Question

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.

Answer 1

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)`.