Question

The molar mass of benzoic acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\right)\) determined by measuring the freezing-point depression in benzene is twice what we would expect for the molecular formula, \(\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{2} .\) Explain this apparent anomaly.

Short answer

Benzoic acid dimerizes in benzene, doubling the molar mass.

Step-by-step solution

Review Freezing Point Depression

Freezing point depression refers to the lowering of a solvent's freezing point due to the addition of a solute. The relationship is described by the formula \[\Delta T_f = i \cdot K_f \cdot m\]where \(\Delta T_f\) is the change in freezing point, \(i\) is the van't Hoff factor, \(K_f\) is the freezing point depression constant, and \(m\) is the molality of the solution.

Determine Expected Molar Mass

The expected molar mass is calculated using the molecular formula \(\mathrm{C}_7\mathrm{H}_6\mathrm{O}_2\). The atomic masses are approximately: C (12 g/mol), H (1 g/mol), O (16 g/mol). Calculate: \[(7\times12) + (6\times1) + (2\times16) = 84 + 6 + 32 = 122 \, \text{g/mol}.\]

Analyze Observed Molar Mass

The observed molar mass based on freezing-point depression in benzene is said to be twice that of the expected molar mass. Hence, the observed molar mass is \[2\times122 = 244 \, \text{g/mol}.\]

Interpret the Anomaly

The increment in molar mass suggests the formation of a dimer of benzoic acid molecules. When benzoic acid is dissolved in a nonpolar solvent like benzene, the molecules dimerize (two molecules join to form a larger structure) due to hydrogen bonding interactions. This effectively doubles the observed molar mass, confirming the dimerization.

Key concepts

Molar Mass Calculation

When understanding how to calculate the molar mass of a compound, it's essential to begin with its molecular formula. Each element in the formula contributes a certain amount to the overall molar mass based on its atomic mass. For example, if we take benzoic acid with the formula \( \text{C}_7\text{H}_6\text{O}_2 \), each carbon atom contributes about 12 g/mol, each hydrogen contributes 1 g/mol, and each oxygen contributes 16 g/mol.
Here's how you would calculate:

• Multiply the atomic mass of carbon, 12, by the number of carbon atoms, 7, to get 84 g/mol.
• Multiply the atomic mass of hydrogen, 1, by the number of hydrogen atoms, 6, to get 6 g/mol.
• Multiply the atomic mass of oxygen, 16, by the number of oxygen atoms, 2, to get 32 g/mol.

Adding these up gives the total molar mass of 122 g/mol for benzoic acid. If any calculations deviate significantly from this expected mass, it might indicate a deeper chemical behavior such as dimerization.

Van't Hoff Factor

The van't Hoff factor \( i \) is a concept used to account for the effect of solutes on the colligative properties of solutions, like freezing point depression. It represents the number of particles a solute forms in a solution. For instance, when a solute dissociates or associates in a solution, the van't Hoff factor helps in understanding the observable changes in properties.
For most electrolytes, \( i \) is greater than 1 because the compound breaks into multiple ions. However, for nonelectrolytes like benzoic acid in benzene, \( i \) may be less than 1, due to dimerization, where two molecules associate, reducing the total number of particles in the solution. This can cause an increase in the molar mass measured by freezing point depression, as seen in the case discussed.

Dimerization

Dimerization is a process where two identical molecules react to form a dimer, which is a compound composed of two similar linked molecules. With benzoic acid in benzene, dimerization occurs due to intermolecular hydrogen bonds, which cause two benzoic acid molecules to form a single dimer unit.
This dimerization effectively doubles the molar mass of benzoic acid from its typical 122 g/mol to the observed 244 g/mol when measured through freezing point depression.

• In nonpolar solvents like benzene, molecular interactions such as hydrogen bonding are less shielded, promoting dimerization.
• The formation of this dimer is crucial in explaining why the van't Hoff factor \( i \) does not simply equal 1, even though no ions are involved.

Recognizing this behavior is essential for accurate interpretation of experimental data.

Nonpolar Solvent

Nonpolar solvents like benzene have unique interactions with solutes. These solvents do not have significant partial charges, making them ideal for dissolving nonpolar substances or facilitating specific molecular behaviors such as dimerization.
This is important because, in a nonpolar environment lacking charge, hydrogen bonding becomes a more prominent interaction between molecules.
Considered in the example of benzoic acid, without the polar influence to interrupt, benzoic acid molecules utilize hydrogen bonding more efficiently to form dimers.

• This reduction in particle number by forming dimers results in unexpectedly high molar mass measurements.
• The phenomenon influences the van't Hoff factor, leading to different impacts on colligative properties like freezing point depression.

Understanding the role of nonpolar solvents is crucial in interpreting experimental results and behaviors in solutions.