Problem 75 - Chemistry The Molecular Science

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 \(0.500 \mathrm{~g}\) anthracene in \(30.0 \mathrm{~g}\) henzene. The boiling point of the solution is \(80.34{ }^{\circ} \mathrm{C}\). Determine the molar mass and molecular formula of anthracene.

Step-by-Step Solution

Answer

The molar mass of anthracene is 175.6 g/mol and its molecular formula is \( \text{C}_{14} \text{H}_{10} \).

Step 1: Analysis and Utilizing Freezing/Boiling Point Elevation

To determine the molar mass of anthracene, we will use the concept of boiling point elevation. The formula for boiling point elevation is \( \Delta T_b = i \cdot K_b \cdot m \), where \( \Delta T_b \) is the change in boiling point, \( i \) is the van't Hoff factor (which is 1 for non-electrolytes like anthracene), \( K_b \) is the ebullioscopic constant for benzene (\(2.53 \, \text{°C kg/mol}\)), and \( m \) is the molality of the solution.

Step 2: Calculating Change in Boiling Point

The normal boiling point of benzene is \( 80.1 \, ^{\circ} \text{C} \). Given the boiling point of the solution is \( 80.34 \, ^{\circ} \text{C} \), the boiling point elevation \( \Delta T_b \) is calculated as follows: \[ \Delta T_b = 80.34 - 80.1 = 0.24 \, ^{\circ} \text{C} \]

Step 3: Calculating Molality

Using the boiling point elevation equation, solve for molality \( m \):\[ 0.24 = 1 \cdot 2.53 \cdot m \] Solve for \( m \):\[ m = \frac{0.24}{2.53} = 0.0949 \, \text{mol/kg} \]

Step 4: Determine Moles of Solute

Convert molality into moles of anthracene using the definition of molality, which is moles of solute per kilogram of solvent. With the mass of benzene being 30.0 g (0.030 kg), the moles of anthracene \( n \) are calculated as: \[ n = 0.0949 \, \text{mol/kg} \times 0.030 \, \text{kg} = 0.002847 \, \text{mol} \]

Step 5: Determine Molar Mass

Find the molar mass \( M \) of anthracene using its given mass and calculated moles.\[ M = \frac{0.500 \, \text{g}}{0.002847 \, \text{mol}} = 175.6 \, \text{g/mol} \]

Step 6: Derive Molecular Formula

With the molar mass known, compare it to the empirical mass of \( \text{C}_7 \text{H}_5 \). The empirical formula mass is \( 12.01 \times 7 + 1.01 \times 5 = 89.11 \, \text{g/mol} \). The ratio between the molar mass and the empirical mass is \( \frac{175.6}{89.11} \approx 2 \).This implies the molecular formula is twice the empirical formula: \( \text{C}_{14} \text{H}_{10} \).

Key Concepts

Empirical Formula

The empirical formula of a compound is the simplest ratio of the elements present in it. It provides a basic idea of the relative number of atoms of each element in the compound, but not the actual numbers present in a molecule. For instance, the empirical formula for anthracene is \(\mathrm{C}_7 \mathrm{H}_5\), which means there are 7 carbon atoms for every 5 hydrogen atoms. This formula doesn't give the total number of atoms in the molecule but represents the simplest proportion between them.

The empirical formula helps in determining fundamental properties such as the type of elements present and their stoichiometric relationships. It's particularly useful in combustion analysis and can be determined through experiments and combining the molar mass information.

Molecular Mass

Molecular mass, also known as molecular weight, is a measure of the total mass of a molecule. It is calculated by adding up the atomic masses of all the atoms in a molecule's chemical formula. For anthracene, once the moles are calculated from a given mass using boiling point elevation, we use them to find the molecular mass. This value is essential for determining how many empirical formula units are present in a complete molecule.

Calculating the molecular mass involves understanding the moles of the substance. This was determined by dividing the given mass of anthracene by the moles of it, found using the concept of boiling point elevation. Thus, molecular mass plays a pivotal role in translating experimental data into molecular formulae.

Molality

Molality is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute present per kilogram of solvent. In this case, anthracene is the solute, and benzene is the solvent. The formula for molality, \( m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \), is used when calculating the molality based on the boiling point elevation of a solution.

Molality is particularly useful because it is independent of temperature changes. Unlike molarity, which depends on the volume of the solution (which can expand or contract with temperature changes), molality relies on the mass of the solvent. This makes it a more reliable measure when conducting experiments involving temperature variations, such as boiling point elevation.

Molecular Formula Determination

The molecular formula provides the actual number of each type of atom in a molecule. To determine the molecular formula from the empirical formula, you first need to calculate the molecular mass and compare it with the empirical formula mass. For anthracene, the empirical formula is \(\text{C}_7 \text{H}_5\). The empirical mass is calculated by summing the atomic masses of all atoms in \(\text{C}_7 \text{H}_5\) which amounts to \(89.11 \, \text{g/mol}\).

With the experimental molecular mass calculated at \(175.6 \, \text{g/mol}\), we find the number of empirical units in the molecule by dividing the molecular mass by the empirical formula mass: \( \frac{175.6}{89.11} \approx 2 \). This indicates the molecular formula is two times the empirical formula, resulting in \(\text{C}_{14} \text{H}_{10}\). This process reveals the full chemical identity of the compound by multiplying each atom count in the empirical formula by this factor.