Problem 94 - Chemistry and Chemical Reactivity

Question

A 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 \(\mathrm{Ti}_{2} \mathrm{O}_{y},\) what is the empirical formula of the new oxide?

Step-by-Step Solution

Answer

The empirical formula of the new oxide is \( \text{TiO} \).

Step 1: Write the Balanced Chemical Equation

Write the reaction to see what's happening: \[ \text{TiO}_2 + \text{H}_2 \rightarrow \text{Ti}_x\text{O}_y + \text{H}_2\text{O} \]. We need the empirical formula of the new oxide \(\text{Ti}_x\text{O}_y\).

Step 2: Determine Moles of \(\text{TiO}_2\)

Calculate the moles of \(\text{TiO}_2\) using its molar mass (79.87 g/mol). \[ \text{Moles of } \text{TiO}_2 = \frac{1.598 \, \text{g}}{79.87 \, \text{g/mol}} \approx 0.0200 \, \text{mol} \].

Step 3: Determine Moles of \(\text{Ti}_x\text{O}_y\)

Using the mass of \(\text{Ti}_x\text{O}_y\) (1.438 g), and setting its mass equation in terms of \(y\) and \(x\), the molar mass would be \((47.87x + 16y) \, \text{g/mol}\). Rearrange and solve for moles: \[ \text{Moles of } \text{Ti}_x\text{O}_y \approx 0.0200 \, \text{mol} \].

Step 4: Find the Moles of Titanium and Oxygen in \(\text{Ti}_x\text{O}_y\)

Assuming conservation of moles for titanium, \(0.0200\) moles of \(\text{Ti}\) from \(\text{TiO}_2\) are mole \(0.0200\times x\) in \(\text{Ti}_x\text{O}_y\). Calculate \(y\) by deducing oxygen's mole difference.

Step 5: Empirical Formula Calculation

By logical simplification, since 0.0200 moles of Ti in \(\text{Ti}_x\text{O}_y\) suggests \(x\approx1\); \(y\) calculated from stoichiometric balance of O2 yields: presumably \(y\approx1\). Therefore empirical formula examines as \(\text{TiO}\).

Key Concepts

Stoichiometry

Stoichiometry is the art of quantifying relationships in chemical reactions. It's about finding out how much of each substance you need or can produce. This concept is fundamental to predicting yields, planning laboratory experiments, and scaling reactions for industrial purposes.
Stoichiometry works on the principle of conservation of mass, where chemical equations are used to deduce significant quantities related to reactive species.

• You identify the molar ratios of reactants and products.
• The reactant quantities are converted to moles.
• Through stoichiometry, you can determine how much product will form when a specific amount of reactant is used.

This process helps chemists figure out all sorts of useful information like how much material is required or expected as a result of a reaction.

Balanced Chemical Equation

A balanced chemical equation is essential in stoichiometry, as it ensures that the equation reflects the conservation of mass. This conservation means that the number of each type of atom is the same on both sides of the equation.
To balance a chemical equation:

• Write down the unbalanced equation.
• Count the atoms of each element on both sides.
• Adjust coefficients to get the same number of atoms for each element on both sides.

A balanced equation not only reflects correct stoichiometric ratios but also relates directly to the empirical formula derivation in exercises like the one given.
For instance, in the original exercise, we know our reaction: \[\text{TiO}_2 + \text{H}_2 \rightarrow \text{Ti}_x\text{O}_y + \text{H}_2\text{O}\] must be balanced first to explore stoichiometric relationships between reactants and products.

Molar Mass Calculation

Calculating the molar mass is a crucial step in finding the moles of a substance, a necessary action in any stoichiometry problem.
The molar mass is the mass of one mole of a given substance and can be found by summing the atomic masses of its constituent elements, as listed in the periodic table.

• For example, \[\text{Molar mass of } \text{TiO}_2 = (47.87 \, \text{g/mol for Ti}) + 2 \times (16.00 \, \text{g/mol for O}) = 79.87 \, \text{g/mol} \]

By dividing the given weight of a substance by its molar mass, we determine how many moles are present. This was used in the problem to find the moles of \(\text{TiO}_2\), essential for calculating the empirical formula of \(\text{Ti}_x\text{O}_y\).

Chemical Reactions

Chemical reactions are processes where reactants are transformed into products. They're the core of chemistry that drive transformation in matter, energy release, and synthesis of new materials.
Each reaction:

• Involves breaking initial bonds in reactants.
• Forms new bonds to create products.
• Conserves atoms, meaning the number of each type of atom in reactants has to equal that in products.

In the context of the original problem, the reduction of \(\text{TiO}_2\) when reacted with \(\text{H}_2\) illustrates a chemical reaction involving a change of state from a solid oxide to another solid with the liberation of water. Here, product formation reflected through empirical formulas offers insights into how substances transform at the atomic level.