How does the nature of solute affect the colligative properties, i.e., boiling point, freezing point, and...?

osmotic pressure of the solution?

Colligative properties depend on the number of paricles in solution and not on characteristics of the particles like size, charge,etc. The nature of the solute affects these properties only at the degree that it affects the number of particles in solution.

Non-electrolytes don't dissociate when they dissolve. Thus the particles in solution will be equal to the number of molecules of the compound that dissolved. So 1M glucose will have 1M particles in solution and e.g. osmotic pressure π=CRT= 1*RT

Electrolytes dissociate into ions when dissolved in solvents like water. Then the number of particles in solution is actually greater than what you would expect according to the molecular formula.
E.g. NaCl dissociates into 1 Na+ and 1 Cl- in water. So if you had 1 M NaCl, you'll have 2 M particles and π= 2*1RT= 2RT, and you see that consequently you have double osmotic pressure than that for the equimolar amount of glucose.

So the proper equation for osmotic pressure is in general

π = iCRT
where i is the van't Hoff coefficient and depends on how many particles are truly in solution. This applies respectively to all colligative properties.

for non-electrolytes i=1

for dilute solutions of strong electrolytes i= the number of ions the electrolyte dissociates into, according to its molecular formula (e.g. NaCl i=2, CaCl2 i=3, CaSO4 i=2, FeCl3 i=4)

for weak electrolytes i=1-a+na where a is the degree of dissociation and n the number of ions that it can dissociate into, according to its molecular formula. E.g. acetic acid CH3COOH is a weak acid. n=2 (CH3COO- and H+) and if we say that a=80%=0.8 then i=1-0.8+2*0.8= 1.8

Notes of caution
1. for a first level chemistry that's all you need to know for calculating i. In reality i is a function of the concentration. Even for strong electrolytes i<n for concentrated solutions. E.g. for concentrated NaCl you could have i=1.9 instead of 2. This is because the solvated ions can interact together forming temporarily ion pairs (Bjerrum ion pairs) and formation of such pairs is dependent on the concentration. When such pairs form, the number of particles decreases a bit (the pair is one particle compared to the 2 particles that you would have for unpaired ions)
2.There are cases where the solute doesn't dissociate. For example benzoic acid is a weak acid that will dissociate in water and thus for studying its colligative properties in water you would need to use i=1-a+2a=1+a.
However, if you dissolve it in benzene, it will not dissociate at all and then i=1. So be careful which solvent you use when deciding the value of i.
Well, colligative properties aren't contingent on the nature of the solute; that is, colligative properties are what they are because they involve solutes changing properties of solutions.

As for the boiling point, an added solute dissolved in the solution will raise the boiling point. This is not a direct relationship, however, since the boiling point will cap out at a certain point.

The freezing point is also affected: added solute makes the solution freeze at a lower temperature.

Both of these properties stem from the fact that added solute changes the respective vapor pressures and molecular organizations of the solutions. Solute particles keep the vapor pressure high: they congregate throughout the solution, and prevent the solvent's particles from evaporating, or boiling. Likewise, they "get in the way" of the solvent particles, when the solvent wants to solidify.

The answers post by the user, for information only, does not guarantee the right.

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