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Q1.22:
At 300 K, 36 g of glucose present in a litre of its solution has an osmotic pressure of 4.98 bar. If the osmotic pressure of the solution is 1.52 bars at the same temperature, what would be its concentration?

Solution :

Step 1: Analyzing the Initial Condition and Determining Constants

We are given the following initial parameters for a glucose solution:

  • Mass of solute ($w_1$) = $36\text{ g}$
  • Volume of solution ($V$) = $1\text{ L}$
  • Temperature ($T$) = $300\text{ K}$
  • Osmotic pressure ($\pi_1$) = $4.98\text{ bar}$

Using the van't Hoff equation for osmotic pressure, $\pi = CRT = \frac{nRT}{V}$, where $n = \frac{w}{M}$. We can calculate the gas constant product ($RT$) utilized in this specific system [Per the van't Hoff Law of Osmotic Pressure]:

First, find the molar mass of glucose ($C_6H_{12}O_6$):

$M = (6 \times 12.01) + (12 \times 1.008) + (6 \times 16.00) = 180.16\text{ g/mol} \approx 180\text{ g/mol}$

Now, calculate concentration ($C_1$):

$C_1 = \frac{n}{V} = \frac{36\text{ g} / 180\text{ g/mol}}{1\text{ L}} = 0.2\text{ mol/L}$

The relationship between osmotic pressure and concentration at constant temperature is $\pi \propto C$. Thus, $\frac{\pi_1}{C_1} = RT$.

$\frac{4.98\text{ bar}}{0.2\text{ mol/L}} = RT \implies RT = 24.9\text{ L}\cdot\text{bar/mol}$

Step 2: Calculating Concentration for the Second Osmotic Pressure

Given the new osmotic pressure $\pi_2 = 1.52\text{ bar}$ at the same temperature ($300\text{ K}$), we apply the linear proportionality derived from the ideal gas law analogy for solutions [By Boyle-van't Hoff Law]:

Since $T$ is constant:

$\pi_1 = C_1 RT$

$\pi_2 = C_2 RT$

Dividing the two equations to isolate $C_2$:

$\frac{\pi_2}{\pi_1} = \frac{C_2}{C_1}$

$C_2 = \frac{\pi_2 \times C_1}{\pi_1}$

Step 3: Numerical Computation

Substitute the values into the derived equation:

$C_2 = \frac{1.52\text{ bar} \times 0.2\text{ mol/L}}{4.98\text{ bar}}$

$C_2 = \frac{0.304}{4.98}\text{ mol/L}$

$C_2 \approx 0.061\text{ mol/L}$

Osmotic Pressure ($\pi$) vs Concentration ($C$) $\pi = CRT$ ($C_2, \pi_2$) ($C_1, \pi_1$)

Final Solution: The concentration of the solution at an osmotic pressure of 1.52 bar is 0.061 mol/L.


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