(a) Show that the following equation is true. Molar mass of solute \(=\frac{\text { (grams of solute) } R T}{\Pi \mathrm{V}}\) (b) An aqueous solution of a compound with a very high molecular mass was prepared in a concentration of \(2.0 \mathrm{~g} \mathrm{~L}^{-1}\) at \(25^{\circ} \mathrm{C}\). Its osmotic pressure was 0.021 torr. Calculate the molecular mass of the compound.

The van't Hoff equation is a fundamental principle in physical chemistry that relates osmotic pressure to solute concentration. The equation is expressed as \[ \Pi = \frac{nRT}{V} \], where \(\Pi\) is the osmotic pressure, \(n\) is the number of moles of solute, \(R\) is the universal gas constant, \(T\) is the absolute temperature in Kelvin, and \(V\) is the volume of the solution in liters.

This equation assumes that the solution behaves ideally, which means the solute-solvent interactions are similar to the solvent-solvent interactions, and the solute's effect on the solution's volume is negligible. By using the van't Hoff equation, you can determine how the concentration of solute in a solvent affects its osmotic pressure and vice versa, which is crucial when studying solutions and their properties.

Molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol). It is a physical property that is fundamental to many aspects of chemistry, including the calculation of osmotic pressure. The molar mass can be used to convert between the mass of a substance and the amount of substance in moles. In practical terms, determining the molar mass of a solute allows for the calculation of its concentration in a solution when given its mass and the solution's volume.

In the context of osmotic pressure, the molar mass (\(M\)) can be rearranged from the van't Hoff equation to find the molecular mass by using the relationship \(M = \frac{\text{(grams of solute)} R T}{\Pi V}\). This expression sets the stage for converting easily measurable quantities like mass and volume into values that characterize the substance, such as its molar mass.

Solute concentration often measured in terms of molarity (moles per liter), is a measure of how much solute is present in a given volume of solvent. It is a key factor in determining the osmotic pressure of a solution. The higher the concentration of solute particles, the higher the osmotic pressure, assuming temperature and gas constant are kept constant.

Understanding the relationship between solute concentration and osmotic pressure is essential in fields like chemistry and biology, where osmosis can affect things like the balance of fluids in cells. Solute concentration also plays a critical role in applications such as dialysis, where it is used to control the transfer of substances between blood and a dialysis fluid based on osmotic pressure.

The universal gas constant (\(R\)), a fundamental parameter in the equations of state for ideal gases, appears prominently in the van't Hoff equation for calculating osmotic pressure. Its value is approximately 0.0821 \(L\cdot atm\cdot K^{-1}\cdot mol^{-1}\), or various equivalents depending on the pressure and volume units.

Why is \(R\) crucial for osmotic pressure calculations? It serves as a proportionality constant that relates the energy scale to the molecular scale when considering a gas or a solution in thermodynamic terms. Thus, knowing the value of \(R\) and how to apply it across different conditions is essential for accurate osmotic pressure calculations and for differentiating between the osmotic behavior of various solutes in solution.