Given the lattice energies and lattice parameters for the alkali halides with the \(\mathrm{NaCl}\) structure listed in the table below, calculate the Madelung energy, assuming each ion has formal charge (i.e. the charges given by the valence, so that the cation charge is 1 unit of positive electron charge, and the anion charge is \(-1\) unit of electron charge), and hence the Born-Mayer energy. $$ \begin{array}{ccccccc} & {c}{\mathrm{F}} & \mathrm{Cl} & \mathrm{Br} & \mathrm{I} & \\ \hline \mathrm{Li} & a: & 4.028 & 5.140 & 5.502 & 6.000 & \\ & E: & 1014.3 & 832.6 & 794.5 & 743.9 & \mathrm{~kJ} \mathrm{~mol}^{-1} \\ & B: & 67.1 & 29.8 & 23.8 & 17.1 & \mathrm{GPa} \\ \mathrm{Na} & a: & 4.634 & 5.640 & 5.978 & 6.474 & \\ & E: & 897.5 & 764.4 & 726.7 & 683.2 & \mathrm{~kJ} \mathrm{~mol}^{-1} \\ & B: & 46.5 & 24.0 & 19.9 & 15.1 & \mathrm{GPa} \\ \mathrm{K} & a: & 5.348 & 6.294 & 6.596 & 7.066 & \\ & E: & 794.5 & 694.0 & 663.5 & 627.5 & \mathrm{~kJ} \mathrm{~mol}^{-1} \\ & B: & 30.5 & 17.4 & 14.8 & 11.7 & \mathrm{GPa} \\ \mathrm{Rb} & a: & 5.630 & 6.582 & 6.890 & 7.342 & \\ & E: & 759.3 & 666.8 & 638.8 & 606.6 & \mathrm{~kJ} \mathrm{~mol}^{-1} \\ & B: & 26.2 & 15.6 & 13.0 & 10.6 & \mathrm{GPa} \\ \hline \end{array} $$ Given the condition for equilibrium, calculate the value of \(\rho\) for each case, and hence the coefficients \(B\) in the BornMayer interaction. For each crystal, calculate the value of the bulk modulus and compare with the experimental value. (Hint: a case for using a spreadsheet program.)

Step by step solution

01

Identify the provided data

The provided data includes lattice energies (\(E\)), lattice parameters (\(a\)), and bulk moduli (\(B\)) for various alkali halides. Each cation and anion pair such as \(\text{LiF}\), \(\text{LiCl}\), \(\text{NaF}\), \(\text{NaCl}\), etc., are given specific values for \(a\) and \(E\) as outlined in the table.

02

Find the Madelung energy

The Madelung constant (\(M\)) for the NaCl structure is approximately 1.7476. The Madelung energy (\(E_M\)) can be calculated using the formula: \(E_M = -M \frac{{e^2}}{a}\), where \(e\) is the electronic charge (\(1.602 \times 10^{-19} \text{C}\)) and \(a\) is the lattice parameter.

03

Calculate \(\rho\)

Use the given equilibrium condition between lattice energy and Madelung energy to solve for \(\rho\). The equilibrium condition is given by \(E = E_M + E_{\text{Born-Mayer}}\), where \(E_{\text{Born-Mayer}} = B e^{-\frac{a}{\rho}}\). By substituting the known values of \(E\) and \(E_M\), we can rearrange to find \(\rho\).

04

Calculate \(B\) in the Born-Mayer interaction

Rearrange the Born-Mayer potential formula to solve for \(B\). Since \(E_{\text{Born-Mayer}} = B e^{-\frac{a}{\rho}}\) and we have calculated \(\rho\), solve for \(B\) using: \(B = (E - E_M) e^{\frac{a}{\rho}}\).

05

Calculate the theoretical bulk modulus

The theoretical bulk modulus (\(K\)) can be given by: \(K = \frac{1}{V} \frac{\text{d}^2E}{\text{d}a^2}\). For simplicity, consider: \(K \rightarrow \frac{2}{3} \frac{B}{a_0}\). Use the experimentally given bulk modulus values for comparison.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Madelung Energy

The Madelung energy is crucial for understanding the stability and properties of ionic crystals such as alkali halides. It arises due to the electrostatic interactions between ions in a crystal lattice. The Madelung constant (\text{M}) reflects the geometrical arrangement of ions. For the \text{NaCl} structure, it’s approximately 1.7476. To calculate the Madelung energy (\text{E_M}), use the relation:
\[ E_M = -M \frac{e^2}{a}\]where \[e\] is the elementary charge (\text{1.602 \times 10^{-19} C}) and \[a\] is the lattice parameter. This equation demonstrates the role of both the charge magnitude and the lattice spacing in determining the interaction energy. Understanding the Madelung energy helps explain why certain crystal structures are more stable than others.

Born-Mayer Interaction

The Born-Mayer interaction complements the Madelung energy by accounting for the repulsive forces between closely packed ions. This interaction’s energy (\text{E_{Born-Mayer}}) can be modeled using an exponential function:
\[ E_{\text{Born-Mayer}} = B e^{-\frac{a}{\rho}}\]where:

• \[a\] \text{is the lattice parameter}
• \[B\] \text{is a fitting parameter related to the strength of the repulsive force}
• \[ \rho\] \text{is the repulsion parameter representing the decay length of the repulsive interaction}

By considering both the Madelung and Born-Mayer energies, we obtain a more complete description of the total lattice energy in ionic solids.

Bulk Modulus Calculation

The bulk modulus (\text{K}) is a measure of a material's resistance to uniform compression. It provides insights into the rigidity and mechanical stability of ionic crystals. To calculate the theoretical bulk modulus, use:
\[ K = \frac{2}{3} \frac{B}{a_0}\]Here, \[B\] is the parameter from the Born-Mayer equation and \[a_0\] is the equilibrium lattice parameter. This relation stems from solid-state physics principles and can be compared with experimental values for validation. It helps in understanding how changes in lattice spacing affect the material's mechanical properties.

Equilibrium Condition

For an ionic crystal to be at equilibrium, the total lattice energy must balance attractive and repulsive forces. The equilibrium condition is given by:
\[ E = E_M + E_{\text{Born-Mayer}}\]where:

• \[ E_M \] is the Madelung energy
• \[ E_{\text{Born-Mayer}} \] is the Born-Mayer energy
• \[ E \] is the total lattice energy, given in the exercise data

By substituting the known values and solving for \[ \rho\], this approach helps us understand the balance of forces in maintaining crystal stability.

Lattice Parameters

Lattice parameters, notably the lattice constant \[ a\], are essential for calculating a crystal’s properties like lattice energy and bulk modulus. They represent the spatial dimensions of the unit cell in the crystal lattice. Different alkali halides have unique lattice parameters, reflecting their structural and energetic characteristics. For instance, larger lattice parameters indicate more extensive unit cells, which affect interaction energies. Understanding lattice parameters is fundamental to predicting material behavior under various conditions.