Problem 15 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Calculate the energy of attraction between a \(\mathrm{Mg}^{2+}\) ion and a CI ion separated by \(0.2 \mathrm{nm}\) in a vacuum. (Section 17.3 )

Step-by-Step Solution

Answer

The energy of attraction is \(2.302 \times 10^{-18}\) joules.

Step 1: Understand the Problem

You need to calculate the energy of attraction between a magnesium ion \(\mathrm{Mg}^{2+}\) and a chloride ion \(\mathrm{Cl}^-\) that are \(0.2\) nm apart in a vacuum.

Step 2: Formula for Energy of Attraction

The energy of attraction \(E\) between two ions separated by distance \(r\) in a vacuum can be calculated using Coulomb's law:\[ E = -\frac{k \cdot q_1 \cdot q_2}{r} \]Where \(k\) is the Coulomb's constant \(8.99 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2\), \(q_1\) and \(q_2\) are the charges of the ions, and \(r\) is the separation distance.

Step 3: Identify the Charges

For \(\mathrm{Mg}^{2+}\), \(q_1 = +2e\). For \(\mathrm{Cl}^-\), \(q_2 = -e\) where \(e = 1.602 \times 10^{-19}\) C (elementary charge).

Step 4: Convert Distance

Convert the distance from nanometers to meters. Since \(1\) nm = \(1 \times 10^{-9}\) meters, \(0.2\) nm = \(0.2 \times 10^{-9}\) meters.

Step 5: Plug Values into the Formula

Substitute the known values into the formula:\[ E = -\frac{(8.99 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2) \times (2 \times 1.602 \times 10^{-19} \mathrm{C}) \times (-1.602 \times 10^{-19} \mathrm{C})}{0.2 \times 10^{-9} \mathrm{m}} \]

Step 6: Calculation of Energy

Perform the calculation. First calculate the product of charges: \[ (2 \times 1.602 \times 10^{-19}) \times (-1.602 \times 10^{-19}) = -5.1264 \times 10^{-38} \mathrm{C}^2 \]Then compute the energy:\[ E = -\frac{(8.99 \times 10^9) \times (-5.1264 \times 10^{-38})}{0.2 \times 10^{-9}} = 2.302 \times 10^{-18} \mathrm{J} \]

Step 7: Result Interpretation

The energy of attraction between the \(\mathrm{Mg}^{2+}\) ion and the \(\mathrm{Cl}^-\) ion is \(2.302 \times 10^{-18}\) joules, which indicates a stable ionic bond.

Key Concepts

Coulomb's law

Coulomb's law helps us understand the force between two charged particles. It's a fundamental principle in physics that describes how electric charges interact. Imagine having two static charges. The force of attraction or repulsion between them can be calculated using this law.

The formula is given by:

• \[ F = \frac{k \cdot q_1 \cdot q_2}{r^2} \]
  
• Here, F is the force, k is Coulomb's constant (\(8.99 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2\)), and q values are the charges of the particles in coulombs.
  
• The distance r is the separation between charges, squared in this context.

In energy calculations, as in the given exercise, the formula changes slightly.
We remove the square from the distance since energy is not about **force** directly,
and the negative sign indicates energy release upon attraction.

Ionic bonds

Ionic bonds form when electrons transfer from one atom to another. This transfer creates ions with opposite charges, which then attract each other. It's one of the key ways atoms come together to form compounds. These bonds are strong and are essential to forming stable structures.

Consider the example in the exercise: a magnesium (Mg) ion and chloride (Cl) ion. Magnesium loses two electrons to achieve a stable electron arrangement, becoming \( \mathrm{Mg}^{2+} \).
Chlorine gains those electrons, forming \( \mathrm{Cl}^- \).
The opposite charges of \( \mathrm{Mg}^{2+} \) and \( \mathrm{Cl}^- \) ions result in a strong electrostatic attraction,
which is the ionic bond.

• Ionic bonds result in the formation of crystals, like salt.
• They have high melting and boiling points due to their strength.

Rich in fascinating chemistry, ionic bonds are foundational in many compounds around us.

Magnesium ion

A magnesium ion, symbolized as \( \mathrm{Mg}^{2+} \), represents a magnesium atom that has lost two electrons. This transforms it from a neutral atom into a positively charged ion with a charge of +2. This type of ion is common in numerous chemical reactions, especially those leading to the formation of ionic compounds.

Understanding the nature of the magnesium ion can be simplified into a few facts:

• Magnesium is in Group 2 of the periodic table.
• It typically loses two electrons to achieve a stable noble gas configuration.
• The resulting \( \mathrm{Mg}^{2+} \) ion is highly reactive and ready to form bonds.

Mg ions are vital in many biological and industrial processes.
They play key roles in things like muscle contraction and are part of materials like magnesium oxide and magnesium sulfate.
In the context of ionic bonds, its positive charge gets attracted to negatively charged ions, forming stable compounds.

Chloride ion

The chloride ion, denoted as \( \mathrm{Cl}^- \), is formed when a chlorine atom gains an extra electron. This ion now carries a negative charge of -1, making it an anion. Chloride ions are ubiquitous in chemistry, forming compounds readily with a variety of metals.

Some essential points about chloride ions include:

• Chlorine belongs to Group 17 (halogens) in the periodic table.
• The gain of one electron allows chlorine to fulfill its desire for a complete outer shell.
• This single-charge anion pairs well with positively charged ions like sodium and magnesium.

In daily life, chloride ions are everywhere. They're found in the table salt (sodium chloride) we use in cooking.
They're also in biological fluids and play critical roles in maintaining proper nerve function.
When paired with \( \mathrm{Mg}^{2+} \) ions in ionic bonds, chloride ions contribute to the compound's overall stability and functionality.