Problem 13 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) What is the electrostatic potential energy (in joules) between an electron and a proton that are separated by \(230 \mathrm{pm}\) ? (b) What is the change in potential energy if the distance separating the electron and proton 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) The electrostatic potential energy between the electron and proton separated by 230 pm is: \(U_1 = \dfrac{(8.99 * 10^9 N m^2/C^2)(1.60 * 10^{-19} C)(-1.60 * 10^{-19} C)}{230 * 10^{-12} m} \approx -4.99 * 10^{-18} J\) b) The electrostatic potential energy between the electron and proton separated by 1.0 nm is: \(U_2 = \dfrac{(8.99 * 10^9 N m^2/C^2)(1.60 * 10^{-19} C)(-1.60 * 10^{-19} C)}{1.0 * 10^{-9} m} \approx -2.30 * 10^{-18} J\) c) The change in potential energy is: \(\Delta U = U_2 - U_1 \approx -2.30 * 10^{-18} J - (-4.99 * 10^{-18} J) \approx 2.69 * 10^{-18} J\) Since \(\Delta U > 0\), the potential energy of the two particles increases when the distance is increased to 1.0 nm.

Step 1: Convert Distances to Meters

First, we need to convert the given distances to meters to be compatible with the units of the electrostatic constant: - 230 pm = 230 * 10^(-12) meters - 1.0 nm = 1.0 * 10^(-9) meters

Step 2: Constants and Initial Values

For solving this problem, we require the following constants and initial values: - \(k_e = 8.99 * 10^9 \ N m^2/C^2\) (electrostatic constant) - Charge of a proton, \(q_p = 1.60 * 10^{-19} C\) - Charge of an electron, \(q_e = -1.60 * 10^{-19} C\)

Step 3: Calculate Potential Energy at 230 pm

Using the electrostatic potential energy formula for a distance of 230 pm (230 * 10^(-12) meters): \(U_1 = \dfrac{(8.99 * 10^9 N m^2/C^2)(1.60 * 10^{-19} C)(-1.60 * 10^{-19} C)}{230 * 10^{-12} m}\) Calculate \(U_1\) to find the potential energy.

Step 4: Calculate Potential Energy at 1.0 nm

Using the electrostatic potential energy formula for a distance of 1.0 nm (1.0 * 10^(-9) meters): \(U_2 = \dfrac{(8.99 * 10^9 N m^2/C^2)(1.60 * 10^{-19} C)(-1.60 * 10^{-19} C)}{1.0 * 10^{-9} m}\) Calculate \(U_2\) to find the potential energy.

Step 5: Change in Potential Energy

To find the change in potential energy, subtract the initial potential energy from the final potential energy: \(\Delta U = U_2 - U_1\) Calculate \(\Delta U\) to find the change in potential energy.

Step 6: Potential Energy Increase or Decrease

Determine if the potential energy increased or decreased by analyzing the sign of \(\Delta U\). If \(\Delta U > 0\), the potential energy increased; if \(\Delta U < 0\), the potential energy decreased.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in physics that describes the electrostatic interaction between electrically charged particles.
It states that the force between two charges is proportional to the product of the charges, and inversely proportional to the square of the distance between them. This can be expressed with the formula:

• F = \( \dfrac{k_e \cdot |q_1 \cdot q_2|}{r^2} \)

Here, \( F \) is the force between the charges, \( k_e \) is the electrostatic constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the centers of the two charges.
In the context of the given exercise, Coulomb's Law helps quantify the attraction between an electron and a proton, which are two charged particles. These interactions are influenced by the magnitude of their charges and the distance separating them. Understanding this principle is crucial for computing potential energy in electrostatic contexts. By integrating Coulomb's law with dimensional analysis, we can determine potential energy values for various distances between particles.

Electron-Proton Interaction

The interaction between an electron and a proton is a classic example of electrostatic attraction. Electrons are negatively charged particles, while protons carry a positive charge. As per Coulomb's Law, opposite charges attract each other, and thus electrons and protons are naturally drawn towards one another.
In the exercise, this principle underpins the calculation of the potential energy between the two particles. The charge of a proton is \( q_p = 1.60 \times 10^{-19} C \), while the charge of an electron is \( q_e = -1.60 \times 10^{-19} C \).
This interaction determines how the potential energy is calculated, as one electron and one proton create a dipole with specific potential energy that depends on their separation distance.

• An electron in proximity to a proton results in a lower potential energy state.
• An increased separation distance would typically lead to a less negatively charged potential energy.

Such interactions are essential in understanding chemical bonds and reactions, as the energy levels between particles can indicate potential molecular arrangements and reactions.

Potential Energy Change

Potential energy change is critical when examining how distances between charged particles influence their energy states. In the context of the exercise, potential energy is calculated for two different separation distances between an electron and a proton.
This concept is highlighted by examining the change in potential energy when the separation increases from 230 pm to 1.0 nm. Using the formula:

• \( U = \dfrac{k_e \cdot q_1 \cdot q_2}{r} \)

we find the potential energy at each distance and the difference between them. The change in potential energy, represented as \( \Delta U = U_2 - U_1 \), tells us about the energy dynamics as distance varies.
For this system:

• If \( \Delta U < 0 \), the system loses energy, often when moving closer.
• If \( \Delta U > 0 \), the system gains energy when moving apart.

Understanding these changes helps in predicting the behavior of atomic and molecular interactions.

Distance in Chemistry

Distance is a fundamental factor in chemistry, affecting the interactions and energy level of atoms and molecules. By analyzing how particles interact over different distances, we can infer a lot about the molecular structure and stability.
In the exercise example, the separation between an electron and a proton is considered at two points: 230 pm and 1.0 nm. Distance affects the potential energy calculated, where a smaller distance typically results in stronger interactions and lower potential energy due to increased attraction.
This concept is especially relevant to chemical bonds, where:

• Shorter distances often lead to stronger bonds.
• Increasing distance can weaken interaction energy, signifying the potential dissociation of bonds.

Understanding these principles is crucial in fields such as organic chemistry and molecular physics, where bonding patterns and chemical reactions rely on the electrostatic interactions dictated by the distances between atoms and molecules.