Question

$100mg$ of a protein is dissolved in just enough water to make $10mL$ of solution. If this solution has an osmotic pressure of $13.3mm$ Hg at $25{}^\circ C$ , what is the molar mass of the protein?
$[\left( R=0.0821Latm\cdot mo{{l}^{-1}}{{K}^{-1}} \right)and\left( 760mmHg=1atm \right)]$

Answer 1

Hint : Osmotic pressure is directly proportional to the amount of solute present in a volume of a solution. Osmotic pressure is the pressure developed on the solution side due to the net flow of the solvent molecules through a semipermeable membrane.

Complete Step By Step Answer:
As we know that a solution of its own has only vapour pressure, it will develop osmotic pressure only when osmosis (osmosis always occur from a dilute solution into concentrated solution) takes place by placing it in contact with a solvent or less concentrated solution through a semipermeable membrane. Thus we can say that the osmotic pressure is the pressure required to prevent osmosis.
According to Boyle-Van’t Hoff Law, the osmotic pressure ( $\pi$ ) of a dilute solution is directly proportional to its molar concentration at constant temperature. In fact, osmotic pressure can be determined at room temperature whereas other colligative properties such as elevation in boiling point and depression in freezing point cannot be evaluated at room temperature. It is particularly useful for biomolecules and polymers with high molecular masses such as proteins because of their dilute solutions.
We have Mass of protein ${{w}_{2}}=0.1g$ , $R=0.082Latm/kmol$ , $T=298K$
And we know that $\pi =13.33mmHg=\dfrac{13.33}{760}=0.0175atm$ and volume of solution is $0.01L$
Using the formula ${{M}_{2}}=\dfrac{{{W}_{2}}RT}{\pi V}=\dfrac{0.1\times 0.082\times 298}{0.0175\times 0.01}=13963.42g/mol$
Thus the molar mass of the protein is $13963.42g/mol$.

Note :
We know that all the colligative properties are helpful in determining the observed molar mass of the solute. Out of these, osmotic pressure is the best although it is sometimes inconvenient to prepare different semi permeable membranes.