Colligative property relationship:\nFor dilute solutions, osmotic pressure is directly proportional to which of the following quantities?

Introduction / Context:
Osmotic pressure is a classic colligative property, meaning it depends on the number of solute particles rather than their identity (for ideal dilute solutions). It connects directly to other colligative phenomena such as vapor-pressure lowering, boiling-point elevation, and freezing-point depression.

Given Data / Assumptions:

• Dilute, ideal solutions (van’t Hoff law applies).
• Non-electrolyte solute unless otherwise specified; for electrolytes, use the van’t Hoff factor i.
• Constant solvent properties aside from colligative effects.

Concept / Approach:
The van’t Hoff equation for ideal dilute solutions is π = i C R T, where π is osmotic pressure, C is molar concentration of solute, R is the gas constant, and T is absolute temperature. Vapor-pressure lowering Δp_v / p_v^0 ≈ x_solute (mole fraction), and for dilute solutions, C is proportional to x_solute at fixed solvent amount. Hence osmotic pressure is proportional to solute concentration and to absolute temperature, and it correlates linearly with the relative lowering of vapor pressure.

Step-by-Step Solution:
Relate π to concentration and temperature: π ∝ C and π ∝ T.Relate π to vapor-pressure lowering: both π and Δp_v are proportional to the number of solute particles (colligative), yielding a direct proportionality under dilute conditions.Therefore, all three statements hold in the ideal, dilute limit.

Verification / Alternative check:
Derivations from chemical potential equality across a semipermeable membrane lead to πV = n_s R T, consistent with van’t Hoff’s law and the proportionalities stated.

Why Other Options Are Wrong:
Picking only one or two ignores the complete set of proportionalities valid for dilute, ideal solutions.

Common Pitfalls:
Applying these relations to concentrated or strongly nonideal solutions without activity corrections; neglecting ionization (van’t Hoff factor i) for electrolytes.

Final Answer:
All of (a), (b), and (c)