\(1 \mathrm{~g}\) of \(\mathrm{Mg}\) atoms in the vapour phase absorbs \(50.0 \mathrm{~kJ}\) of energy. Find the composition of \(\mathrm{Mg}^{+}\) and \(\mathrm{Mg}^{2+}\) formed as a result of absorption of energy. \(\mathrm{IE}_{1}\) and \(\mathrm{IE}_{2}\) for \(\mathrm{Mg}\) are 740 and \(1450 \mathrm{~kJ} \mathrm{~mol}^{-1}\) respectively.

Understanding how to calculate moles is a fundamental skill in chemistry that allows us to quantify the amount of a substance. Moles provide a bridge between the microscopic world of atoms and the macroscopic world we can measure. When given the mass of a substance, we can use the substance's molar mass to find the number of moles. The molar mass is the weight in grams of one mole of a compound, typically listed on the periodic table or a chemical database.

To calculate moles, the formula is:
\[\text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\]
For example, if we have 1 gram of magnesium, which has a molar mass of 24.305 g/mol, we can calculate the number of moles of magnesium. This step is often the starting point for many stoichiometry problems in chemistry, as seen in the textbook problem provided.

Ionization energy is the amount of energy absorbed by an atom to remove an electron from it, resulting in a cation, a positively charged ion. When energy is absorbed, electrons in an atom can be ejected, leading to the formation of ions, a process that is quantified in units of kilojoules per mole (kJ/mol).

The first ionization energy (\(\mathrm{IE}_{1}\)) is typically lower than the second (\(\mathrm{IE}_{2}\)), as it's easier to remove the first electron than the subsequent ones. Knowing the ionization energies of elements is crucial for predicting how energy will be distributed among the ions formed during ionization.

In the exercise, energy is absorbed by magnesium atoms, and we assume it is fully used to remove electrons. By comparing the energy absorbed with the first and second ionization energies of magnesium, we can predict the formation of \(\mathrm{Mg}^{+}\) and \(\mathrm{Mg}^{2+}\) ions.

Stoichiometry involves the calculation of the reactants and products in chemical reactions. It is based on the law of conservation of mass, where the mass of the products in a chemical reaction must equal the mass of the reactants. Stoichiometry allows chemists to predict the quantities of substances consumed and produced in a given reaction.

In the context of the textbook problem, stoichiometry comes into play when determining how the absorbed energy is distributed between forming \(\mathrm{Mg}^{+}\) and \(\mathrm{Mg}^{2+}\). Once you have the number of moles of magnesium and the energy absorbed, you can use stoichiometry to calculate the number of moles of each ion formed based on their respective ionization energies, then set up equations to distribute the energy accordingly and solve for the quantities of each ion produced.

Percentage composition refers to the relative amount of each element within a chemical substance, expressed as a percentage by mass. This concept is particularly useful for determining the purity of a substance or the fraction of a particular component within a mixture.

In our problem, after calculating the moles of ions formed, the next step is to translate those amounts into percentage composition. This involves dividing the moles of each type of ion (\(\mathrm{Mg}^{+}\) and \(\mathrm{Mg}^{2+}\)) by the total moles of magnesium present in the sample, and then multiplying by 100 to obtain a percentage. This gives us a clearer understanding of the sample's composition after ionization. The percentage composition calculation highlights which ion is the predominant form after the absorption of energy and ionization has occurred.