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

The molar mass of a polypeptide is a fundamental concept in chemistry, particularly important in the study of biomolecules. Molar mass, essentially, is the mass of one mole of molecules or atoms of a substance, expressed in grams per mole (g/mol).

For polypeptides, which are chains of amino acids linked by peptide bonds, the molar mass can be quite large, as it is the sum of the molar masses of all the amino acids in the chain. To determine the molar mass of a polypeptide, one must know the mass of the polypeptide sample and the number of moles of the polypeptide present in the solution. If you have a certain mass of polypeptide dissolved in a known volume of solution, you can use osmotic pressure calculations to find the molarity.

Then, with the molarity (M), which is the number of moles of solute per liter of solution, you can use the formula: \[Molar\ Mass\ (Mm) = \frac{\text{mass of the polypeptide}}{\text{volume of solution} \times M}\] to calculate the molar mass. Remember, the molar mass is key in understanding the size and composition of the polypeptide.

The van't Hoff factor (symbolized as i) is crucial when discussing solutions and their properties, including osmotic pressure. It represents the number of particles into which a compound dissociates in solution. For non-electrolytes—substances that do not split into ions when dissolved in water—the van't Hoff factor is 1 because they do not dissociate. Conversely, for electrolytes, which do dissociate into ions, the factor could be greater than 1.

For instance, common table salt (NaCl), when dissolved in water, would generally have a van't Hoff factor close to 2, as it dissociates into one sodium ion (\(Na^+\)) and one chloride ion (\(Cl^-\)). However, the polypeptide in our problem does not dissociate in this manner, hence, we use a van't Hoff factor of 1 to reflect this behavior.

Understanding this factor is essential when using the osmotic pressure equation, as it directly affects the value of osmotic pressure calculated for a given solution.

The ideal gas constant, denoted as R, is a universal constant that appears in the ideal gas law equation. It relates the pressure, volume, temperature, and number of moles of gas within the parameters of an ideal gas. Its value is approximately \(0.0821\ L\cdot atm/(mol\cdot K)\).

In the context of osmotic pressure, the ideal gas constant is also used, though we are not dealing with a gas. The reason R is involved in the osmotic pressure equation is due to the similarity in the behavior of ideal gases and the way solute particles spread out in a solution, which is a reflection of the colligative properties of solutions.

Knowing the value of R is key when applying the osmotic pressure formula \(\Pi = iMRT\). The constant serves as a bridge, allowing us to connect the physical properties of the solution with the theoretical calculations required to find properties like molarity or, through further steps, the molar mass of a solute like our polypeptide.

Temperature conversion between Celsius and Kelvin is an essential skill in chemistry, as most chemical equations, including those dealing with gases or solutions, use the Kelvin scale. The Kelvin scale is an absolute temperature scale, starting at absolute zero, which is the theoretical point where particles have minimal thermal motion.

To convert from Celsius to Kelvin, you add 273.15 to the Celsius temperature. Thus, if you have a temperature of \(27^\circ C\), converting to Kelvin gives you: \[T = 27^\circ C + 273.15 = 300.15 K\].

This step is non-negotiable in calculations involving osmotic pressure because the value of the ideal gas constant (R) is based on the Kelvin scale. Performing this conversion correctly ensures that all components of the osmotic pressure equation are dimensionally consistent, leading to an accurate calculation of the system in question.