Problem 80 - Chemistry The Molecular Science

Question

The molar mass of a polymer was determined by measuring the osmotic pressure, \(7.6 \mathrm{mmHg},\) of a benzene solution containing \(5.0 \mathrm{~g}\) of the polymer dissolved in \(1.0 \mathrm{~L}\) solution. Calculate the molar mass of the polymer. Assume a temperature of \(298.15 \mathrm{~K}\).

Step-by-Step Solution

Answer

The molar mass of the polymer is approximately 12,255 g/mol.

Step 1: Understand the Van't Hoff Equation for Osmotic Pressure

The Van't Hoff equation for osmotic pressure (\(\pi\)) is given by:\[\pi = \frac{n}{V}RT\]where \(\pi\) is the osmotic pressure, \(n\) is the number of moles of solute, \(V\) is the volume of the solution in liters, \(R\) is the universal gas constant \((0.0821\, \text{L atm K}^{-1} \text{mol}^{-1})\), and \(T\) is the temperature in Kelvin.

Step 2: Convert Osmotic Pressure to Atmospheres

Osmotic pressure is initially given in mmHg, and we must convert it to atmospheres for consistency with the gas constant. Use the conversion 1 atm = 760 mmHg:\[7.6 \text{ mmHg} = \frac{7.6}{760} \text{ atm}\]\[\approx 0.01 \text{ atm}\]

Step 3: Rearrange the Equation to Solve for the Number of Moles

Rewrite the Van't Hoff equation to solve for the number of moles \(n\):\[\pi V = nRT \\Rightarrow n = \frac{\pi V}{RT}\]

Step 4: Plug in the Known Values to Find Number of Moles

Substitute the known values into the equation:- \(\pi = 0.01 \text{ atm}\)- \(V = 1.0 \text{ L}\)- \(R = 0.0821\, \text{L atm K}^{-1} \text{mol}^{-1}\)- \(T = 298.15 \text{ K}\)\[n = \frac{(0.01 \text{ atm})(1.0 \text{ L})}{(0.0821\, \text{L atm K}^{-1} \text{mol}^{-1})(298.15 \text{ K})}\]\[\approx 4.08 \times 10^{-4} \text{ moles}\]

Step 5: Calculate Molar Mass with Mass and Moles

The molar mass \(M\) can be calculated using the mass of the polymer and the number of moles:\[M = \frac{\text{mass}}{n}\]where the mass of the polymer is 5.0 g:\[M = \frac{5.0 \text{ g}}{4.08 \times 10^{-4} \text{ moles}}\]\[\approx 12255 \text{ g/mol}\]

Key Concepts

Van't Hoff Equation

The Van’t Hoff equation is essential for understanding osmotic pressure. It allows us to calculate the pressure exerted by solute particles in a solution that has partially permeable membranes. The formula for this is \(\pi = \frac{n}{V}RT\). Here, \(\pi\) represents the osmotic pressure, \(n\) is the number of moles of solute, \(V\) is the volume of the solution (in liters), \(R\) stands for the gas constant \(0.0821\, \text{L atm K}^{-1} \text{mol}^{-1}\), and \(T\) is the temperature in Kelvin.

Using the Van’t Hoff equation lets us translate the osmotic pressure data into valuable molecular insights. It is particularly useful for polymers where other methods might not be as effective.

To ensure results are accurate, it is crucial to use consistent units of measurement. For example, convert osmotic pressure from mmHg to atm to align with the universal gas constant's units. Understanding how each component of the equation correlates helps in solving a variety of solution-based problems.

Molar Mass Calculation

Calculating molar mass becomes straightforward once you know the number of moles from the Van’t Hoff equation. Molar mass \(M\), is found using the formula \(M = \frac{\text{mass}}{n}\). Here, \(\text{mass}\) is the gram weight of the polymer and \(n\) is the number of moles calculated previously.

In this exercise, substituting the given values helps us compute the molar mass of the polymer. With an osmotic pressure of 7.6 mmHg converted to 0.01 atm, and a given mass of 5.0 g, the number of moles comes out to approximately \(4.08 \times 10^{-4}\) moles. Subsequently, dividing the mass by the number of moles yields an approximate molar mass of \(12255 \text{ g/mol}\).

Molar mass calculation is a powerful technique in polymer chemistry, allowing researchers to determine the average molecular weight of complex molecules. It provides essential information about a polymer's physical properties and its potential applications.

Polymer Chemistry

Polymer chemistry involves studying large, complex molecules formed by bonding repeating smaller units known as monomers. The determination of molecular weight or molar mass is critical for understanding how these large molecules, or polymers, will behave under different conditions.

Polymers have a wide variety of applications depending on their molecular structure and molar mass. They are used in everything from plastics to biopolymers crucial in medical applications. Knowing the molar mass helps chemists to tailor and predict the polymer's properties, such as strength, flexibility, and durability.

Within the context of this exercise, determining the molar mass of a polymer using methods like osmotic pressure provides crucial insights into its size and composition.

• Low molar mass polymers often result in flexible, easily moldable substances.
• High molar mass polymers tend to be stronger and more durable.

Polymer chemistry is a dynamic field that continues to innovate and enrich our technological landscape.