An ionic compound made from the metal \(M\) and the diatomic gas \(X_{2}\) has the formula \(M_{a} X_{b},\) in which \(a=1\) or 2 and \(b=1\) or \(2 .\) Use the data provided to determine the most likely values for \(a\) and \(b,\) along with the most likely charges for each of the ions in the ionic compound. Data (in units of \(\mathrm{kJ} / \mathrm{mol} )\) Successive ionization energies of \(\mathrm{M} : 480 ., 4750\) . Successive electron affinity values for \(\mathrm{X} :-175,920\) . Enthalpy of sublimation for \(\mathrm{M}(s) \rightarrow \mathrm{M}(g) : 110\) . Lattice energy for MX $\left(\mathrm{M}^{+} \text { and } \mathrm{X}^{-}\right) :-1200$ . Lattice energy for $\mathrm{MX}_{2}\left(\mathrm{M}^{2+} \text { and } \mathrm{X}^{-}\right) :-3500$ . Lattice energy for $\mathrm{M}_{2} \mathrm{X}\left(\mathrm{M}^{+} \text { and } \mathrm{X}^{2-}\right) :-3600$ . Lattice energy for $\mathrm{MX}\left(\mathrm{M}^{2+} \text { and } \mathrm{X}^{2-}\right) :-4800$ .

We need to find the most likely values for a and b in the formula \(M_{a}X_{b}\). Since a can be 1 or 2 and b can also be 1 or 2, there are four possible combinations: 1. \(M_{1}X_{1}\): \(M^{+}\) and \(X^{-}\) 2. \(M_{1}X_{2}\): \(M^{+}\) and \(X^{2-}\) 3. \(M_{2}X_{1}\): \(M^{2+}\) and \(X^{-}\) 4. \(M_{2}X_{2}\): \(M^{2+}\) and \(X^{2-}\)

For each combination, we need to calculate the energy required for sublimation, ionization, and electron affinity. Then, we'll compare this value to the corresponding lattice energy. Combination 1: \(M_{1}X_{1}\) - Ionization energy: 480 kJ/mol - Electron affinity: -175 kJ/mol - Enthalpy of sublimation: 110 kJ/mol - Overall energy required: 480 + 110 - 175 = 415 kJ/mol - Lattice energy: -1200 kJ/mol Combination 2: \(M_{1}X_{2}\) - Ionization energy: 480 kJ/mol - Electron affinity: -(175+920) kJ/mol - Enthalpy of sublimation: 110 kJ/mol - Overall energy required: 480 + 110 - (175+920)= -505 kJ/mol - Lattice energy: -3600 kJ/mol Combination 3: \(M_{2}X_{1}\) - Ionization energy: 480 + 4750 kJ/mol - Electron affinity: -175 kJ/mol - Enthalpy of sublimation: 2 * 110 kJ/mol - Overall energy required: (480 + 4750) + (2 * 110) - 175 = 5275 kJ/mol - Lattice energy: -3500 kJ/mol Combination 4: \(M_{2}X_{2}\) - Ionization energy: 480 + 4750 kJ/mol - Electron affinity: -(175+920) kJ/mol - Enthalpy of sublimation: 2 * 110 kJ/mol - Overall energy required: (480 + 4750) + (2 * 110) - (175+920) = 4355 kJ/mol - Lattice energy: -4800 kJ/mol

Now we can compare the overall energy required for each combination to its corresponding lattice energy: Combination 1: 415 kJ/mol required, -1200 kJ/mol lattice energy Combination 2: -505 kJ/mol required, -3600 kJ/mol lattice energy Combination 3: 5275 kJ/mol required, -3500 kJ/mol lattice energy Combination 4: 4355 kJ/mol required, -4800 kJ/mol lattice energy

The most favorable combination is the one with the most significant difference between the overall energy required and the lattice energy. In this case, it's combination 2, \(M_{1}X_{2}\), with -505 kJ/mol required and a lattice energy of -3600 kJ/mol. Summary: The most likely formula for the ionic compound is \(M_{1}X_{2}\), which has the metal ion M with a charge of +1 and the diatomic gas ion X with a charge of -2 (\(M^{+}\) and \(X^{2-}\)).