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Coulomb's Law is a foundational principle in physics that describes the electrostatic interaction between two charged particles. The law states that the force (\( F \)) between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance (\( r \)) separating them. It is mathematically represented as:
\[ F = k \cdot \frac{q_1 \cdot q_2}{r^2} \]where:

• \( F \) is the electrostatic force.
• \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \).
• \( q_1 \) and \( q_2 \) are the amounts of the charges involved.
• \( r \) is the distance between the centers of the two charges.

The force described by Coulomb's Law is a vector, meaning it has both magnitude and direction. This force acts along the line joining the centers of the two charges. When the charges are of opposite signs, the force is attractive, pulling the charges toward one another. Conversely, if the charges have the same sign, they repel each other. This principle is crucial for understanding interactions at a molecular level, like those between ions, which we will explore shortly.

Ions are atoms or molecules that have gained or lost electrons, giving them a net electric charge. Positive ions, known as cations, result from losing one or more electrons, whereas negative ions, anions, form by gaining electrons. Understanding ion charges is vital when applying Coulomb's Law, as this charge determines how ions interact with each other.In our example:

• The \( \mathrm{Na}^{+} \) ion has a charge of \(+e \), where \( e \) is the elementary charge, approximately \( 1.6 \times 10^{-19} \text{C} \).
• The \( \mathrm{Cl}^{-} \) ion has a charge of \(-e \, \).

These charges create an electrostatic force between them as described by Coulomb's Law. Because the sodium ion is positively charged and the chloride ion is negatively charged, the force between them will be attractive.Notably, this principle holds for other ions such as lithium (\( \mathrm{Li}^{+} \)) and bromide (\( \mathrm{Br}^{-} \)), because they also carry the charges \(+e\) and \(-e\) respectively, meaning that replacing one set of ions with another that has the same charges will yield the same electrostatic force.

The distance between charged objects significantly affects the magnitude of the electrostatic force experienced between them. According to Coulomb's Law, the force is inversely proportional to the square of the separation distance (\( r \)).This means:

• If the distance between the charges is halved, the force increases by a factor of four.
• If the distance is doubled, the force decreases by a factor of four.

In the given problem, the separation distance between the sodium and chloride ions is \( 0.50 \, \text{nm} \), or \( 0.50 imes 10^{-9} \, \text{m} \). Accurately calculating the force requires converting the distance into meters, as Coulomb's constant is in units that include meters.Thus, the precise calculation is crucial and involves plugging the distance into Coulomb’s Law to understand how strongly these ions attract each other. This relationship between distance and force underscores why even tiny changes in molecular distances can significantly affect chemical interactions and states of matter.