Problem 14 - Chemistry The Central Science In Si Units

Question

(a) What is the electrostatic potential energy (in joules) between two electrons that are separated by \(460 \mathrm{pm} ?\) (b) What is the change in potential energy if the distance separating the two electrons is increased to \(1,0 \mathrm{nm} ?\) (c) Does the potential energy of the two particles increase or decrease when the distance is increased to \(1.0 \mathrm{nm} ?\)

Step-by-Step Solution

Answer

(a) Calculate using the formula for potential energy. (b) The change in potential energy is positive. (c) Potential energy increases with increased distance.

Step 1: Understanding Electrostatic Potential Energy

The electrostatic potential energy between two point charges is given by the formula: \[ U = \frac{k \cdot q_1 \cdot q_2}{r} \] where \( U \) is the potential energy, \( k = 8.99 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2 \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges (for electrons, \( q = -1.60 \times 10^{-19} \mathrm{C} \)), and \( r \) is the distance between the charges.

Step 1: Calculate Potential Energy at 460 pm

Convert the distance from picometers to meters: \(460 \mathrm{pm} = 460 \times 10^{-12} \mathrm{m}\). Substitute the values into the formula: \[U = \frac{(8.99 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2) \cdot (-1.60 \times 10^{-19} \mathrm{C})^2}{460 \times 10^{-12} \mathrm{m}}\] Calculate to find \( U \).

Step 2: Convert to Nano-meters and Calculate New Potential Energy

Convert the distance from nanometers to meters: \(1.0 \mathrm{nm} = 1.0 \times 10^{-9} \mathrm{m}\). Substitute back into the formula with the new distance: \[U' = \frac{(8.99 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2) \cdot (-1.60 \times 10^{-19} \mathrm{C})^2}{1.0 \times 10^{-9} \mathrm{m}}\] Calculate to find the new \( U' \).

Step 3: Calculate Change in Potential Energy

Find the change in potential energy: \( \Delta U = U' - U \). Substitute the values found in Steps 1 and 2 and calculate \( \Delta U \).

Step 5: Conclusion: Potential Energy Behavior with Increasing Distance

The potential energy increases when two like charges are moved further apart. In this case, because the charges are both negative, moving them from \(460 \mathrm{pm}\) to \(1.0 \mathrm{nm}\) leads to an increase in potential energy as \( \Delta U \) will be positive.

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic interaction between two charged particles. It's an essential principle in understanding electric forces in quantum mechanics and classical physics. The law states that the magnitude of the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance separating them. This relationship is mathematically expressed as:

\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]
where:

• \( F \) is the force between the charges,
• \( k = 8.99 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2 \) is Coulomb's constant,
• \( q_1 \) and \( q_2 \) are the point charges,
• \( r \) is the distance between the charges.

Coulomb's Law helps us understand that like charges repel each other, and opposite charges attract. This principle is foundational for calculating forces and understanding how charges interact in electric fields.

Point Charges

In the study of Electrostatic Potential Energy, point charges are considered to be infinitesimally small, with charge concentrated at a single point in space. This simplification allows for easier computation of forces and potential energies without worrying about the physical size or shape of the objects involved.

Point charges are crucial in both physics and quantum mechanics because:

• They simplify the mathematical treatment of charge interactions, making complex problems manageable.
• They help in understanding interactions at atomic and subatomic levels, where standard geometric bodies are not applicable.
• Applications include modeling atoms, electron clouds, and understanding behaviors in electric fields.

For example, when calculating the potential energy between two electrons as in the exercise, they're considered point charges given their negligible size relative to the distance separating them. This approach facilitates the precise application of Coulomb's Law to derive electrostatic energy.

Potential Energy Change

The concept of Potential Energy Change is pivotal when discussing the interactions between charges. Potential energy can be thought of as stored energy that is available to do work at some point in the future. For two point charges, it gives insight into how they influence each other as they move closer or further apart.

The potential energy (\( U \)) between two charges changes with distance as described by:

\[ U = \frac{k \cdot q_1 \cdot q_2}{r} \]
When two like charges are moved apart, the potential energy increases. This behavior is explained by the fact that work must be done against the attractive or repulsive electrostatic forces to change their relative positions. Conversely, when they are brought closer, the potential energy reduces. In the exercise provided, moving electrons (both negative like charges) from 460 pm to 1.0 nm results in an increase in potential energy, as expected from theory.

Quantum Mechanics

Quantum Mechanics is the branch of physics that studies the behavior and interactions of particles at the atomic and subatomic levels. It provides a framework for understanding and predicting the behavior of systems that might not align with classical physics, especially in describing the behavior of point charges.

In the realm of quantum mechanics:

• Electrons, typically modeled as point charges, are described by wave functions, encapsulating probabilities rather than definite locations.
• Quantum mechanics helps explain how electrons form bonds, transition energy states, and interact via forces such as those described by Coulomb's Law.
• It opens the door to phenomena like quantum tunneling and superposition, which have no classical analogs.

Understanding the principles of quantum mechanics allows us to comprehend the calculations involving electrostatic potential energy, as the exercise mentions electrons interacting over minute distances. This quantum perspective is key to applications in advanced technology, energy solutions, and understanding fundamental physics.