Problem 42 - Chemistry and Chemical Reactivity

Question

Anthracene, a hydrocarbon obtained from coal, has an empirical formula of \(\mathrm{C}_{7} \mathrm{H}_{5} .\) To find its molecular formula, you dissolve \(0.500 \mathrm{g}\) in \(30.0 \mathrm{g}\) of benzene. The boiling point of pure benzene is \(80.10^{\circ} \mathrm{C},\) whereas the solution has a boiling point of \(80.34^{\circ} \mathrm{C} .\) What is the molecular formula of anthracene?

Step-by-Step Solution

Answer

The molecular formula of anthracene is \( \text{C}_{14}\text{H}_{10} \).

Step 1: Calculate Boiling Point Elevation

The boiling point elevation is the difference between the boiling point of the solution and the boiling point of pure benzene.\[ \Delta T_b = 80.34^{\circ}C - 80.10^{\circ}C = 0.24^{\circ}C .\]

Step 2: Identify Ebullioscopic Constant

The ebullioscopic constant, \( K_b \), for benzene is given as 2.53 K*kg/mol. This will be used to find the molality and the number of moles of the solute.

Step 3: Calculate Molality of Solution

Using the formula for boiling point elevation: \( \Delta T_b = K_b \times m \), where \( m \) is the molality.\[ m = \frac{\Delta T_b}{K_b} = \frac{0.24}{2.53} = 0.09486 \text{ mol/kg} .\]

Step 4: Determine Moles of Solute

The molality formula \( m = \frac{n_{solute}}{m_{solvent}} \) can be rearranged to find the moles of solute: \( n_{solute} = m \times m_{solvent} \).\[ n_{solute} = 0.09486 \times 0.030 = 0.0028458 \text{ mol} .\]

Step 5: Calculate Molar Mass

Knowing the grams of solute and the moles, use the formula: \( Molarity (M) = \frac{mass}{n} \).\[ M = \frac{0.500 \text{ g}}{0.0028458 \text{ mol}} \approx 175.71 \text{ g/mol} .\]

Step 6: Determine Molecular Formula

The empirical formula weight of \( \text{C}_7\text{H}_5 \) is \( 89 \text{ g/mol} \) (from \( 7 \times 12 + 5 \times 1 = 89 \)). Divide the molar mass by the empirical formula mass: \( \frac{175.71}{89} \approx 2 \). Thus, the molecular formula is twice the empirical formula, \( \text{C}_{14}\text{H}_{10} \).

Key Concepts

Molecular Formula

In chemistry, the molecular formula of a compound provides crucial information about the number and type of atoms in a molecule. It tells us the complete number of each type of atom present. To find the molecular formula, we often start with an empirical formula. The empirical formula represents the smallest whole-number ratio of atoms in a compound, but doesn't give the exact count as the molecular formula does.
For anthracene, we used the empirical formula, \( \text{C}_7\text{H}_5 \), as our starting point. By knowing the empirical formula mass and the molar mass of the substance obtained through experimental means, we can discern the molecular formula.
We calculated the molar mass of anthracene as 175.71 g/mol. Then, we compared this with the empirical formula mass (89 g/mol). By the ratio \( \frac{175.71}{89} \approx 2 \), we recognize that the molecular formula is two times the empirical formula. Therefore, anthracene's molecular formula is \( \text{C}_{14}\text{H}_{10} \). This multiplication factor is critical as it scales the empirical formula to reflect the real number of atoms in each molecule.

Ebullioscopic Constant

The ebullioscopic constant, denoted as \( K_b \), is an essential parameter in calculating the boiling point elevation of solutions. It expresses how much the boiling point of a solvent will increase for each molal concentration of solute added. This constant varies per solvent due to distinct chemical and physical properties.
For benzene, the \( K_b \) value is 2.53 K\( \cdot \)kg/mol. This means that each molal addition of a non-volatile solute raises the boiling point of benzene by 2.53 degrees Kelvin. The boiling point elevation formula \( \Delta T_b = K_b \times m \), where \( m \) is molality (moles of solute per kilogram of solvent), allows us to determine changes in boiling points due to solute addition.
By understanding \( K_b \), students can predict and calculate the effect of solutes on boiling points accurately, a critical skill in both lab settings and real-world applications.

Empirical Formula

The empirical formula plays a fundamental role in understanding compound composition, representing the simplest whole-number ratio of atoms in a compound. Unlike the molecular formula, it doesn't provide specific amounts of atoms, merely their ratio. For example, the empirical formula \( \text{C}_7\text{H}_5 \) for anthracene shows seven carbon atoms to five hydrogen atoms per unit, but not the full molecular quantity.
To derive the molecular formula from the empirical formula, additional information—such as the compound's molar mass—is required. This empirical formula is invaluable in initial chemical analysis, setting the foundation for further exploration and understanding of a compound's complete molecular structure.
By comparing the molar mass obtained via laboratory experiments to the theoretical mass from the empirical formula, we can uncover the full molecular formula. This is achieved by determining how many times the empirical formula fits into the molecular mass, thus giving a clear and complete picture of compound make-up.