Problem 14 - Chemistry: The Central Science in SI Units, Global Edition

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

The electrostatic potential energy between two electrons separated by \(460 \, \mathrm{pm}\) is approximately \(-2.50 \times 10^{-18} \, \mathrm{J}\). When the distance between the electrons is increased to \(1.0 \, \mathrm{nm}\), the change in potential energy is approximately \(2.27 \times 10^{-18} \, \mathrm{J}\), indicating that the potential energy increases when the distance is increased to \(1.0 \, \mathrm{nm}\).

Step 1: Part (a): Electrostatic potential energy between two electrons separated by 460 pm

First, we need to convert the given distance from picometers (pm) to meters (m): \(460 \, \mathrm{pm} = 460 \times 10^{-12} \, \mathrm{m}\) Now, we can use the formula for electrostatic potential energy: \[U = \frac{kq_1q_2}{r}\] Since both charges are electrons, we have: \[U = \frac{k(-e)(-e)}{r}\] Substitute the given values and constants: \[U = \frac{(8.99 \times 10^{9} Nm^2/C^2)((-1.60 \times 10^{-19} C)^2)}{460 \times 10^{-12} m}\] Calculate the electrostatic potential energy: \[U \approx -2.50 \times 10^{-18} \, \mathrm{J}\]

Step 2: Part (b): Change in potential energy

First, convert the new distance from nanometers (nm) to meters (m): \(1.0 \, \mathrm{nm} = 1.0 \times 10^{-9} \, \mathrm{m}\) Calculate the new electrostatic potential energy with the increased distance: \[U' = \frac{k(-e)(-e)}{r'}\] \[U' = \frac{(8.99 \times 10^{9} Nm^2/C^2)((-1.60 \times 10^{-19} C)^2)}{1.0 \times 10^{-9} m}\] Calculate the new electrostatic potential energy: \[U' \approx -2.30 \times 10^{-19} \, \mathrm{J}\] Now, find the change in potential energy: \[\Delta U = U' - U \] \[\Delta U \approx -2.30 \times 10^{-19} \, \mathrm{J} - (-2.50 \times 10^{-18} \, \mathrm{J})\] Calculate the change in potential energy: \[\Delta U \approx 2.27 \times 10^{-18} \, \mathrm{J}\]

Step 3: Part (c): Increase or decrease in potential energy

Compare the initial potential energy \(U\) to the new potential energy \(U'\). Since the initial potential energy was about \(-2.50 \times 10^{-18} \, \mathrm{J}\) and the new potential energy is about \(-2.30 \times 10^{-19} \, \mathrm{J}\), the potential energy has increased. This is because when the distance between two like charges (such as the electrons in this case) increases, the electrostatic force between them decreases, making their potential energy closer to zero.

Key Concepts

Electrostatic force

Electrostatic force is one of the fundamental forces in nature. It arises from the interaction between objects that have electric charge. In essence, it is the force between charged particles. Two key features define how electrostatic force acts:

• Attractive or Repulsive: The electrostatic force can either pull charges towards each other or push them apart. Opposite charges (such as positive and negative) will attract each other, while like charges (either positive-positive or negative-negative) will repel each other.
• Magnitude: The strength (magnitude) of the electrostatic force depends on two factors: the amount of charge that each object possesses and the distance between the two charges. Greater charges or closer distances result in a stronger force.

This concept is crucial in fields ranging from chemistry, where it explains how atoms hold together in molecules, to everyday experiences like how static electricity causes a balloon to stick to a wall.

Electric charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charges, positive and negative. Each type of charge is associated with particles in atoms:

• Electrons: These subatomic particles carry a negative charge. They orbit the nucleus of an atom and are integral in the formation of chemical bonds.
• Protons: Protons reside in the nucleus and carry a positive charge. Their positive charge balances the negative charge of the electrons.

Charge is quantized, meaning it occurs in discrete amounts. The charge on each electron and proton is equal in magnitude, around \(1.60 \times 10^{-19}\ C\). The principle of conservation of charge states that the total charge in an isolated system remains constant. Understanding electric charge is essential because it governs the behavior of electrons in atoms, influencing everything from electrical conductivity to chemical reactivity.

Coulomb's law

Coulomb's Law is a mathematical equation that quantifies the electrostatic force between two point charges. Named after physicist Charles-Augustin de Coulomb, it is a cornerstone of electrostatics. Coulomb’s Law states that the force (\(F\)) between two charges is:

• Directly proportional to the product of the magnitudes of the charges. This means if one charge doubles, the force will double, assuming distance remains constant.
• Inversely proportional to the square of the distance (\(r\)) between them. As the distance increases, the force rapidly decreases.

The formula for Coulomb's Law is given by:\[F = \frac{k|q_1q_2|}{r^2}\]where:

• \(k\) is Coulomb's constant, approximately \(8.99 \times 10^9 \space Nm^2/C^2\).
• \(q_1\) and \(q_2\) are the magnitudes of the charges.

This law provides a basis for understanding electrostatic interactions, offering insight into the behavior of charged particles both in isolated systems and complex materials. It helps explain phenomena from the binding of atoms to the operation of electrical circuits.