Question

One molal solution of a carboxylic acid in benzene shows the elevation of boiling point of \(1.518 \mathrm{~K}\). The degree of association for dimerization of the acid in benzene is \(\left(K_{b}\right.\) for benzene \(\left.=2.53 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right):\) (a) \(60 \%\) (b) \(70 \%\) (c) \(75 \%\) (d) \(80 \%\)

Short answer

The degree of association for dimerization of the acid in benzene is 80%.

Step-by-step solution

Understand the boiling point elevation

The elevation in boiling point, \(\Delta T_b\), is given, and for a solution, it is the product of the molal depression constant, \(K_b\), and the molality, \(m\), times the van't Hoff factor, \(i\): \[\Delta T_b = i \cdot K_b \cdot m\] Given \(\Delta T_b = 1.518 K\) and \(K_b = 2.53 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\), we need to find \(i\) using the degree of association for dimerization.

Determine the Van't Hoff factor, \(i\)

Since the acid dimerizes, we can write the association reaction as: \[2 \mathrm{HA} \rightleftharpoons (\mathrm{HA})_2\] Let \(\alpha\) be the degree of association. The initial number of moles is 1 (since the solution is 1 molal) and after association, the moles of HA remaining are \(1 - \alpha\) and (HA)2 formed are \(\alpha/2\). The expression for \(i\), the van't Hoff factor, which is the ratio of the final number of particles to the initial number is: \[i = \frac{1 - \alpha + \alpha/2}{1} = 1 - \frac{\alpha}{2}\]

Relate \(\Delta T_b\) with \(\alpha\) using the Van't Hoff factor

Substituting \(i\)'s expression in terms of \(\alpha\) into the boiling point elevation equation, we get: \[1.518 = (1 - \frac{\alpha}{2}) \cdot 2.53 \cdot m\] We also know that the solution being 1 molal means \(m = 1 \text{mol/kg}\).

Calculate the degree of association, \(\alpha\)

Now we solve for \(\alpha\): \[1.518 = (1 - \frac{\alpha}{2}) \cdot 2.53\] \[\frac{1.518}{2.53} = 1 - \frac{\alpha}{2}\] \[\alpha = 2(1 - \frac{1.518}{2.53})\] \[\alpha = 2 \left(1 - 0.6\right)\] \[\alpha = 0.8 = 80\%\] Thus, the degree of association for dimerization of the acid in benzene is 80%.

Key concepts

Boiling Point Elevation

When a non-volatile solute is dissolved in a solvent, the boiling point of the resulting solution is higher than that of the pure solvent. This phenomenon, known as boiling point elevation, is a fundamental concept in physical chemistry, and crucial for students preparing for competitive exams like the JEE.

The elevation of boiling point is directly proportional to the molality of the solution and is also affected by the solute's properties, as encapsulated in the van't Hoff factor. Formally, it is expressed as \( \Delta T_b = i \cdot K_b \cdot m \) where \( \Delta T_b \) is the change in boiling point, \( K_b \) is the ebullioscopic constant of the solvent, \( m \) is the molality of the solution, and \( i \) is the van't Hoff factor, indicating the number of particles the solute contributes to the solution.

Understanding this concept not only helps in solving problems related to solution chemistry but also builds a foundation for more advanced topics in thermodynamics and physical chemistry.

Van't Hoff Factor

The van't Hoff factor (i) is a dimensionless quantity that indicates the number of particles into which a compound dissociates or associates in solution. For nonelectrolytes that do not dissociate, \( i = 1 \). For electrolytes, \( i \) corresponds to the number of ions produced per formula unit of solute.

In the case of association or dimerization, such as when carboxylic acids form dimers in a non-polar solvent like benzene, \( i \) is less than 1. The calculation of \( i \) can be challenging and is especially important in competitive exams as it incorporates understanding of equilibrium and stoichiometry. In JEE physical chemistry, mastering the use of \( i \) is essential for solving problems involving colligative properties.

Dimerization

Some molecules can form stable associations with each other, a process known as dimerization. This is particularly common among carboxylic acids, where two acid molecules form a dimer with the aid of hydrogen bonds. The degree of dimerization is significant because it directly influences the van't Hoff factor, which ultimately affects the boiling point elevation of a solution.

Understanding dimerization is not only crucial for mastering colligative properties but also contributes to a deeper understanding of chemical bonding and intermolecular forces - key topics in JEE syllabus. Analysing the dimerization process also encourages students to apply their knowledge of chemical equilibrium to real-world problems in the solutions.

Colligative Properties

Colligative properties are those properties of a solution that depend on the ratio of the number of solute particles to the number of solvent molecules in a solution, and not on the nature of the chemical species present. These properties include boiling point elevation, freezing point depression, osmotic pressure, and vapour pressure lowering.

For JEE aspirants, understanding colligative properties is essential. They represent a category of properties that help students to apply theoretical knowledge of solutions to practical situations, such as in calculating the molecular mass of a solute or understanding how antifreeze works. When approaching problems regarding colligative properties, it is crucial to consider the role of the van't Hoff factor, which can be influenced by dissociation or association (like dimerization) in the solution.

Physical Chemistry for JEE

Physical chemistry forms an integral part of the JEE syllabus, challenging students with its blend of theoretical concepts and their practical applications. It encompasses topics like thermodynamics, chemical kinetics, equilibrium, and colligative properties.

Mastery of concepts such as boiling point elevation and the van't Hoff factor is not only required for excelling in the JEE but also for higher studies in chemistry and related fields. Students are often tested on their ability to apply concepts to solve intricate problems, and a clear understanding of the fundamentals can provide a significant advantage in such competitive exams. Furthermore, proficiency in physical chemistry is a stepping stone to careers in research, pharmaceuticals, and many other science-based industries.