Question

Acrylic resins are polymeric materials used to make warm yet lightweight garments. The osmotic pressure of a solution prepared by dissolving \(47.7\) g of an acrylic resin in enough water to make \(500 . \mathrm{mL}\) of solution is \(0.325\) atm at \(25^{\circ} \mathrm{C}\). (a) What is the average molar mass of the polymer? (b) How many monomers constitute the "average" molecule? The repeating unit of this acrylic resin has the formula \(-\mathrm{CH}_{2} \mathrm{CH}(\mathrm{CN})-\). (c) What would be the vapor pressure of the solution if the vapor pressure of pure water at \(25^{\circ} \mathrm{C}\) is \(0.0313 \mathrm{~atm}\) ? (Assume that the density of the solution is \(1.00 \mathrm{~g}^{\circ} \mathrm{cm}^{-3}\).) (d) Which approach (osmometry or the lowering of vapor pressure) would you prefer for the determination of very high molar masses such as those of acrylic resins? Why?

Short answer

The average molar mass of the polymer is calculated using the osmotic pressure, temperature, and mass of the solute. Use the molar mass of the repeating unit to determine the number of monomers. Vapor pressure can be assumed to stay approximately the same due to the very small change caused by the solute. Osmometry is preferred for determining very high molar masses.

Step-by-step solution

Calculate the Molarity Using Osmotic Pressure

Use the van't Hoff equation for osmotic pressure, which is \( \Pi = MRT \) where \( \Pi \) is the osmotic pressure in atm, M is the molarity of the solution, R is the gas constant in \( \frac{L \cdot atm}{mol \cdot K} \) which is 0.0821, and T is the temperature in Kelvin. Convert the temperature from degrees Celsius to Kelvin by adding 273.15 to 25 to get 298.15 K. Then, solve for M: \( M = \frac{\Pi}{RT} = \frac{0.325}{0.0821 \cdot 298.15} \).

Calculate the Average Molar Mass of the Polymer

Find the number of moles of the solute by using the molarity and the volume of the solution. The volume should be in liters, so convert 500 mL to 0.500 L. Then, calculate the moles of solute: \( n = M \cdot V = M \cdot 0.500 \). Now, use the mass of the acrylic resin (47.7 g) to find the average molar mass of the polymer using the formula: \( \text{Molar Mass} = \frac{\text{mass}}{\text{moles}} = \frac{47.7}{n} \).

Determine the Number of Monomers per Polymer

Find the molar mass of the repeating unit, \( -CH_2CH(CN)- \) by adding the atomic weights of one carbon (12.01), two hydrogen (1.01 each), one carbon (12.01), and one nitrogen (14.01). This gives a molar mass of \( 12.01 + 2(1.01) + 12.01 + 14.01 = 40.05 \text{ g/mol} \). Now, divide the average molar mass of the polymer by the molar mass of the repeating unit to get the number of monomers: \( \text{Number of monomers} = \frac{\text{Average Molar Mass}}{\text{Molar Mass of Repeating Unit}} \).

Calculate the Vapor Pressure of the Solution

Using Raoult's Law, which states that the vapor pressure of a solution is directly proportional to the mole fraction of the solvent, calculate the mole fraction of water (solvent) and use it to find the new vapor pressure. Since the density of the solution is approximately that of pure water, for dilute solutions, we can estimate that the mole fraction of the solvent is almost 1, and thus the vapor pressure of the solution will be very close to the vapor pressure of the pure solvent. Therefore, the vapor pressure of the solution will be approximately equal to 0.0313 atm.

Discuss the Preferred Method for Molar Mass Determination

Osmometry is generally preferred over lowering vapor pressure for the determination of very high molar masses due to its greater sensitivity to large molecules. The change in vapor pressure is less noticeable with very high molar masses, making it difficult to detect and measure accurately, while osmotic pressure can be detected and measured with more sensitivity in such cases.

Key concepts

van't Hoff equation

Understanding the van't Hoff equation is crucial when learning about osmotic pressure calculations. It's an essential tool in physical chemistry for relating the osmotic pressure of a solution to its molarity, temperature, and the ideal gas constant. Specifically, the equation is expressed as \( \Pi = MRT \), where \( \Pi \) represents the osmotic pressure, \( M \) is the molarity of the solution, \( R \) is the ideal gas constant, and \( T \) is the temperature in kelvin.

In practical terms, the van't Hoff equation allows us to calculate the molarity of a solute in solution by measuring the osmotic pressure. This is particularly useful for solutions where the solute concentration is low, and direct measurement of concentration is challenging. In the context of polymers, such as the acrylic resins in the exercise, the equation helps determine the concentration of polymer in solution, which is then used to calculate properties like the average molar mass of the polymer.

average molar mass

The average molar mass is a way of expressing the overall mass of a mole of mixture, taking into account the various species present and their relative proportions. It is particularly significant in the study of polymers, where individual polymer molecules may vary in length, and hence mass. To determine the average molar mass, you can use the mass of solute and the moles of solute in solution, given by \( Molar Mass = \frac{mass}{moles} \).

For example, when studying synthetic polymers, calculating the average molar mass provides insights into the polymer's physical properties, which are influenced by the lengths of the molecular chains. Knowing the average molar mass assists in understanding materials' strength, elasticity, and other significant attributes for various applications, such as textiles or structural materials.

Raoult's Law

Raoult's Law plays a crucial role whenever we delve into the vapor pressure of solutions. This law states that the partial vapor pressure of each component of an ideal mixture of liquids is equal to the vapor pressure of the pure substance multiplied by its mole fraction in the mixture. Expressed mathematically, \( P = X_sP_s \), where \( P \) is the vapor pressure of the solution, \( X_s \) is the mole fraction of the solvent, and \( P_s \) is the vapor pressure of the pure solvent.

For the exercise involving acrylic resins, Raoult's Law provides a means to estimate the vapor pressure reduction due to the solute's presence. However, in the case of solutions with high molar mass solutes like polymers, the mole fraction of the solute is extremely low, meaning the solution’s vapor pressure will be close to that of the pure solvent.

vapor pressure of solutions

The vapor pressure of solutions is an important concept that comes into play when you're considering the evaporative characteristics of a solution. It refers to the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature within a closed system. In simpler terms, it's the pressure created by the molecules that have escaped from the liquid or solid phase to become gas.

When a solute is dissolved in a solvent, the vapor pressure usually decreases because the solute molecules occupy space at the surface of the liquid, making it harder for solvent molecules to escape into the gas phase. This property is essential for understanding distillation, refrigeration, and even determining the molar mass of compounds through methods like osmometry, which uses the colligative property of vapor pressure lowering.

polymers and monomers

Polymers are large, complex molecules composed of repeating structural units called monomers. These monomers can be thought of as the building blocks of polymers, and they are covalently bonded to form long chains. Understanding the relationship between polymers and their monomers is fundamental to material science, as it dictates the properties of the material.

For instance, in the provided exercise involving acrylic resins, each resin molecule is a polymer composed of numerous repeating units represented by the monomer \( -CH_2CH(CN)- \). By calculating the average molar mass of the polymer and knowing the molar mass of a single repeating unit, you can determine how many monomer units are in one polymer chain. This helps predict the material's behavior, including its strength, solubility, and how it interacts with other substances.