Colligative Properties

Dissolve a spoon of sugar in water and the water no longer behaves like plain water: it freezes a little colder, boils a little hotter, and develops a thirst for solvent across a membrane. Colligative properties are exactly these solution behaviors, and the striking part is that they depend mainly on the number of dissolved particles, not on what those particles are made of. The standard formulas below work best for dilute solutions and usually assume the solute is nonvolatile.

The single organizing idea: adding solute particles changes how easily solvent molecules can escape, freeze, or move through a membrane. That is why vapor pressure goes down, boiling point goes up, freezing point goes down, and osmotic pressure appears.

The Four Formulas and Their Symbols

The four standard colligative properties each have a compact formula. In all of them, $i$ is the van't Hoff factor (particle count per formula unit), $m$ is molality, $M$ is molarity, $R$ is the gas constant, $T$ is temperature, and $K_b$ and $K_f$ are solvent-specific constants.

Vapor pressure lowering (Raoult's law, nonvolatile solute):

$$P_{\text{solution}} = X_{\text{solvent}} P^0_{\text{solvent}}$$

where $X_{\text{solvent}}$ is the solvent mole fraction and $P^0_{\text{solvent}}$ is the pure-solvent vapor pressure.

Boiling point elevation:

$$\Delta T_b = i K_b m$$

Freezing point depression:

$$\Delta T_f = i K_f m$$

Osmotic pressure:

$$\pi = i M R T$$

Why These Formulas Hold

Each formula traces back to the same physical picture, so the algebra is not arbitrary. Adding solute makes $X_{\text{solvent}} < 1$, so by Raoult's law the solution's vapor pressure must drop below the pure solvent's. Because the solution now needs more heating before its vapor pressure matches the surroundings, the boiling point rises; because dissolved particles get in the way of the solvent locking into an ordered solid, the freezing point falls. Osmotic pressure is just the pressure that exactly cancels the net tendency of solvent to flow across a semipermeable membrane toward the more crowded side.

The deeper reason all four scale with particle count is that they each depend on the solvent's escaping tendency, which the dissolved particles dilute regardless of identity. That is the whole content of the word "colligative."

Worked Example: Freezing Point Depression

Dissolve glucose in water to make a $0.50\ \mathrm{m}$ solution. For water,

$$K_f = 1.86\ ^\circ\mathrm{C\, m^{-1}}$$

Glucose is a nonelectrolyte here, so

$$i = 1$$

Apply the formula:

$$\Delta T_f = i K_f m = (1)(1.86)(0.50) = 0.93\ ^\circ\mathrm{C}$$

Pure water freezes at $0.00\ ^\circ\mathrm{C}$, so the new freezing point is

$$0.00 - 0.93 = -0.93\ ^\circ\mathrm{C}$$

The size of the change comes entirely from particle count. Keep the same molality but switch to a solute that splits into more particles, and the freezing point drop grows.

Practice It Yourself

1. Take a $1.00\ \mathrm{m}$ glucose solution in water with the same $K_f = 1.86\ ^\circ\mathrm{C\, m^{-1}}$. Predict the new freezing point, then check: $\Delta T_f = (1)(1.86)(1.00) = 1.86\ ^\circ\mathrm{C}$, giving $-1.86\ ^\circ\mathrm{C}$. Compare with the $0.50\ \mathrm{m}$ case to see the linear particle-count relationship directly.
2. Estimate the boiling point elevation of that same $1.00\ \mathrm{m}$ glucose solution using $K_b = 0.512\ ^\circ\mathrm{C\, m^{-1}}$. Answer check: $\Delta T_b = (1)(0.512)(1.00) = 0.512\ ^\circ\mathrm{C}$, so it boils near $100.51\ ^\circ\mathrm{C}$.

Calculation Traps to Watch

Mixing Up Molality and Molarity

Boiling point elevation and freezing point depression use molality. Osmotic pressure uses molarity in the common dilute-solution form. Plugging one into the other quietly corrupts the answer.

Treating Formula Units and Particles as the Same Thing

One mole of dissolved formula units is not always one mole of dissolved particles. Electrolytes split into ions, so their colligative effect is larger than a nonelectrolyte's at the same concentration. That is what the factor $i$ corrects for.

Using the Formulas Outside Their Best Conditions

These formulas are most reliable for dilute solutions. Concentrated or strongly non-ideal solutions drift away from the simple estimates.

Assuming Every Solute Is Nonvolatile

The clean vapor-pressure-lowering picture assumes the solute does not evaporate much. If both components are volatile, a more careful model is needed.

Where the Idea Shows Up

Colligative properties run through antifreeze, road salting, food preservation, cell water balance, reverse osmosis, and some molar-mass measurements. In every case the same lever is being pulled: dissolved particles change how the bulk solvent behaves.