Question

In a study designed to prepare new gasoline-resistant coatings, a polymer chemist dissolves \(6.053 \mathrm{~g}\) of poly(vinyl alcohol) in enough water to make \(100.0 \mathrm{~mL}\) of solution. At \(25^{\circ} \mathrm{C},\) the osmotic pressure of this solution is 0.272 atm. What is the molar mass of the polymer sample?

Short answer

The molar mass of the polymer sample is 5453 g/mol.

Step-by-step solution

Write down the formula for osmotic pressure

The formula for osmotic pressure (Pi) is \( \Pi = nRT \). Rewrite the equation to find the molar mass (M), \(\Pi = \frac{n}{V}RT\), where \(\Pi \) is the osmotic pressure in atm, \(n\) is the number of moles of solute, \(R\) is the gas constant, \(T\) is the absolute temperature, and \(V\) is the volume of the solution in liters.

Convert temperature to Kelvin

Convert the given temperature in Celsius to Kelvin. \[ T(K) = T(^{\circ}C) + 273.15 \] Substituting the value, 25^{\circ}C + 273.15 = 298.15 K.

Rearrange the equation to solve for molarity

Rearrange the osmotic pressure equation to solve for molarity (\(C\)). The equation becomes \[\Pi = CRT \]. So, \[ C = \frac{\Pi}{RT}\].

Insert known values into equation

Insert the known values into the equation to find the molarity. Given: \(\Pi = 0.272 \, \text{atm}\), and \(R = 0.0821 \frac{\text{L atm}}{\text{mol K}}\), and temperature \(T = 298.15 \text{K}\).Thus, \[C = \frac{0.272 \, \text{atm}}{0.0821 \, \text{L atm}/(\text{mol} \cdot K) \times 298.15 \, \text{K}} = 0.0111 \frac{\text{mol}}{\text{L}}=0.0111 M\]

Calculate the number of moles of solute

Find the number of moles of solute by multiplying molarity (\(C\)) by the volume of the solution (\(V\) in liters). Given: \(C = 0.0111 \frac{\text{mol}}{\text{L}}\), and \(V = 0.100 \text{L}\). So, \[ n = C \times V = 0.0111 \, \frac{\text{mol}}{\text{L}} \times 0.100 \, \text{L} = 0.00111 \, \text{mol}\]

Determine the molar mass

The molar mass (M) can be calculated by dividing the mass of the polymer by the number of moles. Given: mass (\(m\)) = 6.053 \, \text{g}\ and the number of moles (\(n\))= 0.00111 \text{mol}\.So,\[M = \frac{m}{n} = \frac{6.053 \text{g}}{0.00111 \text{mol}} = 5453 \text{g/mol}\]

Key concepts

Polymer Chemistry

Polymer Chemistry studies large molecules made up of repeating subunits, called monomers. These monomers can form a variety of structures with unique properties. For example, Poly(vinyl alcohol) is a type of polymer used in this exercise. It is water-soluble and often used in coatings because of its film-forming capabilities. Understanding the behavior and properties of these polymers helps scientists develop new materials with specific traits, such as gasoline-resistant coatings.

Molar Mass Determination

Molar mass is crucial in chemistry for converting between mass of a substance and the amount in moles. To find the molar mass, the mass of the sample divided by the number of moles is used. Here, the exercise calculates the molar mass of Poly(vinyl alcohol) by using osmotic pressure. First, we find molarity from osmotic pressure data. Then, by multiplying the molarity with the volume, we get the moles of the polymer. Finally, dividing the given mass by the number of moles gives us the molar mass.

Gas Constant

The gas constant (\(R\)) is a fundamental constant in chemistry. It appears in various equations, including the ideal gas law and the osmotic pressure formula used here. Its value, 0.0821 L atm / (mol K), helps relate the pressure, volume, and temperature of a gas or solution. By using this constant, we can accurately determine relationships among these variables, as shown in the step-by-step solution where it helps us find the molarity and subsequently the molar mass.