A metal oxide has a lattice energy in the range of \(3300 \mathrm{~kJ} /\) mol. Is the charge on the metal ion M likely to be \(1^{+}, 2^{+},\) or \(3^{+} ?\) Explain.

Lattice energy is the energy required to break down an ionic solid into isolated gaseous ions. This energy value can be calculated using Coulomb's law, which states that the force between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between the particles. Based on Coulomb's law, lattice energy can be expressed as: Lattice energy (U) = \( \frac{k * Q_1 * Q_2}{r} \) where Q1 and Q2 represent the charges of the ions involved, r is the distance between the ions, and k is a proportionality constant. From this equation, we can see that lattice energy is directly proportional to the product of the charges of the ions involved (i.e., metal cation and non-metal anion). Therefore, a metal ion with a higher charge will result in higher lattice energy.

The given lattice energy value is 3300 kJ/mol. It is important to note that lattice energy increases with the increase in charge on the ions involved. Comparing the possible charges, 1+, 2+, and 3+, we can analyze which charge would result in lattice energy close to the given value.

Since lattice energy is directly proportional to the product of the charges of the ions involved (Q1 * Q2), we can assume that the metal ion's charge (Q1) and the charge on the oxide ion (Q2 = 2-) are responsible for the lattice energy of 3300 kJ/mol. Comparing the possibilities for the metal ion with charges 1+, 2+, and 3+: 1. If the charge on the metal ion (Q1) is 1+, the product of charges (Q1 * Q2) would be 1 * (-2) = -2. 2. If the charge on the metal ion (Q1) is 2+, the product of charges (Q1 * Q2) would be 2 * (-2) = -4. 3. If the charge on the metal ion (Q1) is 3+, the product of charges (Q1 * Q2) would be 3 * (-2) = -6. The comparison shows that for lattice energy of 3300 kJ/mol, the higher the product of charges (Q1 * Q2), the greater the lattice energy. Therefore, we can conclude that it is most likely that the charge on the metal ion (M) would be 2+ in this case.