At 300 K, an ideal dilute solution of a macromolecule exerts osmotic pressure that is expressed in terms of the height (h) of the solution (density = 1.00 g cm$$^{-3}$$) where h is equal to 2.00 cm. If the concentration of the dilute solution of the macromolecule is 2.00 g dm$$^{-3}$$, the molar mass of the macromolecule is $$X \times 10^4$$ g mol$$^{-1}$$. The value of $$X$$ is ______.

Use: Universal gas constant (R) = 8.3 J K$$^{-1}$$ mol$$^{-1}$$ and acceleration due to gravity (g) = 10 m s$$^{-2}$$

The osmotic pressure $$ \pi $$ exerted by the solution equals the hydrostatic pressure of a liquid column of height $$ h $$.

Hydrostatic pressure formula: $$\pi = \rho g h$$

Given data (convert to SI units):
  • Density, $$ \rho = 1.00 \text{ g cm}^{-3} = 1000 \text{ kg m}^{-3} $$
  • Acceleration due to gravity, $$ g = 10 \text{ m s}^{-2} $$
  • Height, $$ h = 2.00 \text{ cm} = 0.02 \text{ m} $$

Hence
$$\pi = 1000 \times 10 \times 0.02 = 200 \text{ Pa}$$

For an ideal dilute solution, van’t Hoff equation applies:
$$\pi = c R T$$

Here $$ c $$ is the molar concentration (mol m$$^{-3}$$). The solution contains 2.00 g of solute per dm$$^{3}$$.
1 dm$$^{3} = 0.001 \text{ m}^{3}$$, so mass concentration is
$$\frac{2.00 \text{ g}}{0.001 \text{ m}^{3}} = 2000 \text{ g m}^{-3}$$

If the molar mass of the macromolecule is $$ M \text{ g mol}^{-1} $$, then
$$c = \frac{2000}{M} \text{ mol m}^{-3}$$

Substituting $$ c $$ into van’t Hoff equation:
$$\pi = \frac{2000}{M} R T$$

Solve for $$ M $$:
$$M = \frac{2000 R T}{\pi}$$

Insert the given constants $$ R = 8.3 \text{ J K}^{-1} \text{ mol}^{-1} $$ and $$ T = 300 \text{ K} $$:
$$M = \frac{2000 \times 8.3 \times 300}{200}$$

Calculate step by step:
$$8.3 \times 300 = 2490$$
$$2000 \times 2490 = 4\,980\,000$$
$$\frac{4\,980\,000}{200} = 24\,900$$

Thus
$$M \approx 24\,900 \text{ g mol}^{-1} = 2.49 \times 10^{4} \text{ g mol}^{-1}$$

Comparing with the required form $$ X \times 10^{4} \text{ g mol}^{-1} $$, we have $$ X \approx 2.49 $$.

Therefore, the value of $$ X $$ lies in the required range 2.4 - 2.55.