How to Calculate Force of Attraction Between Ions

By Jack Brubaker; Updated April 25, 2017
Ionic compounds such as salt are held together by Coulombic forces.

When metals and nonmetals form compounds, the metal atoms donate electrons to the nonmetal atoms. The metal atoms thereby assume positive charges due to their loss of negatively charged electrons, and the nonmetal atoms assume negative charges. Chemists refer to charged atoms as “ions.” Ions exhibit attractive forces for ions of opposite charge -- hence the adage that “opposites attract.” The force of attraction between oppositely charged ions, or of repulsion between like-charged ions, follows Coulomb’s law, expressed mathematically as F = k * q1 * q2 / d^2, where F represents the force of attraction in Newtons, q1 and q2 represents the charges of the two ions in coulombs, d represents the distance between the ions’ nuclei in meters and k is a proportionality constant of 8.99 x 10^9 Newton square meters per square coulomb.

Refer to a table of ions to determine the charges of the positive and negative ions in the compound in question. Chemical formulas, by convention, list the positive ion first. In the compound calcium bromide, or CaBr2, for example, the calcium represents the positive ion and exhibits a charge of +2. Bromine represents the negative ion and exhibits a charge of -1. Therefore, q1 = 2 and q2 = 1 in the Coulomb’s law equation.

Convert the charges on the ions to coulombs by multiplying each charge by 1.9 x 10^-19. The +2 calcium ion therefore exhibits a charge of 2 * 1.9 x 10^-19 = 3.8 x 10^-19 coulombs, and bromine exhibits a charge of 1.9 x 10^-19 coulombs.

Determine the distance between the ions by referring to a table of ionic radii. When they form solids, the ions normally sit as close to each other as possible. The distance between them is therefore determined by adding together the radii of the positive and negative ions. In the example calcium bromide, Ca2+ ions exhibit a radius of about 1.00 angstroms and Br- ions exhibit a radius of about 1.96 angstroms. The distance between their nuclei is therefore 1.00 + 1.96 = 3.96 angstroms.

Convert the distance between the ions’ nuclei to units of meters by multiplying the value in angstroms by 1 x 10^-10. Continuing the previous example, the distance of 3.96 angstroms converts to 3.96 x 10^-10 meters.

Calculate the force of attraction according to F = k * q1 * q2 / d^2. Using the previously obtained values for calcium bromide and using 8.99 x 10^9 as the value for k gives F = (8.99 x 10^9) * (3.8 x 10^-19) * (1.9 x 10^-19) / (3.96 x 10^-10)^2. Under the rules of the scientific order of operations, the squaring of the distance must be carried out first, which gives F = (8.99 x 10^9) * (3.8 x 10^-19) * (1.9 x 10^-19) / (1.57 x 10^-19). Performing the multiplication and division then gives F = 4.1 x 10^-9 Newtons. This value represents the force of attraction between the ions.

Tip

Numbers such as 1.9 x 10^-19 represent scientific notation. In this case, the number reads as “one point nine times ten to the negative nineteenth power.” You can easily enter these values into a scientific calculator using the scientific notation button, usually labeled EE.