Problem 96 - Chemistry and Chemical Reactivity

Question

Titanium(IV) oxide, \(\mathrm{TiO}_{2}\), is heated in hydrogen gas to give water and a new titanium oxide, \(\mathrm{Ti}_{x} \mathrm{O}_{y},\) If \(1.598 \mathrm{g}\) of \(\mathrm{TiO}_{2}\) produces \(1.438 \mathrm{g}\) of \(\mathrm{Ti}_{x} \mathrm{O}_{y},\) what is the empirical formula of the new oxide?

Step-by-Step Solution

Answer

The empirical formula of the new titanium oxide is \( \mathrm{Ti_2O} \).

Step 1: Determine the Mass of Oxygen Lost

Calculate the mass of oxygen that was removed from TiO₂ by subtracting the mass of TiₓOᵧ from the mass of TiO₂.Mass of oxygen lost = Mass of TiO₂ - Mass of TiₓOᵧ\[ 1.598 \, \text{g} - 1.438 \, \text{g} = 0.160 \, \text{g} \]

Step 2: Determine the Moles of Titanium

To find the moles of titanium, note that all the titanium from TiO₂ is in TiₓOᵧ. Use the difference in atomic masses to calculate moles.Molar mass of TiO₂ is approximately 79.9 g/mol (Ti = 47.9 g/mol, O = 16 g/mol per O).Moles of Ti in 1.598 g of TiO₂:\[ \text{Moles of Ti} = \frac{1.598}{79.9} \approx 0.0200 \text{ mol} \]

Step 3: Determine the Moles of Oxygen

Moles of oxygen lost can be found by dividing the mass of oxygen by its molar mass (16 g/mol).Moles of oxygen lost = \( \frac{0.160 \, \text{g}}{16 \, \text{g/mol}} \approx 0.0100 \, \text{mol} \)Since these moles were lost, the moles of oxygen remaining in TiₓOᵧ are:\( 0.0200 \, \text{mol} - 0.0100 \, \text{mol} = 0.0100 \, \text{mol} \)

Step 4: Calculate the Empirical Formula

The moles ratio of Ti to O in TiₓOᵧ is 2:1 derived from Step 2 and Step 3. \( 0.0200 \, \text{mol} : 0.0100 \, \text{mol} \approx 2:1 \).Thus, the empirical formula of the new oxide is \( \mathrm{Ti_2O} \).

Key Concepts

Understanding Titanium Oxide

Titanium oxide, known chemically as \( \mathrm{TiO}_2 \), is a compound that plays a significant role in various chemical reactions. It is composed of one titanium atom and two oxygen atoms. This oxide is known for its stability and high refractive index, making it valuable in applications like pigments, sunscreens, and even as a photocatalyst. When heating \( \mathrm{TiO}_2 \) in hydrogen gas, we can influence its crystalline structure or even change its stoichiometry. This process allows us to explore different compounds of titanium and oxygen, leading to the formation of new titanium oxides that can exhibit unique properties.

The Role of Stoichiometry in Reactions

Stoichiometry is fundamental when analyzing chemical reactions. It allows us to calculate the quantities of reactants and products involved, based on their molar relationships derived from balanced chemical equations. When \( \mathrm{TiO}_2 \) reacts to form a new titanium oxide, stoichiometric calculations help us to understand how much of each element is involved. By knowing the initial and final masses of compounds, and using molar masses—like in our exercise where \( 1.598 \, \mathrm{g} \) of \( \mathrm{TiO}_2 \) produces \( 1.438 \, \mathrm{g} \) of new oxide—we determine how the elements combine to form products. This accurate calculation is crucial in finding ratios and thus deducing empirical formulas, like \( \mathrm{Ti_2O} \), in the given reaction.

Understanding Chemical Reactions

Chemical reactions involve the transformation of substances through breaking and forming of bonds. In our example, \( \mathrm{TiO}_2 \) is heated in the presence of hydrogen gas. As a result, oxygen is removed from the titanium dioxide, forming a new compound and generating water as a by-product. In stoichiometric changes such as these, it's important to remember the law of conservation of mass—unless we lose products to the surroundings, the mass of the reactants will equal the mass of the products. Thus, knowing the amounts and identities of reactants allows us to predict the products and calculate transformations fully. This understanding helps us not only in laboratory settings but also in fields such as material science, where new compounds are synthesized for specific properties.