Problem 196 - Chemistry Matter and Change

Question

Mothballs Naphthalene, commonly found in mothballs, is composed of 93.7\(\%\) carbon and 6.3\(\%\) hydrogen. The molar mass of naphthalene is 128 \(\mathrm{g} / \mathrm{mol}\) . Determine the empirical and molecular formulas for naphthalene.

Step-by-Step Solution

Answer

The empirical formula is \( \text{C}_5\text{H}_4 \) and the molecular formula is \( \text{C}_{10}\text{H}_8 \).

Step 1: Calculate Moles of Each Element

Convert the percentage by mass of each element into moles. For carbon, with a molar mass of 12.01 g/mol, the moles are:\[\text{Moles of } C = \frac{93.7 \text{ g}}{12.01 \text{ g/mol}} = 7.80 \text{ moles}\]For hydrogen, with a molar mass of 1.008 g/mol, the moles are:\[\text{Moles of } H = \frac{6.3 \text{ g}}{1.008 \text{ g/mol}} = 6.25 \text{ moles}\]

Step 2: Find the Simplest Ratio of Moles

Divide each element's moles by the smallest number of moles found (6.25 in this case):\[\text{Ratio for C} = \frac{7.80}{6.25} \approx 1.25\]\[\text{Ratio for H} = \frac{6.25}{6.25} = 1\]The ratio 1.25:1 can be further simplified by multiplying through by 4 to achieve whole numbers:

Step 3: Determine Empirical Formula

Multiply the simplest ratio by 4 to clear the fraction and obtain whole numbers:\[\text{C ratio: } 1.25 \times 4 = 5\]\[\text{H ratio: } 1 \times 4 = 4\]Thus, the empirical formula is \( \text{C}_5\text{H}_4 \).

Step 4: Determine Molecular Formula

Compute the empirical formula's molar mass:\[\text{Empirical formula mass} = (5 \times 12.01) + (4 \times 1.008) = 60.05 + 4.032 = 64.082 \text{ g/mol}\]Divide the given molar mass by the empirical molar mass to find the multiplier:\[n = \frac{128}{64.082} \approx 2\]Thus, the molecular formula is two times the empirical formula: \( \text{C}_{10}\text{H}_8 \).

Key Concepts

Molar Mass Calculation

To determine the molar mass of a compound, you need to know the atomic masses of each element within the compound. These atomic masses are found on the periodic table. The process of calculating molar mass involves multiplying the atomic mass of each element by the number of times the element appears in the compound, and then summing up all these values.
To illustrate, let's use naphthalene as an example. This compound has a formula of \( \text{C}_{10}\text{H}_8 \), which means it contains 10 carbon atoms and 8 hydrogen atoms.

• The atomic mass of carbon is approximately 12.01 g/mol.
• The atomic mass of hydrogen is approximately 1.008 g/mol.

The molar mass of naphthalene is thus calculated by:

• Carbon: \( 10 \times 12.01 = 120.1 \) g/mol
• Hydrogen: \( 8 \times 1.008 = 8.064 \) g/mol
• Total molar mass: \( 120.1 + 8.064 = 128.164 \) g/mol

This confirms the given molar mass of 128 g/mol. Molar mass is a fundamental concept used in chemistry to relate the mass of a sample to the number of moles of particles it contains.

Empirical Formula Determination

The empirical formula of a compound represents the simplest whole-number ratio of the elements within it. This contrasts with the molecular formula, which shows the actual number of each type of atom in a molecule. To determine an empirical formula, follow these general steps:
First, convert the percentage composition of each element to grams. Assume you have a 100 g sample, so the percentage becomes grams directly. For naphthalene:

• Carbon: 93.7 g
• Hydrogen: 6.3 g

Next, convert these masses into moles by dividing by each element's molar mass (12.01 g/mol for carbon, 1.008 g/mol for hydrogen):

• \( \text{Moles of C} = \frac{93.7}{12.01} \approx 7.80 \) moles
• \( \text{Moles of H} = \frac{6.3}{1.008} \approx 6.25 \) moles

Then, divide by the smallest number of moles to obtain a ratio:

• \( \frac{7.80}{6.25} \approx 1.25 \) for carbon
• \( \frac{6.25}{6.25} = 1 \) for hydrogen

To convert to whole numbers, multiply each ratio by 4, resulting in an empirical formula of \( \text{C}_5\text{H}_4 \). This method ensures the most accurate representation of the simplest ratio.

Molecular Formula Calculation

Once you have the empirical formula, you can determine the molecular formula by relating the empirical formula mass to the actual molar mass of the compound. If you know the molar mass of the compound, you can find a multiplier that scales the empirical formula to the molecular formula.
First, calculate the molar mass of the empirical formula. For naphthalene's empirical formula \( \text{C}_5\text{H}_4 \):

• \( 5 \times 12.01 = 60.05 \) g/mol for carbon
• \( 4 \times 1.008 = 4.032 \) g/mol for hydrogen

This results in an empirical formula mass of \( 64.082 \) g/mol. Next, divide the compound's actual molar mass by this empirical mass:

• \( n = \frac{128}{64.082} \approx 2 \)

This factor tells us to multiply the subscripts in the empirical formula by 2 to find the molecular formula: \( \text{C}_{10}\text{H}_8 \). The relationship between empirical and molecular formulas is crucial in understanding the structure and composition of chemical compounds.