Problem 131 - Chemistry The Central Science In Si Units

Question

One method to synthesize ionic solids is by the heating of two reactants at high temperatures. Consider the reaction of \(\mathrm{MgO}\) with \(\mathrm{TiO}_{2}\), to form \(\mathrm{MgTiO}_{3}\). Determine the amount of each of the two reactants to prepare \(3.250 \mathrm{~g} \mathrm{MgTiO}_{3}\), assuming the reaction goes to completion. (a) Write a balanced chemical reaction. (b) Calculate the formula weight of \(\mathrm{MgTiO}_{3}\). (c) Determine the moles of \(\mathrm{MgTiO}_{3}\). (d) Determine moles and mass (g) of MgO required. (e) Determine moles and mass (g) of \(\mathrm{TiO}_{2}\) required.

Step-by-Step Solution

Answer

To synthesize 3.250 g of \(\mathrm{MgTiO}_3\), use approximately 1.089 g of \(\mathrm{MgO}\) and 2.156 g of \(\mathrm{TiO}_2\).

Step 1: Write the Balanced Chemical Equation

First, we need to write the chemical equation for the reaction between magnesium oxide (\(\mathrm{MgO}\)) and titanium dioxide (\(\mathrm{TiO}_2\)) to form magnesium titanate (\(\mathrm{MgTiO}_3\)). The balanced chemical reaction is:\[ \mathrm{MgO} + \mathrm{TiO}_2 \rightarrow \mathrm{MgTiO}_3 \]

Step 2: Calculate the Formula Weight of MgTiO3

Next, compute the molar mass of \(\mathrm{MgTiO}_3\). Using the atomic masses from the periodic table, magnesium (Mg) is approximately 24.31 g/mol, titanium (Ti) is approximately 47.87 g/mol, and oxygen (O) is approximately 16.00 g/mol. The formula weight is:\[ \text{Formula Weight} = 24.31 + 47.87 + 3 \times 16.00 = 120.38 \text{ g/mol} \]

Step 3: Determine the Moles of MgTiO3

Given the desired mass of \(\mathrm{MgTiO}_3\) is 3.250 g, we calculate the moles using its molar mass calculated above. \[ \text{Moles of } \mathrm{MgTiO}_3 = \frac{3.250 \text{ g}}{120.38 \text{ g/mol}} \approx 0.027 \text{ moles} \]

Step 4: Determine Moles and Mass of MgO Required

Since \(\mathrm{MgO}\) and \(\mathrm{TiO}_2\) react in a 1:1 molar ratio to produce \(\mathrm{MgTiO}_3\), the moles of \(\mathrm{MgO}\) required is equal to the moles of \(\mathrm{MgTiO}_3\). Thus, moles of \(\mathrm{MgO}\) needed = 0.027 moles. The molar mass of \(\mathrm{MgO}\) is 24.31 (Mg) + 16.00 (O) = 40.31 g/mol.\[ \text{Mass of } \mathrm{MgO} = 0.027 \text{ moles} \times 40.31 \text{ g/mol} \approx 1.089 \text{ g} \]

Step 5: Determine Moles and Mass of TiO2 Required

Since \(\mathrm{MgO}\) and \(\mathrm{TiO}_2\) react in a 1:1 molar ratio to produce \(\mathrm{MgTiO}_3\), the moles of \(\mathrm{TiO}_2\) required is equal to the moles of \(\mathrm{MgTiO}_3\). Thus, moles of \(\mathrm{TiO}_2\) needed = 0.027 moles.The molar mass of \(\mathrm{TiO}_2\) is 47.87 (Ti) + 2 \times 16.00 (O) = 79.87 g/mol.\[ \text{Mass of } \mathrm{TiO}_2 = 0.027 \text{ moles} \times 79.87 \text{ g/mol} \approx 2.156 \text{ g} \]

Key Concepts

Molar Mass Calculation

Finding the molar mass of a compound is critical in stoichiometry. It helps us understand the mass-to-molecule relationship necessary for reacting substances. To find the molar mass, you need the individual atomic masses of each element present, which are found in the periodic table. For the compound \( \mathrm{MgTiO}_3 \), calculate the molar mass by adding up the atomic masses of one magnesium atom, one titanium atom, and three oxygen atoms.

• Magnesium (Mg) has an atomic mass of about 24.31 g/mol.
• Titanium (Ti) has an atomic mass of about 47.87 g/mol.
• Oxygen (O), with three atoms in \( \mathrm{MgTiO}_3 \), has a collective mass of \( 3 \times 16.00 = 48.00 \) g/mol.

Adding these values gives the molar mass: \( 24.31 + 47.87 + 48.00 = 120.18 \) g/mol. This value allows you to convert between grams and moles, which is crucial for reacting appropriate amounts of substances.

Balancing Chemical Equations

Balancing chemical equations is all about ensuring mass and charge conservation. In every chemical reaction, the number of atoms for each element should be the same on both sides of the equation. For the reaction of magnesium oxide (\( \mathrm{MgO} \)) with titanium dioxide (\( \mathrm{TiO}_2 \)), forming magnesium titanate (\( \mathrm{MgTiO}_3 \)), we write the balanced chemical equation as:
\[ \mathrm{MgO} + \mathrm{TiO}_2 \rightarrow \mathrm{MgTiO}_3 \]
This is a one-to-one reaction, meaning one mole of \( \mathrm{MgO} \) reacts with one mole of \( \mathrm{TiO}_2 \) to yield one mole of \( \mathrm{MgTiO}_3 \). Balancing equations ensures the reactants perfectly convert to the products without any leftover atoms or uncreated ones.

Chemical Synthesis

Chemical synthesis involves the creation of new substances through controlled reactions. When synthesizing \( \mathrm{MgTiO}_3 \), react magnesium oxide and titanium dioxide at high temperatures. This synthesis process requires accurately measured reactants based on calculated chemical stoichiometry. Here's how:

• Know the stoichiometric ratio, which here is 1:1:1 for \( \mathrm{MgO} \), \( \mathrm{TiO}_2 \), and \( \mathrm{MgTiO}_3 \).
• Calculate moles needed of the product from desired mass.
• Convert moles of reactants back to grams using their molar masses.

This way, each step in synthesis, from measuring reactants to heating, is guided by precise calculations.

Ionic Compounds

Ionic compounds are formed from positive and negative ions attracted to each other. \( \mathrm{MgTiO}_3 \) is an ionic compound because it contains ions formed from magnesium, titanium, and oxygen. Understanding ionic compounds involves the following:

• Ionic bonds: These occur between metals (like Mg) and nonmetals (like O), holding the compound's structure due to forceful electrostatic interactions.
• High melting points: Ionic compounds tend to form crystalline structures, making them hard and having high melting points, which is why heat is needed to form \( \mathrm{MgTiO}_3 \).

Each ion contributes to the compound's property, allowing reactions, such as the synthesis of \( \mathrm{MgTiO}_3 \), to occur efficiently and effectively at high temperatures.