, \mathrm{J} \].Solve for \(r'\):\[ r' = \frac{2 \times k_e (e^2)}{-3.26 \times 10^{-18}} \].

Step 7: Solve for r' Expression

Substitute known values into the equation:\[ r' = \frac{2 (8.9875 \times 10^9)(1.60 \times 10^{-19})^2}{3.26 \times 10^{-18}} \approx 1.07 \times 10^{-10} \, \mathrm{m} \].Thus, the electron can reach a maximum distance of approximately \(1.07 \times 10^{-10} \, \mathrm{m}\) from the midpoint between the protons.

Key Concepts

Electron Dynamics

In the realm of quantum mechanics, "electron dynamics" refers to how electrons behave and move. This movement can be influenced by forces acting on the electron, which, in this scenario, is by electric forces generated by the protons of the \(\mathrm{H}_2^+\) ion. When examining an electron's dynamics, it's essential to consider both its kinetic energy, resulting from its velocity, and the electric forces it experiences.
For the \(\mathrm{H}_2^+\) ion, the electron starts out at the midpoint between two protons. It is subject to the attractive forces from both positively charged protons. As it moves due to its initial velocity, these forces dictate how far and where it can travel. The calculated initial kinetic energy helps predict how the electron will behave as it relates to energy conservation.
Understanding electron dynamics involves:

• Calculating the forces on the electron.
• Deriving the electron's trajectory.
• Balancing kinetic energy with potential energy changes.

This understanding is crucial for predicting the electron's path and how far it can move from the original position.

Potential Energy

Potential energy in this context comes from the electrostatic interaction between the electron and protons. This interaction is a key part of the problem. The potential energy between two point charges (proton and electron) can be calculated using the formula \(U = \frac{k_e Q q}{r}\), where \(k_e\) is Coulomb's constant, \(Q\) and \(q\) are the charges, and \(r\) is the separation distance.
Here, since the electron is equidistant from both protons in the \(\mathrm{H}_2^+\) ion, each electron-proton pair contributes equally to the total potential energy. This total potential energy is calculated by multiplying the single contribution by two as the electron interacts symmetrically with both protons. These interactions result in a negative potential energy value, indicating an attractive interaction.
The concept of potential energy allows us to understand:

• How electrostatic forces govern the interaction between charged particles.
• Why the potential energy is negative, thus decreasing as attraction strengthens.
• How changes in potential energy affect electron movement and dynamics.

Through these electrostatic potential energy considerations, we can predict how the electron will behave under given conditions.

Conservation of Energy

The principle of conservation of energy is crucial in analyzing how the electron moves within the \(\mathrm{H}_2^+\) ion system. Essentially, the total energy of the system—composed of kinetic and potential energy—remains constant as the electron moves. This means that the initial energy calculated at the midpoint is equal to the final energy at any other point.
Initially, the electron has some kinetic energy due to its velocity and potential energy from its position between the protons. As the electron moves, its kinetic energy will change due to acceleration or deceleration caused by the electrostatic forces it experiences.
This principle helps:

• Establish the relationship between potential and kinetic energy.
• Predict maximum displacement by balancing energy forms.
• Understand the limitations and range of motion based on initial energy.

By employing the conservation of energy, we determine how far the electron can move before its kinetic energy becomes zero—marking the point at which the potential energy is at a maximum.

H2+ Ion

The \(\mathrm{H}_2^+\) ion, known as the simplest molecule ion, consists of two protons and one electron. Understanding this ion provides insight into fundamental concepts of chemistry and physics, particularly about quantum mechanics.
The significance of the \(\mathrm{H}_2^+\) ion centers around its simplicity, which makes it an ideal subject for studying electron behavior in a molecular context. In this system, you have a basic yet profound situation: one electron bound to two nuclei, shared between them.
Key aspects of the \(\mathrm{H}_2^+\) ion include:

• The two protons act as fixed positive charges, enforcing symmetrical potential energy fields.
• The single electron experiences an electrostatic force, attracted to the protons.
• Its motion and interaction are simplified models to understand complex molecular systems.

Studying the \(\mathrm{H}_2^+\) ion helps in exploring core quantum mechanical concepts like electronic wave functions, bonding, and the behavior of particles in a probability field.