# How to Calculate a Lattice Energy Equation

An ionic compound like sodium chloride (better known as salt) forms a crystalline lattice featuring atoms arranged in well-defined positions with respect to each other. The difference in energy between ions in gas phase and crystal lattice is called lattice energy. You can calculate lattice energy using the Born-Haber cycle. More directly, you can use the Born-Meyer equation, although the calculation will require you to find a number specific for each a given type of crystal structure, the Madelung constant.

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## Instructions

1. Find the Madelung constant for your type of crystal (the University of Waterloo, listed in Resouces, has this information online). Sodium chloride, for instance, has a Madelung constant of 1.74756.

2. Take the absolute value of the charge of the first ion in your ionic compound and multiply it by the absolute value of the charge of the other ion. Absolute value disregards the positive or negative sign, so plus-1 and minus-1 have the same absolute value. In the case of sodium chloride, for example, the sodium ion has a charge of plus-1, while the chloride ion has a charge of minus-1. Multiplying the absolute value of these two numbers gives 1 as a result.

3. Multiply the Madelung constant by the combined absolute value of the ions. In the case of sodium chloride, 1 x 1.74756 = 1.74756.

4. Multiply the result by Avogadro's number, 6.022 x 10^23 mol^-1. For example, the figure for sodium chloride is 1.052 x 10^24 mol^-1.

5. Multiply this result by the square of the fundamental charge, 1.602 x 10^-19 Coulombs. For example: (1.602 x 10^-19)^2 = 2.566 x 10^-38; then, (2.566 x 10^-38) x (1.052 x 10^24) = 2.699 x 10^-14 Coulombs squared / mole.

6. Divide by 4π. (Remember, you can be retrieve pi by punching the π button on your calculator.) In the case of sodium chloride, your equation is (2.699 x 10^-14) / 4π = 2.148 x 10^-15 Coulombs squared / mole.

7. Divide by the vacuum permittivity constant, a standard constant in physics of 8.854 x 10^-12 Coulombs squared / Newton meters squared. In the case of sodium chloride, (2.148 x 10^-15) / (8.854 x 10^-12) = 2.426 x 10^-4 Newton meters squared / mole, since the Coulombs squared in numerator and denominator cancel each other.

8. Find the atomic radius for each element in your ionic compound (the University of Cal-Santa Barbara, listed in resources, maintains the information online). Add the figures together. Use the radii corresponding to the charge on your ion. In the case of sodium chloride, sodium has a radius of 0.99 angstroms while the value for chlorine is 1.81 angstroms. An angstrom is 1 x 10^-10 meters.

9. Divide the answer from step 7 by the distance between the nuclei of the two atoms from step 8.Sodium chloride, for example: (2.426 x 10^-4 Newton meters squared) / (2.8 x 10^-10 meters) = 866400 Newton meters / mole. A Newton meter is equivalent to a joule, so you now have 866400 joules / mole.

10. Divide this result by 1000 to convert it to kilojoules. In the case of sodium chloride: 866400 / 1000 = 866.4 kJ / mole.

11. Divide 0.3 angstroms by the nuclei distance obtained in step 8, and subtract that result from 1. Continuing the example: 0.3 / 2.8 = 0.107; then, 1 - 0.107 = 0.893. The 0.3 angstroms is a constant taking into account the repulsion of electron shells by neighboring ions, and repulsion is a necessary consideration when calculating lattice energy. The constant depends on the crystal, but 0.3 is a very good approximation for most crystals.

12. Multiply the result from step 11 by the result from step 10 for the approximate lattice energy of your compound. Sodium chloride's figure, then, is: 0.893 x 866.4 = 773 kJ/mol.

## Tips & Warnings

• Your answer is an approximate value and should not be considered exact. Nonetheless, the Born-Meyer values have been found to agree closely with experimental data, and this is usually a good approximation for most purposes.
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