Problem 52 - Chemistry The Central Science In Si Units

Question

What is the molecular formula of each of the following compounds? (a) empirical formula \(\mathrm{CH}_{3} \mathrm{O},\) molar mass \(=62.0 \mathrm{~g} / \mathrm{mol}\) (b) empirical formula \(\mathrm{NH}_{2}\), molar mass \(=32.0 \mathrm{~g} / \mathrm{mol}\)

Step-by-Step Solution

Answer

(a) \(\mathrm{C}_2\mathrm{H}_6\mathrm{O}_2\); (b) \(\mathrm{N}_2\mathrm{H}_4\).

Step 1: Understand the Relationship

First, recognize that an empirical formula gives the simplest whole-number ratio of atoms in a compound, while the molecular formula shows the exact number of each type of atom in a molecule. The molecular formula can be a multiple of the empirical formula.

Step 2: Calculate Molar Mass of Empirical Formula (a)

Calculate the molar mass of the empirical formula \(\mathrm{CH}_{3} \mathrm{O}\). The atomic masses are approximately: C = 12.01 g/mol, H = 1.008 g/mol, O = 16.00 g/mol. Thus, \( \mathrm{CH}_{3} \mathrm{O} \) has a mass of \( 12.01 + (3 \times 1.008) + 16.00 = 31.025 \, \mathrm{g/mol} \).

Step 3: Determine Multiplier for Empirical Formula (a)

Find the ratio of the given molar mass to the empirical formula molar mass: \( \frac{62.0 \, \mathrm{g/mol}}{31.025 \, \mathrm{g/mol}} \approx 2 \). Therefore, the molecular formula is two times the empirical formula.

Step 4: Write the Molecular Formula (a)

Using the multiplier calculated, the molecular formula is \(\mathrm{C}_2\mathrm{H}_6\mathrm{O}_2\). Multiply each subscript in \(\mathrm{CH}_3\mathrm{O}\) by 2.

Step 5: Calculate Molar Mass of Empirical Formula (b)

Calculate the molar mass of the empirical formula \(\mathrm{NH}_2\). The atomic masses are approximately: N = 14.01 g/mol, H = 1.008 g/mol. Thus, \( \mathrm{NH}_2 \) has a mass of \( 14.01 + (2 \times 1.008) = 16.026 \, \mathrm{g/mol} \).

Step 6: Determine Multiplier for Empirical Formula (b)

Find the ratio of the given molar mass to the empirical formula molar mass: \( \frac{32.0 \, \mathrm{g/mol}}{16.026 \, \mathrm{g/mol}} \approx 2 \). Therefore, the molecular formula is two times the empirical formula.

Step 7: Write the Molecular Formula (b)

Using the multiplier calculated, the molecular formula is \(\mathrm{N}_2\mathrm{H}_4\). Multiply each subscript in \(\mathrm{NH}_2\) by 2.

Key Concepts

Empirical Formula

An empirical formula represents the simplest whole-number ratio of atoms in a compound. Imagine you have a bag of blocks of different colors. The empirical formula would tell you the simplest way to arrange these blocks to show their proportion, without considering the total number.

• For instance, the empirical formula \( \mathrm{CH}_3\mathrm{O} \) suggests that for every carbon atom, there are three hydrogen atoms and one oxygen atom.
• The empirical formula does not give actual numbers of atoms—just their simplest ratio.

This concept is crucial because it serves as a baseline or starting point for determining the more complex molecular formula. That formula provides detailed information about the exact number of each type of atom in a compound.

Molar Mass Calculation

To convert from an empirical formula to a molecular formula, it's important to understand molar mass. You can think of molar mass as the weight of a mole of molecules in grams.To find the molar mass of a compound:

• Identify the elements in the compound and find their atomic masses (usually given on the periodic table).
• Multiply each element's atomic mass by the number of atoms of that element in the empirical formula.
• Add these values together to get the empirical formula's molar mass.

For example, in \( \mathrm{CH}_3\mathrm{O} \), the molar mass is calculated as follows: the mass of one carbon (\(12.01\ \mathrm{g/mol}\)), three hydrogens (\(3 \times 1.008\ \mathrm{g/mol}\)), and one oxygen (\(16.00\ \mathrm{g/mol}\)), which adds up to \(31.025\ \mathrm{g/mol}\). This mass becomes essential when comparing it to the compound's total molar mass to find the molecular formula.

Ratio Determination

Determining the ratio between the compound's given molar mass and the empirical formula's molar mass is key to finding the molecular formula.Here's how the ratio acts as a multiplier in this process:

• Calculate the compound's experimental molar mass.
• Divide the experimental molar mass by the empirical formula molar mass to find how many times the empirical units fit into the molecular formula.

For instance, if the molar mass is \(62.0\ \mathrm{g/mol}\) for \( \mathrm{CH}_3\mathrm{O} \), you divide \(62.0\ \mathrm{g/mol}\) by \(31.025\ \mathrm{g/mol}\). The result is approximately \(2\), indicating that the molecular formula is \(2\) times the empirical formula. Hence, the molecular formula derived from \( \mathrm{CH}_3\mathrm{O} \) is \( \mathrm{C}_2\mathrm{H}_6\mathrm{O}_2 \). This step ensures the right scaling, helping us understand the actual number of atoms in the compound.