Answer
mass percentage of $TiO_{2}$ = 82.3 %
Work Step by Step
The strategy: 1- find the mass of $TiO_{2}$ that produce 35.4 g of titanium tetracloride
2- Find mass percentage of $TiO_{2}$ in the ore.
$TiO_{2}$ (s) + C(s) + 2 $Cl_{2}$ (g) ${\longrightarrow} $ $TiCl_{4}$ + C$O_{2}$ (g)
Step 1: to find the mass of $TiO_{2}$ when we know the amount 35.4 g of $TiCl_{4}$ :
g $TiCl_{4}$ ${\rightarrow}$ mol $TiCl_{4}$ ${\rightarrow}$ mol $TiO_{2}$${\rightarrow}$ g $TiO_{2}$
The conversion factors are:
$\frac{1 mol (TiCl_{4})}{Molar mass (TiCl_{4})}$ to convert to moles $TiCl_{4}$, where Molar mass $TiCl_{4}$:
Molar mass $TiCl_{4}$= 1 x 48 + 4 x 35.5 = 190.0 g
$\frac{1 mol (TiO_{2})}{1 mol (TiCl_{4})}$, to convert the moles of $TiCl_{4}$ to mol of $TiO_{2}$.
$\frac{Molar mass(TiO_{2})}{1 mol(TiO_{2})}$ to convert to g of $TiO_{2}$, where the molar mass of $TiO_{2}$ is:
Molar mass $TiO_{2}$ = 1 x 48 + 2 x 16 = 80.0 g
Now we use the conversion factors to find the mas of $TiO_{2}$:
35.4 g $TiCl_{4}$ x $\frac{1 mol (TiCl_{4})}{Molar mass (TiCl_{4})}$ x $\frac{1 mol (TiO_{2})}{1 mol (TiCl_{4})}$ x $\frac{Molar mass(TiO_{2})}{1 mol(TiO_{2})}$=
35.4 g $TiCl_{4}$ x $\frac{1 mol (TiCl_{4})}{190 g (TiCl_{4})}$ x $\frac{1 mol (TiO_{2})}{1 mol (TiCl_{4})}$ x $\frac{80 g(TiO_{2})}{1 mol(TiO_{2})}$= 14.9 g $TiO_{2}$.
So the mass of $TiO_{2}$ is 14.9 g
Step 2 - Find the mass percentage of $TiO_{2}$ in ore.
mass percentage of $TiO_{2}$ = $\frac{mass _(TiO_{2})}{mass_(ore)}$ x 100%
mass percentage of $TiO_{2}$ = $\frac{14.9 g}{18.1}$ x 100%
mass percentage of $TiO_{2}$ = 82.3 %
