A 0.50 g sample of a protein is dissolved in water to give 10.0 mL of solution. The osmotic pressure is found to be 39 torr at 310 K. What is the molar mass (g/mol) of the protein?

Question

Quizplus · Accepted Answer

From the given information:
- Mass of protein = 0.50 g
- Volume of solution = 10.0 mL = 0.01 L
- Osmotic pressure (π) = 39 torr
- Temperature (T) = 310 K

First, we need to convert the osmotic pressure from torr to atm:
1 torr = 1/760 atm
Therefore, π = 39 torr * (1/760) = 0.0513 atm

Next, we can use the osmotic pressure equation to find the molar mass (M) of the protein:
π = MRT i
where R is the gas constant and i is the van 't Hoff factor (which is assumed to be 1 for a protein in water).

Substituting the given values and solving for M:
M = π / (RT)
M = (0.0513 atm) / [(0.0821 L atm mol^-1 K^-1) * (310 K) * (0.01 L)]
M = 20,688.7 g/mol

However, this value is for the molecular weight (MW) of the protein, which includes all of its non-protein components such as water molecules. To find the molar mass of the protein alone, we need to subtract the molar mass of water from MW.

The molecular formula of water is H2O, which has a molar mass of 18.015 g/mol. The number of water molecules can be calculated by dividing the mass of water by its molar mass:
Number of water molecules = (0.01 L) * (1000 mL/L) * (1 g/mL) * (1 mol/18.015 g) = 0.5547 mol

Since each protein molecule has multiple water molecules surrounding it, we need to divide the number of water molecules by Avogadro's number (6.022 × 10^23) to find the number of protein molecules:
Number of protein molecules = 0.5547 mol / (6.022 × 10^23/mol) = 9.21 × 10^-22 mol

Finally, we can divide the mass of protein by the number of protein molecules to find the molar mass of the protein alone:
Molar mass = 0.50 g / (9.21 × 10^-22 mol) = 2.41 × 10^21 g/mol

Therefore, the best choice is C, which is the closest value to our calculated molar mass of the protein alone. The other choices are all significantly different, so they can be eliminated.

A 050 G Sample of a Protein Is Dissolved in Water