Question

The osmotic pressure of a solute is \[{\text{600}}\,{\text{mm}}\] at \[{\text{300K}}\]. The solution is diluted and the temperature is raised to \[{\text{400K}}\]and the solution shows an osmotic pressure of \[{\text{200}}\,{\text{mm}}\]. The solution was diluted to x times x is:

Answer 1

Hint: In the case of the solution it is important to study the colligative properties of the solution. These properties mainly depend on the amount of solute present in the solution, not on the nature of the solute particles. Elevation in boiling point, depression in freezing point, relative lowering of vapor pressure, osmotic pressure are the colligative properties.

Formula used: The osmotic pressure is given as follows:
\[\pi {\text{ = CRT}}\] (i)
Here, osmotic pressure is \[\pi \], concentration is C, gas constant is R, and the temperature is T.

Complete step-by-step answer:
Osmotic pressure is one of the colligative properties represented by a letter \[\pi \]. It is the pressure applied on the pure solvent that prevents the flow of the pure solvent through the semipermeable membrane which allows passage for only the solvent molecules.

In the given question two osmotic pressures and the temperatures are given and we have to determine the concentration change compared to the initial concentration.

Rearrange the equation (i) for concentration as follows:
\[\pi {\text{ = CRT}}\]
\[{\text{C = }}\dfrac{\pi }{{{\text{RT}}}}\]
Now, for initial conditions equation becomes,
\[{{\text{C}}_1}{\text{ = }}\dfrac{{{\pi _1}}}{{{\text{R}}{{\text{T}}_1}}}\]
For final conditions equation becomes,
\[{{\text{C}}_2}{\text{ = }}\dfrac{{{\pi _2}}}{{{\text{R}}{{\text{T}}_2}}}\]
Now, take the ratio of the initial to final concentration.
\[\dfrac{{{{\text{C}}_1}}}{{{{\text{C}}_2}}}{\text{ = }}\dfrac{{{\pi _1} \times {\text{R}}{{\text{T}}_2}}}{{{\text{R}}{{\text{T}}_1} \times {\pi _2}}}\]

Here, substitute \[{\text{600}}\,{\text{mm}}\] for \[{\pi _1}\], \[{\text{300K}}\] for \[{{\text{T}}_1}\],\[{\text{200}}\,{\text{mm}}\] for \[{\pi _2}\], and\[{\text{400K}}\] for\[{{\text{T}}_2}\].
\[\dfrac{{{{\text{C}}_1}}}{{{{\text{C}}_2}}}{\text{ = }}\dfrac{{{\text{600}}\,{\text{mm}} \times 4{\text{00K}}}}{{{\text{300K}} \times {\text{200}}\,{\text{mm}}}}\]
\[\dfrac{{{{\text{C}}_1}}}{{{{\text{C}}_2}}}{\text{ = 4}}\]
\[{{\text{C}}_1}{\text{ = }}{{\text{C}}_2} \times {\text{4}}\]
Here, the final concentration is four times the initial concentration indicates the solution is diluted four times the initial solution.

Therefore, the value of x is 4.

Note:
i) Osmosis is the phenomenon in which solvent molecules are moved from lower solute concentration to higher solute concentration through the semipermeable membrane.
ii) Osmotic pressure is given as \[\pi {\text{ = CRT}}\], indicates osmotic pressure is directly proportional to the concentration of the solute.
iii) The higher the concentration of the solute higher is the osmotic pressure and vice versa.