Question

A solution containing \(4.0 \mathrm{~g}\) of PVC in 2 litre of dioxane (industrial solvent) was found to have an osmotic pressure \(3.0 \times 10^{-4}\) atm at \(27^{\circ} \mathrm{C}\). The molar mass of the polymer \((\mathrm{g} / \mathrm{mol})\) will be : (a) \(1.6 \times 10^{4}\) (b) \(1.6 \times 10^{5}\) (c) \(1.6 \times 10^{3}\) (d) \(1.6 \times 10^{2}\)

Short answer

The molar mass of the polymer is approximately \(1.6 \times 10^{5}\) g/mol (Option (b)).

Step-by-step solution

Understanding the Problem

We need to calculate the molar mass (molecular weight) of PVC using the osmotic pressure of its solution. For this, we can use the formula for osmotic pressure, which is given by van't Hoff's law: \( \Pi = cRT \), where \( \Pi \) is the osmotic pressure, \( c \) is the molar concentration of the solution, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Convert the Temperature to Kelvin

Convert the temperature from Celsius to Kelvin using the formula: \( T(\text{Kelvin}) = T(\text{Celsius}) + 273.15 \). In this case, \( 27^\circ\text{C} = 27 + 273.15 = 300.15\text{K} \).

Calculate the Molar Concentration

The molar concentration \( c \) is the number of moles of solute (PVC in this case) per liter of solution. Since the mass of the PVC is given, we can express molar concentration as \( c = \frac{n}{V} \), where \( n \) is the number of moles of PVC and \( V \) is the volume of the solvent in liters.

Apply van't Hoff's Law

Rearrange van't Hoff's law to solve for \( n \), the number of moles of PVC: \( n = \frac{\Pi \cdot V}{R \cdot T} \). Now we can plug in the values of \( \Pi = 3.0 \times 10^{-4} \) atm, \( V = 2 \) liters, \( R = 0.0821 \) atm\( \cdot \)L\( \cdot \)mol\(^{-1}\)\( \cdot \)K\(^{-1}\), and \( T = 300.15 \) K.

Calculate the Number of Moles of PVC

Using the rearranged van't Hoff's law, calculate the number of moles: \( n = \frac{3.0 \times 10^{-4} \cdot 2}{0.0821 \cdot 300.15} \).

Calculate the Molar Mass of PVC

The molar mass of the polymer is the mass of polymer divided by the number of moles: \( M = \frac{m}{n} \). Given that the mass (\( m \)) of PVC is 4.0 g, we can now calculate the molar mass.

Determine the Correct Answer

After calculating the molar mass using the previous steps, match the result to the closest option given to find the right answer.

Key concepts

van't Hoff's Law

Van't Hoff's law is foundational for understanding osmotic pressure in solutions. It states that the osmotic pressure (\( \text{\text{Pi}} \) of a dilute solution is directly proportional to the molar concentration (\( c \) and the absolute temperature (\( T \) of the solution, with the equation \( \text{\text{Pi}} = cRT \). In this equation, \( R \) is the ideal gas constant.

Osmotic pressure is a crucial concept when dealing with solutions, especially those containing polymers like PVC. The pressure developed across a semipermeable membrane due to a difference in the concentration of solutes on either side of the membrane is what we call osmotic pressure. This principle is used in a variety of applications, including the purification of water, biological cell behavior, and in this case, the determination of molar mass of a polymer.

Understanding van't Hoff's law enables students preparing for competitive exams like the JEE (Joint Entrance Examination) to apply principles of physical chemistry to solve real-world problems. Because it ties together concentration, temperature, and pressure, students can calculate one variable if the others are known, which is exemplified in the calculation of molar mass from osmotic pressure.

Molar Mass Determination

Molar mass determination is a key concept in chemistry, especially when dealing with solutions and colligative properties. Molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol). It's essentially the molecular weight of a compound, which can be determined through various methods, including the use of osmotic pressure.

In the exercise provided, we're given the mass of PVC and the osmotic pressure of its solution and asked to find its molar mass. The process involves calculating the number of moles of PVC using the van't Hoff's equation, which then allows us to find the molar mass by dividing the mass of the substance by the number of moles. Molar mass is a fundamental property used not only in chemistry but also in molecular biology, pharmacology, and materials science. Accurate molar mass values are essential for stoichiometry, preparation of solutions, and understanding the properties of substances.

Physical Chemistry for JEE

Physical chemistry is an integral part of the chemistry syllabus for the Joint Entrance Examination (JEE), which is an engineering entrance assessment conducted in India. It combines principles of physics and chemistry to explain how chemical systems work at a molecular and atomic level. Topics like thermodynamics, chemical kinetics, equilibrium, and colligative properties, such as osmotic pressure, are crucial for JEE aspirants to master.

The ability to apply these concepts, especially for molar mass determination via osmotic pressure calculations, can make a significant difference in the performance of students in this competitive exam. The understanding and application of van't Hoff's law, as seen in the calculation exercise, demonstrate the type of problem-solving skills that JEE candidates are expected to have.

Students preparing for JEE must emphasize conceptual clarity and problem-solving techniques in physical chemistry, as it will aid them not only in the examination but also in their future academic pursuits in fields related to chemistry and engineering.