Molecular mass from osmotic pressure: Pepsin is an enzyme present in the human digestive tract. A solution of a `0.500 gram` sample of purified pepsin in `30.0 mL` of aqueous solution exhibits an osmotic pressure of 892 torr at `27^(@)C`. Estimate the molecular mass of pepsin.
Strategy: An enzyme is a protein that acts as a biological catalyst. Pepsin catalyzes the metabolic cleavage of amino acid chain (called peptide chains) in other proteins. To determine molecular mass, we must first find the number of moles `(n)` of pepsin represented by the `0.500 g` sample. We can do this by first rearranging the equation `pi=CRT` for osmotic pressure to find the molarity of the pepsin solution and then multiplying it by the volume of the solution to obtain the number of moles of pepsin.

Explanation

Step 1. Calculate molarity of pepsin
Since `pi =CRT`, then `C=pi/(RT)`
`C=((8.92 "torr" xx(1 "atm")/(760 "torr")))/((0.082 (L "atm")/(K mol))(300 K))`
`=0.000477 mol L^(-1)`
`=4.77xx10^(-4) mol L^(-1)`
Note that we convert 8.92 torr to atmospheres to be consistent with units of R.
Step 2. Calculate moles of pepsin
`Molarity (C )=n_("pepsin")/V_(mL)xx(1000 mL)/L`
Since the volume of the solution is `30.0 mL`, the number of moles of pepsin is
`n_("pepsin")=(4.77xx10^(-4) mol L^(-1))(30.0 mL)((1L)/(1000 mL))`
`=143.1xx10^(-7) mol`
`1.43xx10^(-5) mol`
Step 3. Calculation the molar mass
Knowing both the mass and the number of moles of pepsin, we can calculate the molar mass and hence molecular mass
`"moles"_("pepsin")=("mass"_("pepsin"))/("molar mass"_("pepsin"))`
or `"molar mass"_("pepsin")=("mass"_("pepsin"))/("moles"_("pepsin"))`
`=(0.500 g)/(1.43xx10^(-5) mol)`
`=0.349xx10^(5) g mol^(-1)`
`=3.50xx10^(4) g mol^(-1)`
Thus, the molecular mass of pepsin is approximately 35,000 amu. This is typical for medium-sized proteins.
Alternatively, we can calculate the molar mass of pepsin through the equation `(2.74)`.