The approximate molecular weight of the polypeptide can be calculated by using the following formula:
Molecular weight = (RT) / (πVn)
where R is the gas constant (0.0821 L·atm/(mol·K)), T is the temperature in Kelvin (31.40 + 273.15 = 304.55 K), π is the osmotic pressure in mmHg (2.012 mmHg), V is the volume of the solution in liters (274 mL = 0.274 L), and n is the number of moles of solute in the solution.
To calculate n, we need to first calculate the number of grams of the polypeptide in the solution:
1.8274 g / 1000 g/mg = 0.0018274 g
Then we can calculate the number of moles of the polypeptide:
0.0018274 g / (molecular weight / 1000 g/mol) = n
Substituting the values into the formula, we get:
Molecular weight = (0.0821 L·atm/(mol·K)) x (304.55 K) / (2.012 mmHg) x (0.274 L) x n
Solving for n, we get:
n = 0.0018274 g / (molecular weight / 1000 g/mol)
Substituting n into the formula, we get:
Molecular weight = (0.0821 L·atm/(mol·K)) x (304.55 K) / (2.012 mmHg) x (0.274 L) x (0.0018274 g / (molecular weight / 1000 g/mol))
Simplifying the equation, we can solve for the molecular weight:
Molecular weight = (0.0821 x 304.55 x 1000) / (2.012 x 0.274 x 0.0018274)
Molecular weight = 12170.2 g/mol (rounded to four significant figures)
Therefore, the approximate molecular weight of the polypeptide is 12170 g/mol.