Final answer:
The molar mass of the protein is calculated using the osmotic pressure equation II = MRT, by rearranging it to solve for molar concentration first, and then using the mass of the protein and the calculated moles to find the molar mass which is 3478 g/mol.
Step-by-step explanation:
To determine the molar mass of the protein from the given osmotic pressure, you can use the formula for osmotic pressure (II) which is equal to the product of the molarity (M) of the solution, the gas constant (R), and the temperature (T), given as II = MRT.
First, convert the osmotic pressure from mmHg to atm:
2.82 mmHg * (1 atm / 760 mmHg) = 0.00371 atm
Next, convert the volume from mL to L:
50.0 mL * (1 L / 1000 mL) = 0.0500 L
Now you can calculate the molarity (M) which is moles of solute per liter of solution using the provided osmotic pressure, the gas constant in atm (R = 0.0821 L.atm/(K.mol)), and the given temperature in Kelvin (T = 25 + 273.15 = 298.15 K).
The formula restructuring to solve for molarity (M) is:
M = II / (RT)
M = 0.00371 atm / (0.0821 L.atm/(K.mol) * 298.15 K) = 1.53 x 10^-4 mol/L
Now, calculate the moles of the protein in the solution:
Moles of protein = Molarity (M) x Volume (L)
Moles of protein = 1.53 x 10^-4 mol/L * 0.0500 L = 7.65 x 10^-6 mol
Finally, find the molar mass of the protein:
Molar mass = Mass of the protein (g) / Moles of protein
Molar mass = 0.0266 g / 7.65 x 10^-6 mol = 3478 g/mol