Question

A solution containing 500g of a protein per litre is isotonic with a solution containing $3.42g$ of sucrose per litre. The molecular mass of protein is:
A. $5$
B. $146$
C. $34200$
D. $50000$

Answer 1

Hint:Two solutions of different substances having same osmotic pressure at same temperature are called isotonic solutions. When isotonic solutions are separated by semi-permeable membranes, no osmosis takes place. Isotonic solutions have equal molar concentrations.

Complete step by step answer:
As already mentioned, when two solutions have the same osmotic pressure across both the sides of the semi-permeable membrane that separates both the solutions, we say that they both are isotonic solutions. Osmotic pressure is the minimum pressure which needs to be applied to a solution to prevent the inward flow of its pure solvent across a semipermeable membrane. Mathematically, it can be represented as follows:
$\pi = CRT$
Where, $\pi = $ osmotic pressure of the solution
$C = $ Concentration of the substance given
$R = $ universal gas constant
$T = $ temperature of the solution
Among the above given terms, the values of temperature and universal gas constant remains constant. Thus, the osmotic pressure is directly proportional to the concentration of the substance given.
For an isotonic solution,
${\pi _1} = {\pi _2}$
$ \Rightarrow {C_1} = {C_2}$
We know that, $C = \dfrac{n}{V} = \dfrac{{\dfrac{w}{{{M_w}}}}}{V}$
In the given question, the volume ($V$ ) remains constant = $2l$
Thus, the final equation becomes:
$\dfrac{{{w_1}}}{{{{({M_w})}_1}}} = \dfrac{{{w_2}}}{{{{({M_w})}_2}}}$
Where, ${w_1} = $ weight of protein
${({M_w})_1} = $ Molar mass of protein
${w_2} = $ weight of sucrose
${({M_w})_2} = $ Molar mass of sucrose
Substituting these values in the main equation, we have:
$ \Rightarrow \dfrac{{500}}{{{{({M_w})}_1}}} = \dfrac{{3.42}}{{342}}$
$ \Rightarrow {({M_w})_1} = \dfrac{{342 \times 500}}{{3.42}} = 50000$
Thus, the correct option is D. $50000$

Note:
Two solutions having the same osmotic pressure across a semipermeable membrane is referred to as an isotonic solution. It has the same osmolarity (solute concentration), as another solution. A solution is isotonic when its effective osmole concentration is the same as that of another solution. This state provides the free movement of water across the membrane without changing the concentration of solutes on either side.