Question

An ionic substance of formula MX has a lattice energy of \(6 \times 10^{3} \mathrm{k} / \mathrm{mol} .\) Is the charge on the ion M likely to be \(1+, 2+,\) or \(3+?\) Explain.

Short answer

It is most likely that the charge on the ion M is 2+ because, among the possible charge combinations, this one yields the best match to the given lattice energy of \(6 \times 10^3 k / mol\).

Step-by-step solution

Consider the proportional factor of the lattice energy

The formula for lattice energy is given by: \[E_{L} \propto \frac{Q_1 Q_2}{r}\] where \(E_{L}\) is the lattice energy, \(Q_1\) and \(Q_2\) are the charges of the two ions, and \(r\) is the distance between the ions.

Determine the possible charge combinations for MX

We need to consider the possible combinations of charges for the M and X ions in the compound MX. According to the question, M can be 1+, 2+, or 3+. Since we have assumed that the formula for the substance is MX, the charge on X must be equal but opposite in sign to the charge on M. Therefore, the possible combinations are: 1. M(1+) and X(1-) 2. M(2+) and X(2-) 3. M(3+) and X(3-)

Find the proportional factor of the lattice energy for each charge combination

Now, we need to calculate the proportional factor of the lattice energy for each charge combination: 1. For M(1+) and X(1-): \(\frac{(1)(-1)}{r} = -\frac{1}{r}\) 2. For M(2+) and X(2-): \(\frac{(2)(-2)}{r} = -\frac{4}{r}\) 3. For M(3+) and X(3-): \(\frac{(3)(-3)}{r} = -\frac{9}{r}\)

Compare the proportional factors to the given lattice energy

The given lattice energy is \(6 \times 10^3 k / mol\). Since we know lattice energy is directly proportional to the product of the charges of the ions, we can compare the proportional factors we calculated in step 3 to the given lattice energy. Considering the proportional factors calculated in Step 3, we find that: 1. For M(1+) and X(1-): Proportional factor is -1. Thus, the lattice energy is too high for this combination. 2. For M(2+) and X(2-): Proportional factor is -4. It is closer to the given lattice energy. Hence, it is more likely. 3. For M(3+) and X(3-): Proportional factor is -9. In this case, the lattice energy seems to be too low compared to the given value. Based on this comparison:

Conclude the charge on the ion M

It is most likely that the charge on the ion M is 2+ because, among the possible charge combinations, this one yields the best match to the given lattice energy of \(6 \times 10^3 k / mol\).

Key concepts

Ionic Bonding

Ionic bonding is a type of chemical bond that involves the electrostatic attraction between oppositely charged ions. These ions are atoms that have lost or gained one or more electrons, resulting in a net positive or negative charge. This attraction forms when one atom donates electrons to another atom, creating ions.

For instance, when sodium (Na) donates an electron to chlorine (Cl), they form the ions Na⁺ and Cl⁻. The opposite charges attract, resulting in the ionic bond that forms the compound NaCl, or common table salt. Ionic bonds are typically formed between metals and non-metals because metals tend to lose electrons easily, whereas non-metals tend to gain electrons.

The strength of ionic bonding is indicated by the lattice energy, which is the energy required to separate one mole of a solid ionic compound into its gaseous ions. While the specific strength of ionic bonding in each compound can vary, the general principle remains the same: greater charges and smaller distances between ions lead to stronger ionic bonds and thus greater lattice energy.

Charge of Ions

The charge of an ion is essentially the difference between the number of protons (positively charged particles) and electrons (negatively charged particles) in an atom. When an atom loses electrons, it becomes positively charged, while gaining electrons results in a negative charge.

To predict the charge of an ion in a compound, it's crucial to consider the number of electrons that an atom tends to lose or gain to achieve a stable, full outer shell like that of noble gases. For example, in the exercise, the ion M can achieve charges of 1+, 2+, or 3+ depending on how many electrons it loses:

• M(1+): Loses one electron
• M(2+): Loses two electrons
• M(3+): Loses three electrons

Conversely, the counter ion X would gain the same number of electrons as M loses, resulting in charges of 1-, 2-, or 3- respectively.

Being aware of these charge tendencies allows us to predict how different elements will behave in ionic bonding and to calculate the lattice energy, an essential factor in understanding the formation of ionic compounds.

Proportionality Factor

The proportionality factor is a concept that helps explain the relationship between the lattice energy of an ionic compound and the charges of the ions that form the compound. The formula used is:\[E_{L} \propto \frac{Q_1 Q_2}{r}\]
Here:

• \(E_{L}\): Lattice energy
• \(Q_1\) and \(Q_2\): Charges of the two ions
• \(r\): Distance between the ions

This relationship indicates that the lattice energy is directly proportional to the product of the ion charges and inversely proportional to the distance between them. In simpler terms, higher charges lead to stronger attractions and therefore more energy is required to separate them, increasing the lattice energy. Conversely, as the distance between ions increases, the attraction between them decreases, reducing the lattice energy.

In the exercise, this factor is used to determine the most probable charge on ion M by comparing the expected lattice energy based on different charge combinations against the given lattice energy. The closer the calculated value is to the actual given value, the more likely the charge combination explains the compound's structure.