Question

200 mL of an aqueous solution of a protein contains its $1.26$g. the osmotic pressure of this solution at 300 K is found to be $2.57 \times {10^{ - 3}}$bar. The molar mass of protein will be: (R = $0.083$ ${\text{L bar mo}}{{\text{l}}^{{\text{ - 1}}}}{\text{ }}{{\text{K}}^{{\text{ - 1}}}}$)
A.61038 ${\text{g mo}}{{\text{l}}^{{\text{ - 1}}}}$
B.51022 ${\text{g mo}}{{\text{l}}^{{\text{ - 1}}}}$
C.122044 ${\text{g mo}}{{\text{l}}^{{\text{ - 1}}}}$
D.31011 ${\text{g mo}}{{\text{l}}^{{\text{ - 1}}}}$

Answer 1

The osmotic pressure is the minimum amount of pressure that needs to be applied to a solution to stop the inward flow of its pure solvent across a semipermeable membrane. A semipermeable membrane is one that allows the passage of only a certain number of substances across it.
Formula Used : $\pi {\text{v}} = \dfrac{{\text{w}}}{{\text{M}}}{\text{RT}}$
where pi is the osmotic pressure, C is the concentration of a solution which is equal to the number of moles of solute present per litre of the solution, R is universal gas constant, and T is the absolute temperature in kelvin scale.

Complete step by step answer:
The osmotic pressure of a solution can be mathematically represented by:
${{\pi = CRT}}$
The concentration of a solution can be expressed as the
${\text{C = }}\dfrac{{\text{w}}}{{{\text{vM}}}}$,
where n is the number of moles of a substance and is equal to $\dfrac{{\text{w}}}{{\text{M}}}$.
According to the question, v = 200 mL, w = $1.26$g, R = $0.083$ ${\text{L bar mo}}{{\text{l}}^{{\text{ - 1}}}}{\text{ }}{{\text{K}}^{{\text{ - 1}}}}$, T = 300 K, and ${\text{\pi }}$= $2.57 \times {10^{ - 3}}$bar.
Putting the values in the above equation we get,
$\Rightarrow {\text{M = }}\dfrac{{{w}}}{{{{\pi v }}}}{\text{RT}}$
Substituting:
$\Rightarrow {\text{M = }}\dfrac{{1.26}}{{2.57 \times {{10}^{ - 3}}{\text{ }} \times {\text{1000}}}} \times {\text{0}}{\text{.083}} \times {\text{ 300 }} \times {\text{ 200}}$ = 61038 ${\text{g mo}}{{\text{l}}^{{\text{ - 1}}}}$

So, the correct answer is option A.

Note:
The osmotic pressure of a solution is a colligative property which depends only on the number of the solute particles present in the solution and not on the identity or the quality of the solute in the solution. The other colligative properties are elevation in the boiling point of a solvent, depression in the freezing point of the solvent, and relative lowering of vapour pressure of the solvent.