Madelung Constant Formula Explanation

The binding energy ($E_b$) per molecule for an ionic crystal is related to the Madelung constant ($M$) via the Born-Landé equation. This equation considers the electrostatic attraction and short-range repulsion between ions.

The formula for binding energy is:

$ E_b = \frac{M \alpha e^2}{4 \pi \epsilon_0 r_0} \left( 1 - \frac{1}{n} \right) $

Where:

• $M$ is the Madelung constant.
• $\alpha$ is the magnitude of the ionic charge (for NaCl, $\alpha=1$).
• $e$ is the magnitude of the elementary charge.
• $\epsilon_0$ is the permittivity of free space.
• $r_0$ is the shortest distance between adjacent ions (interionic distance).
• $n$ is the exponent in the repulsive potential term ($K/r^n$).

Madelung Constant Calculation Steps

Given Information:

• Binding energy per molecule, $E_b = 7.95$ eV.
• Lattice parameter, $a = 0.563$ nm. For NaCl structure, the interionic distance $r_0 = a/2$.
• Repulsive potential term is $K/r^5$, so the exponent $n = 5$.
• Elementary charge magnitude, $e = 1.6 \times 10^{-19}$ C.
• Permittivity of free space, $\epsilon_0 = 8.854 \times 10^{-12}$ $C^2N^{-1}m^{-2}$.
• Ionic charge magnitude for NaCl, $\alpha = 1$.

Step 1: Determine the interionic distance ($r_0$)

The interionic distance $r_0$ is half the lattice parameter $a$.

$r_0 = \frac{a}{2} = \frac{0.563 \text{ nm}}{2} = 0.2815 \text{ nm}$

Convert $r_0$ to meters:

$r_0 = 0.2815 \times 10^{-9} \text{ m}$

Step 2: Convert Binding Energy ($E_b$) to Joules

$ E_b = 7.95 \text{ eV} \times (1.6 \times 10^{-19} \text{ J/eV}) = 1.272 \times 10^{-18} \text{ J} $

Step 3: Rearrange the Born-Landé formula to solve for $M$

$ M = \frac{4 \pi \epsilon_0 r_0 E_b}{\alpha e^2 \left( 1 - \frac{1}{n} \right)} $

Step 4: Substitute the known values into the rearranged formula

$ M = \frac{4 \pi (8.854 \times 10^{-12} \text{ C}^2\text{N}^{-1}\text{m}^{-2}) (0.2815 \times 10^{-9} \text{ m}) (1.272 \times 10^{-18} \text{ J})}{(1) (1.6 \times 10^{-19} \text{ C})^2 \left( 1 - \frac{1}{5} \right)} $

Step 5: Perform the calculation

Calculate the numerator:

$ \text{Numerator} = 4\pi \times (8.854 \times 10^{-12}) \times (0.2815 \times 10^{-9}) \times (1.272 \times 10^{-18}) \approx 3.987 \times 10^{-38} $

Calculate the denominator:

$ \text{Denominator} = (1.6 \times 10^{-19})^2 \times (1 - 0.2) = (2.56 \times 10^{-38}) \times 0.8 = 2.048 \times 10^{-38} $

Compute $M$:

$ M = \frac{3.987 \times 10^{-38}}{2.048 \times 10^{-38}} \approx 1.947 $

The calculated value for the Madelung constant is approximately 1.947. The question states the correct value lies between 1.745 and 1.751.