Problem 18 - Chemistry The Central Science

Question

A magnesium ion, Mg \(^{2+},\) with a charge of \(3.2 \times 10^{-19} \mathrm{Cand}\) an oxide ion, \(\mathrm{O}^{2-},\) with a charge of \(-3.2 \times 10^{-19} \mathrm{C}\) , are separated by a distance of 0.35 \(\mathrm{nm}\) . How much work would be required to increase the separation of the two ions to an infinite distance?

Step-by-Step Solution

Answer

To separate a magnesium ion (Mg\(^{2+}\)) and an oxide ion (O\(^{2-}\)) from a distance of \(0.35 \,\text{nm}\) to an infinite distance, we need to do work equal to the potential energy but with the opposite sign. Using the given charges and the potential energy formula, we calculate the potential energy as \(-3.264 \times 10^{-18} \,\text{J}\). Therefore, the work required to separate the ions is \(3.264 \times 10^{-18}\) Joules.

Step 1: Understand the problem and identify the given values

We are given the following information: - Charge of magnesium ion (Mg\(^{2+}\)): \(Q_1 = 3.2 \times 10^{-19} \,\text{C}\) - Charge of oxide ion (O\(^{2-}\)): \(Q_2 = -3.2 \times 10^{-19} \,\text{C}\) - Distance between the ions: \(r = 0.35 \,\text{nm}\) (convert it to meters) - The work required to increase the separation of the two ions to an infinite distance.

Step 2: Convert the distance from nm to meters

To use the SI units in our calculation, we need to convert the distance of separation (\(r\)) from nm to meters: $$ r = 0.35 \,\mathrm{nm} \times \frac{1 \times 10^{-9} \,\mathrm{m}}{1 \,\mathrm{nm}} = 3.5 \times 10^{-10} \,\text{m} $$

Step 3: Use the potential energy formula

We will calculate the potential energy \(U\) between the two charged particles (ions) using the following formula: $$ U = k \frac{Q_1 \times Q_2}{r} $$ where \(k\) (the Coulomb constant) is \(8.98755 \times 10^9 \mathrm{Nm^2/C^2}\), \(Q_1\) and \(Q_2\) are the charges, and \(r\) is the separation distance.

Step 4: Calculate the potential energy

Now, substitute the given values into the formula to calculate the potential energy between the two ions: $$ U = (8.98755 \times 10^9 \,\mathrm{Nm^2/C^2}) \times \frac{(3.2 \times 10^{-19}\,\text{C}) \times (-3.2 \times 10^{-19} \,\text{C})}{3.5 \times 10^{-10} \,\text{m}} $$ Calculating the value, we get: $$ U = -3.264 \times 10^{-18} \,\text{J} $$

Step 5: Calculate the work required

To separate the two ions to an infinite distance, we need to do work equal to the potential energy but with the opposite sign (as potential energy at infinite distance is zero). Hence, the work required \(W\) to separate them is: $$ W = -U = 3.264 \times 10^{-18} \,\text{J} $$ Therefore, the work required to increase the separation of the two ions to an infinite distance is \(3.264 \times 10^{-18}\) Joules.

Key Concepts

Coulomb's Law

Coulomb's Law is crucial for understanding the electrostatic potential energy between charged particles. It gives us a way to calculate the force between two charged objects. The law states that the force (\( F \)) between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

• The formula is: \[ F = k \frac{Q_1 \times Q_2}{r^2} \]
• Where \( k \) is Coulomb's constant, approximately \( 8.9875 \times 10^9 \text{ Nm}^2/\text{C}^2 \).
• \( Q_1 \) and \( Q_2 \) represent the magnitudes of the charges, and \( r \) is the distance between the charges.

Understanding Coulomb's Law helps to predict how charges will interact at different distances. In our exercise, it is the foundation for calculating the potential energy, which tells us the work involved in changing the distance between the ions.

Ion Separation

Ion separation refers to the physical distance between charged particles such as ions. In the context of electrostatics, changing this separation alters the electrostatic interactions between ions. When calculating potential energy, the distance between ions is crucial: as the distance increases, the potential energy changes, typically requiring work to move the ions.

• Consider ion separation as the key variable in electrostatic potential calculations.
• The smaller the distance, the stronger the interaction due to the inverse relationship in Coulomb's Law.
• Increasing separation to infinity implies no interaction, as forces diminish over greater distances.

In our exercise, the initial separation is given as 0.35 nm (nanometers), which needs to be converted to meters for further calculations. Understanding ion separation helps in visualizing how far the ions must be moved to calculate the work required in overcoming their electrostatic attraction.

Work Energy Principle

The Work Energy Principle connects the concept of work with changes in energy, specifically potential energy in the context of electrostatics. When we separate two charged ions to an infinite distance, we are effectively calculating how much work is needed to overcome their natural electrostatic attraction.

• Work is defined as energy transferred by a force acting over a distance.
• In electrostatics, potential energy ( \( U \)) is the energy stored due to positions of charges.
• To separate charges to infinity, work done by an external force is equal in magnitude and opposite in sign to the potential energy at finite separation.

In our exercise, the work required to separate the ions gives us insight into the energy stored in the electrostatic configuration of the ions. The calculation shows how overcoming this stored energy is essential to move the ions apart to an infinite distance, signifying no mutual interaction.