Question

Some ionic compounds contain a mixture of different charged cations. For example, some titanium oxides contain a mixture of \(\mathrm{Ti}^{2+}\) and \(\mathrm{Ti}^{3+}\) ions. Consider a certain oxide of titanium that is \(28.31 \%\) oxygen by mass and contains a mixture of \(\mathrm{Ti}^{2+}\) and \(\mathrm{Ti}^{3+}\) ions. Determine the formula of the compound and the relative numbers of \(\mathrm{Ti}^{2+}\) and \(\mathrm{Ti}^{3+}\) ions.

Short answer

The formula of the compound is \(Ti_2(Ti^{2+})_1(Ti^{3+})_1O_4\) and the relative numbers of \(Ti^{2+}\) and \(Ti^{3+}\) ions are 1:1.

Step-by-step solution

Determine mass percentages of elements

Given that the oxide of titanium is 28.31% oxygen by mass. The percentage of Titanium in the oxide can be found by the subtraction from 100% since there are only two elements; Titanium and Oxygen. Percentage of Titanium = 100% - Mass percentage of Oxygen

Calculate the moles of Oxygen and Titanium in the compound

Assuming we have 100 grams of the compound, we can calculate the moles of Oxygen and Titanium by dividing their respective masses by their molar masses. The molar mass of Oxygen is 16.00 g/mol and the molar mass of titanium is 47.87 g/mol. Moles of Oxygen = (28.31 g O) / (16.00 g/mol O) Moles of Titanium = (100 g - 28.31 g Ti) / (47.87 g/mol Ti)

Calculate the oxygen to titanium ratio in the compound

Divide the number of moles of Oxygen by the number of moles of Titanium to get the oxygen to titanium ratio: Oxygen to Titanium ratio = Moles of Oxygen / Moles of Titanium

Calculate the ratio of Ti²⁺ and Ti³⁺ ions

We can use the system of equations to determine the ratio of x moles of Ti²⁺ ions and y moles of Ti³⁺ ions: x + y = Moles of Titanium (Equation 1, from moles of Titanium) 2x + 3y = Moles of Oxygen * Charge of Oxygen ion (Equation 2, from the charge balance) Solve this system of equations by multiplying equation 1 by 3 and subtracting equation 2. This will give us the values of x and y.

Determine the empirical formula of the compound

Write the empirical formula using the calculated ratio of Ti²⁺ and Ti³⁺ ions and their corresponding oxygen ratios. Divide all the ratios by the smallest ratio value if necessary to get whole-number subscripts. The empirical formula of the compound will be Ti_x(Ti²⁺)_yO_z where x, y, and z are the whole-number subscripts.

Determine the relative numbers of Ti²⁺ and Ti³⁺ ions

The relative numbers of Ti²⁺ and Ti³⁺ ions can be determined by comparing the values of x and y from step 4.