The concept we are utilizing to solve this problem is osmotic pressure. Osmotic pressure is the pressure required to stop the flow of solvent into a solution through a semipermeable membrane. For dilute solutions, osmotic pressure behaves similarly to an ideal gas, hence the formula we use is similar to the ideal gas law:

$$\Pi = \frac{n}{V}RT$$

where:

• $\Pi$ is the osmotic pressure,
• $n$ is the number of moles of solute,
• $V$ is the volume of the solution,
• $R$ is the ideal gas constant, and
• $T$ is the temperature in Kelvin.

First, we convert all given quantities to the appropriate units.

• $\Pi = 1.29$ mbar is equivalent to $1.29 \times 10^{-3}$ bar (since 1 bar = 1000 mbar),
• $V = 300$ cm³ is equivalent to $0.3$ L (since 1 L = 1000 cm³),
• $T = 300$ K,
• $R = 0.083$ L bar K$^{-1}$ mol$^{-1}$.

Instead of the number of moles, we are given the mass of the solute. However, we can replace the moles with mass using the relationship $n = \frac{m}{M}$, where $M$ is the molar mass, and $m$ is the mass of the solute.

So the equation becomes:

$$\Pi = \frac{m}{MV}RT$$

We are trying to solve for $M$, so rearrange the equation to isolate $M$:

$$M = \frac{mRT}{\Pi V}$$

Finally, substitute the given values into the equation:

$$M = \frac{0.63 \times 0.083 \times 300}{1.29 \times 10^{-3} \times 0.3} \approx 40535 \, \text{g/mol}$$

So the molar mass of the protein is approximately 40535 g/mol.