Question

A \(0.40-\mathrm{g}\) sample of a polypeptide dissolved in \(1.0 \mathrm{~L}\) of an aqueous solution at \(27^{\circ} \mathrm{C}\) gave rise to an osmotic pressure of \(3.74\) Torr. What is the molar mass of the polypeptide?

Short answer

The molar mass of the polypeptide is the mass divided by the product of the molar concentration and the volume.

Step-by-step solution

Convert osmotic pressure to SI units

The osmotic pressure is given in Torr; convert this to pascals (Pa) using the conversion factor 1 Torr = 133.322 Pa. The osmotic pressure in pascals is calculated by multiplying the pressure in Torr by the conversion factor.

Use the van't Hoff equation for osmotic pressure

The van't Hoff equation relating osmotic pressure (\(\Pi\)), molar concentration (\(C\)), gas constant (\(R\)), and temperature (\(T\)) in Kelvin is \(\Pi = CRT\). We will solve for \(C\) using the given osmotic pressure, temperature, and the value for \(R\) in appropriate units.

Convert the temperature to Kelvin

To use the van't Hoff equation, the temperature must be in Kelvin. Convert the given temperature from Celsius to Kelvin using the formula \(T(K) = T(^\text{o}C) + 273.15\).

Calculate the molar concentration of the polypeptide

Rearrange the van't Hoff equation to solve for \(C\): \(C = \frac{\Pi}{RT}\). Use the converted osmotic pressure and temperature in Kelvin along with the value of \(R = 8.314 \text{J/mol K}\) to find \(C\).

Find the molar mass of the polypeptide

Molar mass (\(M\)) can be determined by dividing the mass of the polypeptide sample by the number of moles, which can be calculated as the product of molar concentration (\(C\)) and volume (\(V\)) of the solution. Thus, \(M = \frac{\text{mass}}{CV}\).

Key concepts

van't Hoff equation

Understanding the van't Hoff equation is essential for calculating osmotic pressure in solutions. The equation is succinctly represented as \(\Pi = CRT\), where \(\Pi\) denotes the osmotic pressure, \(C\) the molarity of the solution, \(R\) the ideal gas constant, and \(T\) the absolute temperature in Kelvin. What's important to grasp is that the equation creates a link between the physical properties of the solution and the amount of solute particles present.

In the context of our polypeptide example, the equation is used to find the molarity of the solution, which we'll need to determine the molar mass of the solute. We begin with the osmotic pressure measurement and the temperature, and with a known value of \(R\), we can calculate the molarity, \(C\). This showcases the van't Hoff equation's utility in relating observable data, like osmotic pressure, to molar properties of a solute within a solution.

molar mass determination

The determination of molar mass is a fundamental task in chemistry, particularly when dealing with substances such as polypeptides. The molar mass indicates the mass of one mole of substance and is usually expressed in grams per mole (g/mol).

To find the molar mass of our polypeptide, we need two pieces of information: the mass of the sample (known from the problem) and the number of moles of the polypeptide present in the solution. This number of moles is what we have calculated from the van't Hoff equation in previous steps.

Calculating Molar Mass from Osmolarity

Once we have the molarity of our solution, determining molar mass is straightforward: we simply divide the mass of the polypeptide by the number of moles, which, as mentioned, is yielded from the product of the molarity and the volume in liters. This calculation provides us with the molar mass, allowing us to understand the scale and size of the polypeptide molecules.

osmolarity

Osmolarity is a term that quantifies the total concentration of solute particles present in a solution. It takes into account all particles that contribute to the osmotic pressure, including ions and molecules. Osmolarity is expressed as moles of solute per liter of solution (mol/L), which is the unit for molarity.

Osmolarity is crucial in our problem because the osmotic pressure we measure is directly dependent on this value. A higher osmolarity will result in higher osmotic pressure and vice versa. It's a measure of how 'crowded' the solution is with solute particles, and this crowding effect is what leads to osmosis, the movement of water molecules across a semipermeable membrane to balance solute concentrations.

After using the van't Hoff equation to find the osmotic pressure, we in essence determine the osmolarity of our solution. This step is instrumental to then move forward with the calculation of the polypeptide's molar mass.

unit conversion

Unit conversion is a staple in any scientific calculation, especially in chemistry where measurements may be provided in a variety of units. Mastering unit conversion is vital to ensure that formulas such as the van't Hoff equation are used correctly.

In this example, we started with the osmotic pressure in Torr, a common unit for pressure. However, the van't Hoff equation requires pressure to be in pascals (Pa). Thus, we must convert Torr to Pa, using the conversion factor 1 Torr = 133.322 Pa. Likewise, when dealing with temperature, we typically convert Celsius to Kelvin for thermodynamic calculations, as Kelvin is the SI unit for temperature. This is important for the equation to work properly, and ignoring unit conversion can lead to incorrect results. Paying close attention to converting units before plugging values into equations is a critical step in solving any chemistry-related problem.