A \(0.020 \mathrm{M} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(\mathrm{aq})\) solution is separated from a \(0.050 \mathrm{M} \mathrm{CO}\left(\mathrm{NH}_{2}\right)_{2}(\mathrm{aq})\) solution by a semipermeable membrante at \(25^{\circ} \mathrm{C}\). (a) Which solution has the higher osmotic pressure? (b) Which solution becomes more dilute with the passage of \(\mathrm{H}_{2} \mathrm{O}\) molecules through the membrane? (c) To which solution should an external pressure be applied in order to maintain an equilibrium flow of \(\mathrm{H}_{2} \mathrm{O}\) molecules across the membrane? (d) What external pressure (in atm) should be applied in (c)?

Colligative properties are unique in that they depend on the number of solute particles in a solution, rather than the specific type of chemical species present. They provide essential information about a solution's properties that arise from solute concentration. The four main colligative properties include vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure.

Osmotic pressure, in particular, is particularly important when dealing with solutions separated by a semipermeable membrane. It directly influences the movement of solvent molecules across the membrane and is vital in biological systems such as nutrient absorption and waste filtration.

A semipermeable membrane is a barrier that allows certain substances, typically solvents like water, to pass through but blocks others, usually solutes. This critical feature is employed by cells in our bodies to regulate the passage of substances in and out, which is essential for maintaining homeostasis.

In the context of the given exercise, this membrane is the reason why water moves from one solution to another. It allows for the selective diffusion of water, driven by a concentration gradient, which ultimately leads to osmosis.

The van't Hoff factor, symbolized by the letter 'i', is pivotal in considering how many particles a compound forms when it dissolves in solution. It affects the magnitude of colligative properties. For non-electrolytes like glucose (C₆H₁₂O₆), which do not dissociate into ions, the van't Hoff factor is 1, meaning each molecule of solute yields exactly one particle in solution.

For strong electrolytes, which completely dissociate into ions, the factor would be equivalent to the number of ions formed. It’s a straightforward concept, but its implications are vast, having a direct effect on calculations for osmotic pressure and other colligative properties.

Molarity is a measure of concentration used in chemistry to represent the number of moles of solute per liter of solution. It's given the symbol 'M', and its precise calculation is fundamental for accurately determining colligative properties, including osmotic pressure.

Higher molarity typically signifies a higher number of solute particles per unit volume, which correlates with greater osmotic pressure. This principle is illustrated in the exercise problem, where the solutions' molarities are directly used to calculate osmotic pressures.

The ideal gas constant, symbolized as 'R', is a universal constant used in the equation of state for an ideal gas. In the context of osmosis, it's also employed in the formula to calculate osmotic pressure. The constant has a value of 0.0821 L atm/mol K under standard conditions, offering a bridge between the microscopic world of atoms and molecules and the macroscopic properties of gases.

While the gas constant is derived from gas behavior, it plays an essential role in explaining and calculating properties of solutions, including osmotic pressure, as it relates the effects of solute concentration to physical pressure.