Question

The solution containing \(4.0 \mathrm{~g}\) of \(\mathrm{PVC}\) in \(1 \mathrm{~L}\) of dioxane was found to have osmotic pressure of \(0.006\) atm at \(300 \mathrm{~K}\). The molecular mass of the polymer PVC is (a) 16,420 (b) 1642 (c) \(1,64,200\) (d) 4105

Short answer

The molecular mass of the polymer PVC is 16,420 g/mol.

Step-by-step solution

Identify the formula to use

To find the molar mass of the polymer, we can use the van't Hoff equation for osmotic pressure \( \Pi = \frac{n}{V}RT \) where \( \Pi \) is the osmotic pressure, \( n \) is the number of moles of solute, \( V \) is the volume of the solution, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Calculate the number of moles of solute (n)

The number of moles (\(n\)) can be determined by \( n = \frac{m}{M} \) where \( m \) is the mass of the solute, and \( M \) is the molar mass of the solvent. Here, mass \( m \) is given as 4.0 g, and \( M \) is what we are trying to find.

Rearrange the van't Hoff equation to solve for M

Let's rewrite the van't Hoff equation to solve for \( M \): \( \Pi = \frac{m}{MV}RT \). We can solve for \( M \) by rearranging the formula: \( M = \frac{mRT}{\Pi V} \) where \( m = 4.0 \, \text{g} \), \( R = 0.0821 \, \text{L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1} \), \( T = 300 \, \text{K} \) and \( V = 1 \, \text{L} \).

Calculate the molar mass of PVC

Substitute the known values into the rearranged equation to calculate \( M \): \( M = \frac{4.0 \, \text{g} \cdot 0.0821 \, \text{L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1} \cdot 300 \, \text{K}}{0.006 \, \text{atm} \cdot 1 \, \text{L}} \).

Perform the calculation

Calculate the molar mass \( M \) by performing the multiplication and division: \( M = \frac{4.0 \cdot 0.0821 \cdot 300}{0.006} \).

Final Answer

After performing the calculation, we find that \( M \) equals 16420 g/mol, which corresponds to option (a).

Key concepts

van't Hoff Equation

The van't Hoff equation is one of the fundamental tools in physical chemistry, providing insights into osmotic pressure – a property essential to understanding solutions. It is expressed as:
\[ \Pi = \frac{n}{V}RT \]
Here, \( \Pi \) denotes the osmotic pressure, \( n \) is the number of moles of solute, \( V \) is the volume of the solution in liters, \( R \) is the ideal gas constant, and \( T \) is the absolute temperature in Kelvin.

Physical chemistry problems often require the manipulation of this formula to determine unknown variables. For instance, if we're given osmotic pressure, temperature, and volume, we can rearrange the equation to solve for the number of moles. From there, by knowing the total mass of solute, we can also calculate the molar mass.

Real-world applications of the van't Hoff equation are vast and include investigating biological systems like cells, where osmotic pressure must be regulated finely, or designing industrial processes that involve osmosis, such as desalination plants.

Molar Mass Determination

Determination of molar mass is a critical process in chemistry, particularly for substances with large complex molecules like polymers. Calculating the molar mass involves dividing the mass of substance by the number of moles contained in that mass:
\[ M = \frac{m}{n} \]
In the described exercise, we leveraged the van't Hoff equation to ascertain the molar mass of polyvinyl chloride (PVC). The osmotic pressure grants us the ability to compute the number of moles, especially when the mass (m) of the solute is known and other variables like temperature (T) and the volume (V) of the solvent are provided.

This method is particularly useful for substances that may not readily form a gas or do not easily dissolve, where traditional methods such as vapor density determination or titration may not be applicable.

Physical Chemistry Problems

Tackling physical chemistry problems requires a robust understanding of the underlying principles and a strategic approach to applying formulas and constants. For example, in osmotic pressure calculations, precision is paramount; overlooking units or misinterpreting the temperature can lead to errors. In our solution, careful unit conversion and systematic arrangement of the van't Hoff equation made the process of finding the molar mass of PVC straightforward.

Physical chemistry problems often involve multiple steps and the conversion of units, which is why breaking down the problem into smaller, manageable components—as we see in the determination of molar mass—is a valuable technique for solving complex scenarios.

Polyvinyl Chloride (PVC) Properties

Polyvinyl chloride, commonly known as PVC, is a solid plastic material made up of repeating units of vinyl chloride. It's known for its durability, chemical resistance, and versatility. PVC properties make it suitable for a wide range of applications, from construction materials like pipes and window frames to medical devices and clothing.

In the context of our problem, understanding the chemical properties of PVC isn't required for the calculation. However, the physical parameters, such as mass and volume, directly influence the osmotic pressure in a solution and are necessary for determining the molar mass. PVC, like other polymers, has a high molecular weight, which contributes to its strength and resistance to abrasion and impacts.