# Why do colligative properties depend only on number of solute particles?

Colligative properties depend solely on the number of even though the interactive forces are different for different solute-solvent pairs. So why is the dependence only on the number of solute?

• I'm a little uncertain what you are asking about specifically. But, in a solution (taking a per mole of solution basis for the thermodynamics), if there is a concentration $x$ of solute, there is a concentration $1-x$ of solvent. You only need one number ($x$) to specify the composition of the solution. Jan 30 '19 at 13:53

It's a lie. Colligative properties do depend on the chemical nature of the solute and solvent - their interaction. The trick is: an ideal solution, in which the solute-solute intermolecular forces and solute-solvent interactions are more or less of the same character, has the property that colligative properties are dictated by composition given only in terms of amount of moles. However, any sufficiently diluted solution will behave somehow ideally, that is where the statement goes: for diluted solutions, colligative properties depend only on the number of moles of solute.

• This explanation is no better than just believing in the definition. If it's about the nature of the interaction, why isn't the property proportional to at least the effective surface area (whatever that exactly is) of the particle? Why $n$, no matter if the particle is a Li ion or a 100kg/mol polymer molecule?
– Karl
Jan 30 '19 at 20:37
• You are correct. A more rigorous explanation would require a very deep look into the thermodynamics and why it is usually formulated in terms of molar fractions instead of mass. One would have to analyse very carefully the derivations of equations such as Raoul's law to understand why molar fractions show up - we can get a hint from the simple ideal gas equation pV = nRT, which uses number of moles and not mass (or surface area like you said). By working with Raoul's law it is easy to deduce that boiling point rise for example of a nonvolatile solute depends on solvent molar fraction. Jan 31 '19 at 3:43

A very simple, qualitative explanation:

After your solute has dissolved, there are no more enthalpic effects to take into account. The solvation enthalpy is converted to a temperature change, and that's it. For colligative properties, the solute ideally has no significant own vapour pressure, does not precipitate, it just stays in solution.

The solute molecule moves around (Brownian motion), the solvent molecules around it exchange places, but are all the same, and that's that.

Now everything in solution is about entropy. In a solution, there is only orientational and translational entropy. (Molecular) Vibrational excitation is only relevant at high temperatures, rotation is practically imposible due to the many interactions.

That's why every solute particle, irrespective of its size, has the same contribution to the entropy.

If the solute concentration inceases, its particles start interacting. Their coming close to each other and detaching again makes local energy fluctuations, which add entropy, but depend strongly on the kind of interaction. That's when the identity of the solute starts to matter.

• So, basically the statement valid as long as the entropy changes due to solute-solute interactions are negligible? Jan 31 '19 at 17:17
• solute-solute interactions do not lead to a violation of colligative property expressions, unless their occurrence leads to a violation of Raoult's law behavior for the solvent. Jan 31 '19 at 23:09

Colligative properties are defined as solvent properties that display a linear dependence on the mole fraction $$x_2$$ of solute in solution, but are otherwise blind to the specific nature of intermolecular interactions. The key condition for such properties to be observed is that Raoult's law (one definition of an ideal solution) holds. Colligative behavior is purely statistical. For colligative properties - including freezing point depression, boiling point elevation, and osmotic pressure - the change in the property of the solvent can be expressed as $$\Delta J=K_J x_2$$, where $$K_J$$ is regarded as a constant for the solvent at the given temperature and pressure, independently of the identity of the solute. The relations assume explicitly that the solute concentration is low ($$x_2\lt\lt1$$). In practice, the requirement of dilute solutions is often necessary to ensure that Raoult's law is observed.

Raoult's law implies that the chemical potential of the solvent, $$\mu_1(l)$$, can be written as

$$\mu_1 (l) = \mu_1^*(l)+RTlnx_1$$

where $$\mu_1^*$$ is the chemical potential of the pure solvent at the same temperature and pressure. This equation is the basis for deriving the colligative property equations.

It should be emphasized that it is necessary to ensure ideal solution behavior - that Raoult's law is observed - in order to implement the mathematical formalism associated with colligative properties. In that sense this is a functional definition: one searches for systems that match the definition and allow application of the formalism.

When are these conditions met?

Quoting Atkins' physical chemistry textbook (Freeman, 4th Ed.):

some mixtures obey Raoult's law very well, especially when the components are chemically similar. [emphasis mine]

And while I'd rather you took my word for it, you can always check another tertiary source, the wikipedia:

In chemistry, colligative properties are properties of solutions that depend on the ratio of the number of solute particles to the number of solvent molecules in a solution, and not on the nature of the chemical species present.1 The number ratio can be related to the various units for concentration of solutions. The assumption that solution properties are independent of nature of solute particles is only exact for ideal solutions, and is approximate for dilute real solutions. In other words, colligative properties are a set of solution properties that can be reasonably approximated by assuming that the solution is ideal.

• From what I read at chemistry.stackexchange.com/questions/33896/…, it would be sufficient to consider an ideal-dilute solution. Jan 31 '19 at 21:23
• Raoult's law sometimes holds in the dilute regime even when it doesn't over the entire range of mixing. That is why I emphasize the dilute part, even though the ideal part is also important. Jan 31 '19 at 21:48