Problem 88 - Chemistry The Molecular Nature of Matter

Question

A saturated solution is made by dissolving \(0.400 \mathrm{~g}\) of a polypeptide (a substance formed by joining together in a chainlike fashion a number of amino acids) in water to give \(1.00 \mathrm{~L}\) of solution. The solution has an osmotic pressure of 3.74 torr at \(27{ }^{\circ} \mathrm{C}\). What is the approximate molecular mass of the polypeptide?

Step-by-Step Solution

Answer

Molecular mass of the polypeptide is approximately 17565.37 g/mol.

Step 1: Recall the formula for osmotic pressure

Use the van't Hoff equation for osmotic pressure, which is given by \(\Pi = i M R T\). Here, \(\Pi\) is the osmotic pressure, \(i\) is the van't Hoff factor (which is 1 for non-electrolytes such as polypeptides), \(M\) is the molarity of the solution, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

Step 2: Convert the temperature to Kelvin

Convert the temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature. For this exercise, \(T = 27 + 273.15 = 300.15 \text{K}\).

Step 3: Calculate the molarity of the solution

The molarity \(M\) is found by dividing the mass of the solute by its molar mass \(M_w\) and the volume of the solution in liters. Since we're solving for \(M_w\), we rearrange the equation: \(M = \frac{mass}{M_w \times volume}\) or \(mass = M \times M_w \times volume\). We will use this relationship in a later step.

Step 4: Convert osmotic pressure to SI units

Convert the osmotic pressure from torr to pascals because the gas constant \(R\) is in SI units. Use the conversion 1 torr = 133.322 Pa, hence \(3.74 \text{torr} = 3.74 \times 133.322 \text{Pa}\).

Step 5: Solve for the molar mass

Now that we have \(T\), \(R\) (which is \(0.0821 \text{L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}\) or \(8.3145 \text{J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}\)), and \(\Pi\) in SI units, we can solve for \(M_w\) using the rearranged van't Hoff equation: \(mass = \Pi \times M_w \times volume/ (i \times R \times T)\).

Step 6: Insert the values and calculate the molecular mass

Insert the known values into the rearranged equation and solve for the molecular mass \(M_w\): \(0.400 \text{g} = (3.74 \times 133.322 \text{Pa}) \times M_w \times 1.00 \text{L} / (1 \times 8.3145 \text{J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1} \times 300.15 \text{K})\). Solve for \(M_w\) to find the approximate molecular mass of the polypeptide.

Key Concepts

Van't Hoff Equation

The van't Hoff equation is a relationship that can describe various phenomena involving solutions, such as osmotic pressure, which is the focus of our exercise. In its most basic form, the equation is expressed as \( \Pi = i M R T \), where \( \Pi \) represents osmotic pressure, \( i \) is the van't Hoff factor, \( M \) is the molarity of the solution, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. For non-electrolyte solutions like polypeptides, \( i = 1 \), which simplifies the equation as these molecules don't dissociate into ions when dissolved.

The practical use of the equation allows us to calculate one of these variables if the others are known. In our case, solving for the molecular mass is indirectly achieved by first determining molarity and then using the relationship between mass, molar mass, and volume of the solution.

Molarity of Solution

Understanding molarity is essential when working with solutions. It defines the concentration, indicating how many moles of a solute are present in a given volume of solution. The unit for molarity is moles per liter (M), and the formula is \( M = \frac{moles \: of \: solute}{volume \: of \: solution \: in \: liters} \).

To find the molarity of a solution, you need to know the amount of substance (in moles) and the solution volume into which the substance is dissolved. In our textbook problem, the molarity is part of the path to finding the molecular mass of the polypeptide. By knowing the mass of the dissolved substance and its volume, we can use the molarity in the van't Hoff equation to connect all the dots.

Gas Constant

The gas constant, often symbolized as \( R \), is a key piece in many equations in chemistry, including the ideal gas law and our relevant van't Hoff equation for calculating osmotic pressure. It's a bridge between the world of microscopic particles, like atoms and molecules, and the macroscopic world we measure in the lab. Its value depends on the units used for pressure, volume, and temperature.

In the van't Hoff equation, \( R \) can be commonly expressed in two units: \( 0.0821 L \cdot atm \cdot K^{-1} \cdot mol^{-1} \) if pressures are in atmospheres, or \( 8.3145 J \cdot K^{-1} \cdot mol^{-1} \) for SI units. This constant plays a crucial role as we use it alongside the other variables to solve for molecular mass from osmotic pressure measurements.

Temperature Conversion

Temperature conversion is an often-overlooked but essential step in solving many chemistry problems, including ones involving osmotic pressure. Scientific measurements require temperatures to be in Kelvin because of its absolute scale, which starts at absolute zero, unlike Celsius or Fahrenheit.

To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. This is crucial because the Kelvin temperature scale directly relates to the kinetic energy of particles. For our calculation of osmotic pressure, it's important to perform this conversion correctly to ensure that the resulting molecular mass is accurate. Remember, inaccurate temperature conversion can lead to errors in the final calculations of chemical problems.