Distance of Closest Approach using Born-Lande Equation without Madelung Constant Calculator

Born-Lande Equation without Madelung Constant:

\[ r_0 = -\frac{N_A \cdot N_{\text{ions}} \cdot 0.88 \cdot z^+ \cdot z^- \cdot e^2 \cdot (1 - \frac{1}{n_{\text{born}}})}{4\pi \varepsilon_0 U} \]

Number of Ions:

ions

Charge of Cation:

Charge of Anion:

Born Exponent:

unitless

Lattice Energy:

J/mol

Distance of Closest Approach (r₀):

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1. What is the Born-Lande Equation without Madelung Constant?

The Born-Lande equation without Madelung constant calculates the distance of closest approach between ions in a crystal lattice. It provides a theoretical estimate of the equilibrium distance where the attractive and repulsive forces between ions balance.

2. How Does the Calculator Work?

The calculator uses the Born-Lande equation without Madelung constant:

\[ r_0 = -\frac{N_A \cdot N_{\text{ions}} \cdot 0.88 \cdot z^+ \cdot z^- \cdot e^2 \cdot (1 - \frac{1}{n_{\text{born}}})}{4\pi \varepsilon_0 U} \]

Where:

\( N_A \) — Avogadro's number (6.022 × 10²³ mol⁻¹)
\( N_{\text{ions}} \) — Number of ions per formula unit
\( z^+ \) — Charge of cation (C)
\( z^- \) — Charge of anion (C)
\( e \) — Elementary charge (1.602 × 10⁻¹⁹ C)
\( n_{\text{born}} \) — Born exponent (5-12)
\( \varepsilon_0 \) — Permittivity of vacuum (8.85 × 10⁻¹² C²/J·m)
\( U \) — Lattice energy (J/mol)

Explanation: The equation calculates the equilibrium distance between ions based on electrostatic interactions and Born repulsive forces.

3. Importance of Distance of Closest Approach Calculation

Details: Calculating the distance of closest approach is crucial for understanding crystal structures, predicting lattice parameters, and studying ionic bonding in solids.

4. Using the Calculator

Tips: Enter the number of ions, cation charge, anion charge, Born exponent (between 5-12), and lattice energy. All values must be positive and within valid ranges.

5. Frequently Asked Questions (FAQ)

Q1: What is the significance of the Born exponent?
A: The Born exponent represents the repulsive forces between electron clouds of adjacent ions and typically ranges from 5 to 12 depending on the ionic compound.

Q2: Why is the Madelung constant not included in this equation?
A: This simplified version assumes a symmetric arrangement where the Madelung constant is incorporated into the 0.88 factor for certain crystal structures.

Q3: What are typical values for distance of closest approach?
A: For ionic compounds, r₀ typically ranges from 2-4 Å (2-4 × 10⁻¹⁰ m) depending on the ionic sizes and charges.

Q4: How does lattice energy affect the distance of closest approach?
A: Higher lattice energy generally results in smaller interionic distances as ions are held more tightly together.

Q5: What are the limitations of this equation?
A: This simplified equation may not be accurate for complex crystal structures and doesn't account for covalent character or specific geometric arrangements.