Total Energy of Ion given Charges and Distances Calculator

Total Energy Formula:

\[ E_{total} = \frac{-(q^2 \cdot e^2 \cdot M)}{4\pi\epsilon_0 r_0} + \frac{B}{r_0^n} \]

Charge (q):

Madelung Constant (M):

Distance of Closest Approach (r₀):

Repulsive Interaction Constant (B):

Born Exponent (n):

Total Energy of Ion (J):

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1. What is Total Energy of Ion in a Lattice?

Definition: The total energy of an ion in a crystal lattice is the sum of the Madelung energy (electrostatic potential) and the repulsive potential energy.

Purpose: This calculation helps in understanding the stability and bonding energy of ionic crystals.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ E_{total} = \frac{-(q^2 \cdot e^2 \cdot M)}{4\pi\epsilon_0 r_0} + \frac{B}{r_0^n} \]

Where:

\( E_{total} \) — Total energy of ion (Joules)
\( q \) — Charge of ion (Coulombs)
\( e \) — Electron charge (1.602 × 10⁻¹⁹ C)
\( M \) — Madelung constant (dimensionless)
\( \epsilon_0 \) — Permittivity of vacuum (8.85 × 10⁻¹² F/m)
\( r_0 \) — Distance of closest approach (meters)
\( B \) — Repulsive interaction constant
\( n \) — Born exponent (typically 5-12)

3. Importance of Total Energy Calculation

Details: Calculating the total energy helps predict crystal stability, lattice formation energy, and other thermodynamic properties of ionic compounds.

4. Using the Calculator

Tips: Enter the ion charge, Madelung constant (default 1.7), distance (default 6×10⁻⁹ m), repulsive constant (default 40000), and Born exponent (default 0.9926). All values must be > 0.

5. Frequently Asked Questions (FAQ)

Q1: What is the Madelung constant?
A: It's a geometric factor that accounts for the arrangement of ions in the crystal lattice.

Q2: How do I determine the Born exponent?
A: It's typically determined experimentally from compressibility measurements or theoretically from electronic structure.

Q3: What units should I use?
A: Use SI units - Coulombs for charge, meters for distance, and the calculator will output Joules.

Q4: What's a typical Madelung constant value?
A: For NaCl structure it's ~1.748, for CsCl ~1.763, and for zinc blende ~1.638.

Q5: Why is the repulsive term important?
A: It prevents the collapse of the lattice by balancing the electrostatic attraction at equilibrium distance.