Problem 35 - Fundamentals of Physics

Question

In crystals of the salt cesium chloride, cesium ions \(\mathrm{Cs}^{+}\) form the eight corners of a cube and a chlorine ion \(\mathrm{Cl}^{-}\) is at the cube's center (Fig. \(21-36\) ). The edge length of the cube is \(0.40 \mathrm{nm}\). The \(\mathrm{Cs}^{+}\) ions are each deficient by one electron (and thus each has a charge of \(+e\) ), and the \(\mathrm{Cl}^{-}\) ion has one excess electron (and thus has a charge of \(-e\) ). (a) What is the magnitude of the net electrostatic force exerted on the \(\mathrm{Cl}^{-}\) ion by the eight \(\mathrm{Cs}^{+}\) ions at the corners of the cube? (b) If one of the Cs \(^{+}\) ions is missing, the crystal is said to have a defect; what is the magnitude of the net electrostatic force exerted on the \(\mathrm{Cl}^{-}\) ion by the seven remaining \(\mathrm{Cs}^{+}\) ions?

Step-by-Step Solution

Answer

(a) The net force is 0 N due to symmetry. (b) The net force with one missing Cs\(^{+}\) is non-zero (calculate numerically as described).

Step 1: Identify Charges and Positions

In this crystal structure, each Cs\(^{+}\) ion has a charge of \(+e\) and is located at the corners of a cube, while the Cl\(^{-}\) ion has a charge of \(-e\) and is located at the center. The edge length of the cube is \(0.40\, \text{nm}\) or \(0.40 \times 10^{-9}\, \text{m}\). The distance between the center and each corner is \(\frac{\sqrt{3}}{2}\) times the edge length.

Step 2: Calculate Cube Diagonal

The distance from the center of the cube to a corner (i.e., the diagonal distance) is found using the formula for the diagonal of a cube. If \(l\) is the edge length, then the diagonal \(d\) is \(\frac{\sqrt{3}}{2}l\). Thus, for \(l = 0.40 \times 10^{-9}\, \text{m}\), \(d = \frac{\sqrt{3}}{2} \times 0.40 \times 10^{-9}\). Solve to find \(d = 0.346 \times 10^{-9}\, \text{m}\).

Step 3: Use Coulomb's Law for Each Force

Coulomb's Law gives the force between two charges as \(F = k \frac{|q_1 \cdot q_2|}{r^2}\), where \(k\) is the electrostatic constant \(8.99 \times 10^9\, \text{N}\cdot\text{m}^2/\text{C}^2\). Each Cs\(^{+}\) and the Cl\(^{-}\) are separated by \(d = 0.346 \times 10^{-9}\, \text{m}\), with \(q_1 = e\) and \(q_2 = -e\). Calculate \(F\) for one pair.

Step 4: Calculate Net Force With All Ions Present

Since the cube is symmetric, all force components in directions parallel to the cube's faces cancel out, leaving the force components that point towards the center. This means that only the diagonal components that combine constructively towards the center add up. There are 8 Cs\(^{+}\) ions, each exerting the same magnitude of force. Thus, the effective net force on Cl\(^{-}\) is 0 N, because the vector sum of these symmetrical forces results in an exact cancellation.

Step 5: Net Force With One Cs Ion Missing

With one Cs\(^{+}\) missing, the symmetry breaks. Calculate the net force acting on the Cl\(^{-}\): 7 corner ions exert force along their respective diagonal components. Using vector components, calculate the resultant force magnitude. Each component (from remaining Cs\(^{+}\)) is given by \(k \frac{e^2}{d^2}\) pointing towards the missing ion space. Vectorially add these components, now they no longer completely cancel.

Step 6: Calculate Force Magnitude With One Missing Ion

Simplify the vector sum for the system with a defect. Calculate the net force by summing contributions from the remaining 7 ions. Note that with correct calculation, the force results in a net nonzero value towards the defect's original direction. Obtain numerical magnitude by addressing only the contributing 7 components affecting through symmetry.

Key Concepts

Coulomb's law

Coulomb's Law is fundamental to understanding electrostatic interactions. This law gives the quantitative value of the electrostatic force between two point charges. The formula is: \[ F = k \frac{|q_1 q_2|}{r^2} \]where:

• \( F \) is the magnitude of the electrostatic force between the charges.
• \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \) N·m²/C².
• \( q_1 \) and \( q_2 \) are the charges.
• \( r \) is the distance between the center of the two charges.

Coulomb's Law illustrates that electrostatic force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This means that an increase in distance drastically reduces the force, governing how forces in ionic crystals like cesium chloride are calculated.

crystal structure

Crystal structures provide a consistent arrangement of ions or atoms in a repeating lattice throughout the material. For cesium chloride, the ion arrangement is cubic, which greatly influences the forces acting within the structure.In this example, cesium ions \( \text{Cs}^+ \) occupy the cube's corners while a chlorine ion \( \text{Cl}^- \) resides at the center. The symmetry of this structure is essential; when all ions are present, it creates a balance, effectively canceling all net external electrostatic forces.The crystal's cubic structure causes each vectorial component of the electrostatic force from the corner ions to balance out, maintaining structural stability. This, however, changes with a missing ion, where the symmetry is distorted, causing a resulting net force to act on the central ion.

vector addition

Vector addition is crucial when dealing with forces that act in multiple directions, as in the cesium chloride crystal. In simpler terms, vector addition is used to sum up multiple forces that act at an angle to one another.In the original arrangement where all eight \( \text{Cs}^+ \) ions are in place, every force vector from these corner ions points towards the \( \text{Cl}^- \) ion. However, due to symmetry, these vectors cancel each other out, resulting in a net force of zero.If one \( \text{Cs}^+ \) ion is missing, symmetry is disrupted. The remaining forces no longer cancel perfectly, creating a net force directed towards the location of the missing ion. By adding these vectors, we determine the magnitude and direction of the net electrostatic force.

charge interaction

Charge interaction in physics involves understanding how charges of different signs exert forces on each other. In ionic crystals like cesium chloride, we deal with positive \( \text{Cs}^+ \) ions and negative \( \text{Cl}^- \) ions.Positive and negative charges attract each other, a key principle that explains the stability of the crystal structure. Each positive charge of \( q_1 = +e \) interacts with the negative charge \( q_2 = -e \) at the center, generating attractive forces.These interactions ensure the proper ionic bonding strength to maintain the solid structure. If defects or missing ions occur, the interaction dynamics change, increasing the net force on specific ions. This information is crucial in fields like materials science, where the strength and properties of crystalline materials need to be understood and utilized.