Problem 26 - Fundamentals of Physics

Question

What is the magnitude of the electrostatic force between a singly charged sodium ion \(\left(\mathrm{Na}^{+},\right.\) of charge \(\left.+e\right)\) and an adjacent singly charged chlorine ion \(\left(\mathrm{Cl}^{-},\right.\) of charge \(\left.-e\right)\) in a salt crystal if their separation is \(2.82 \times 10^{-10} \mathrm{~m} ?\)

Step-by-Step Solution

Answer

The force is approximately \(8.18 \times 10^{-9} \text{ N}\).

Step 1: Understanding the Problem

We need to find the magnitude of the electrostatic force between two charged ions: a sodium ion \(\text{Na}^+\) with charge \(+e\) and a chlorine ion \(\text{Cl}^-\) with charge \(-e\), separated by a distance of \(2.82 \times 10^{-10} \text{ m}\).

Step 2: Applying Coulomb's Law

Coulomb's law gives the electrostatic force \( F \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) as \[ F = \frac{k |q_1 q_2|}{r^2} \] where \( k \) is Coulomb's constant, \( k = 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2 \).

Step 3: Substitution of Known Values

Substitute the values into Coulomb's law: \( q_1 = +e = 1.6 \times 10^{-19} \text{ C} \), \( q_2 = -e = -1.6 \times 10^{-19} \text{ C} \), and \( r = 2.82 \times 10^{-10} \text{ m} \). The force \( F \) is thus \[ F = \frac{8.99 \times 10^9 \times (1.6 \times 10^{-19})^2}{(2.82 \times 10^{-10})^2} \].

Step 4: Calculating the Force

Perform the calculations. First, compute \((1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38}\). Then the denominator \((2.82 \times 10^{-10})^2 = 7.9524 \times 10^{-20}\). Finally, \( F = \frac{8.99 \times 10^9 \times 2.56 \times 10^{-38}}{7.9524 \times 10^{-20}}\approx 8.18 \times 10^{-9} \text{ N}\).

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle of physics that helps us understand the interaction between electric charges. It states that the electrostatic force (\( F \)) between two point charges (\( q_1 \) and \( q_2 \)) is directly proportional to the magnitude of the charges and inversely proportional to the square of the distance (\( r \)) between them. This is mathematically expressed as:

• \[ F = \frac{k |q_1 q_2|}{r^2} \]

Here, \( k \) is Coulomb's constant, valued at \( 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2 \). The law implies that the force is stronger when charges are closer and weaker as they move farther apart. This concept is critical in understanding electrostatic forces and interactions in materials like salt crystals. Coulomb's Law can be applied to analyze forces in various physical systems, including chemical bonds and electrical circuits. Whether we are dealing with atoms or ions, Coulomb's formula allows us to predict how electric forces will influence particle behavior.

Ionic Bond

An ionic bond is a type of chemical bond commonly formed between metal and non-metal atoms. It occurs when there is a transfer of electrons from one atom to another, resulting in the creation of ions:

• Cations (positively charged ions)
• Anions (negatively charged ions)

These ions are held together by the strong electrostatic forces outlined by Coulomb's Law. For example, in a sodium chloride (NaCl) compound, ionic bonds form as sodium loses an electron to become a sodium ion (\( \text{Na}^+ \)), while chlorine gains an electron to become a chloride ion (\( \text{Cl}^- \)).

The ionic bond is characterized by:

• High melting and boiling points
• Good electrical conductivity in molten or aqueous states
• Formation of crystalline structures, like salt crystals

In essence, the ionic bond is fundamental in the formation of many compounds, imparting specific physical properties that are important for various industrial and biological processes.

Sodium and Chlorine Ions

Sodium ions (\( \text{Na}^+ \)) and chlorine ions (\( \text{Cl}^- \)) are key components in forming sodium chloride, a common ionic compound. The sodium ion is created when a sodium atom loses an electron, achieving a stable electronic configuration similar to noble gases. In contrast, a chlorine atom gains an electron to form a chloride ion, also attaining a stable electron shell.

These ions play crucial roles beyond just forming salt:

• Sodium ions are essential in nerve function and muscle contraction
• Chloride ions help maintain fluid balance and are involved in digestion

Thus, understanding how these ions interact through electrostatic forces enhances our comprehension of both their roles in biological systems and their contribution to the stability of salts. The force calculated using Coulomb's Law exemplifies the strength of the interaction between sodium and chlorine ions, which stabilizes the NaCl lattice.