Definition: Solutions are homogeneous mixtures of two or more components whose composition can be varied within certain limits. Colligative properties are those physical properties of dilute solutions containing non-volatile solutes that depend solely on the number of solute particles present in a given amount of solvent, rather than on the chemical nature or identity of the solute particles.
Raoult’s Law and Relative Lowering of Vapour Pressure
When a non-volatile solute is dissolved in a volatile liquid solvent, the escaping tendency of the solvent molecules at the liquid surface decreases. This happens because the non-volatile solute particles occupy a fraction of the surface area. Consequently, the vapour pressure of the solution is always lower than that of the pure solvent at any given temperature.
This phenomenon forms the foundational understanding of Raoult’s Law. Quantitatively, Raoult’s law states that for a solution of a non-volatile solute, the partial vapour pressure of each volatile component in the solution is directly proportional to its mole fraction present in the solution.
Expressed mathematically for a binary solution where component 1 is the solvent and component 2 is the non-volatile solute:
$P_1 = P_1^\circ \cdot x_1$
Here, $P_1$ is the vapour pressure of the solvent above the solution, $P_1^\circ$ is the vapour pressure of the pure solvent, and $x_1$ is the mole fraction of the solvent. Since the sum of mole fractions $x_1 + x_2 = 1$, we can rearrange the expression to find the relative lowering of vapour pressure.
- $\Delta P = P_1^\circ – P_1 = P_1^\circ(1 – x_1) = P_1^\circ \cdot x_2$
- Relative lowering of vapour pressure = $\frac{P_1^\circ – P_1}{P_1^\circ} = x_2$
- For dilute solutions where $n_2 \ll n_1$, the expression simplifies to $\frac{P_1^\circ – P_1}{P_1^\circ} \approx \frac{n_2}{n_1} = \frac{w_2 / M_2}{w_1 / M_1}$
Competitive exams frequently test this relationship by asking you to determine the molar mass of an unknown solute ($M_2$). This is done by measuring the change in vapour pressure or by using boiling-point elevation techniques derived from it.
Elevation in Boiling Point and Depression in Freezing Point
The addition of a non-volatile solute to a pure solvent alters its phase transition temperatures. The boiling point of a liquid is defined as the temperature at which its vapour pressure equals the external atmospheric pressure. Because a non-volatile solute lowers the vapour pressure, the solution must be heated to a higher temperature to reach a vapour pressure of one atmosphere. This increase is known as the elevation in boiling point ($\Delta T_b$).
Similarly, the freezing point of a liquid is the temperature at which the solid and liquid phases exist in dynamic equilibrium, possessing identical vapour pressures. The presence of solute particles lowers the vapour pressure curve of the liquid solvent relative to the solid solvent. Therefore, the solution must be cooled to a lower temperature for freezing to occur, a phenomenon called the depression in freezing point ($\Delta T_f$).
- Boiling Point Elevation Formula: $\Delta T_b = T_b – T_b^\circ = K_b \cdot m$, where $K_b$ is the ebullioscopic constant (molal elevation constant) and $m$ is the molality of the solution.
- Freezing Point Depression Formula: $\Delta T_f = T_f^\circ – T_f = K_f \cdot m$, where $K_f$ is the cryoscopic constant (molal depression constant).
- Units for $K_b$ and $K_f$ are typically $\text{K kg mol}^{-1}$. Note that molality is preferred over molarity in these expressions because molality is independent of temperature fluctuations.
In analytical chemistry and physical chemistry problem-solving, these linear relationships allow scientists to determine molecular weights of polymers, proteins, and organic compounds accurately. This is achieved by observing small temperature shifts.
Osmotic Pressure
Osmosis is the spontaneous net movement of solvent molecules through a semipermeable membrane from a region of lower solute concentration (or pure solvent) to a region of higher solute concentration. The hydrostatic pressure built up across the membrane that just stops this osmotic flow is defined as osmotic pressure ($\pi$).
Scientist Jacobus Henricus van ‘t Hoff demonstrated that for dilute solutions, osmotic pressure obeys a law analogous to the ideal gas equation. The governing mathematical expression is:
$\pi = CRT = \left(\frac{n}{V}\right)RT$
Here, $\pi$ is the osmotic pressure, $C$ is the molar concentration of the solution, $R$ is the universal gas constant, and $T$ is the absolute temperature in Kelvin. This equation can also be expanded to calculate the molar mass ($M_2$) of macromolecules.
- $\pi = \left(\frac{w_2}{M_2 \cdot V}\right)RT$
- Rearranging gives: $M_2 = \frac{w_2 R T}{\pi V}$
Osmotic pressure is considered the most reliable colligative property for determining the molecular weights of large biomolecules, such as proteins and polymers. This is because even very dilute solutions generate measurable, high-magnitude osmotic pressures at room temperature, minimizing experimental error.
Abnormal Colligative Properties and van ‘t Hoff Factor
Discrepancies often arise when experimental colligative property values are compared against theoretical values calculated assuming ideal behavior. These deviations occur because certain solutes undergo association or dissociation when dissolved in a solvent, altering the total number of particles in the solution.
To account for these deviations, Swiss chemist van ‘t Hoff introduced a correction coefficient known as the van ‘t Hoff factor ($i$). This factor normalizes all colligative equations to handle electrolytes and associating molecules properly.
- $i = \frac{\text{Observed Colligative Property}}{\text{Calculated (Normal) Colligative Property}}$
- $i = \frac{\text{Total number of moles of particles after dissociation/association}}{\text{Number of moles of formula units dissolved}}$
- For strong electrolytes like $\text{NaCl}$ that dissociate completely into two ions ($n=2$), the ideal $i = 2$. For $\text{CaCl}_2$, the ideal $i = 3$.
When association occurs, such as benzoic acid dimerizing in benzene, the value of $i$ is less than 1. When dissociation occurs, such as $\text{K}_4[\text{Fe}(\text{CN})_6]$ breaking into 5 ions, $i$ is greater than 1. For non-electrolytes like glucose or urea that remain intact, $i = 1$.
The modified colligative property equations incorporating the van ‘t Hoff factor are:
- Relative lowering of vapour pressure: $\frac{P_1^\circ – P_1}{P_1^\circ} = i \cdot x_2$
- Elevation in boiling point: $\Delta T_b = i \cdot K_b \cdot m$
- Depression in freezing point: $\Delta T_f = i \cdot K_f \cdot m$
- Osmotic pressure: $\pi = i \cdot C R T$
Degree of Dissociation and Association
For a substance undergoing dissociation with a dissociation constant or degree $\alpha$ and yielding $n$ particles:
$i = 1 + \alpha(n – 1) \implies \alpha = \frac{i – 1}{n – 1}$
For a substance undergoing association where $n$ molecules associate to form a single aggregated particle with a degree of association $\alpha$:
$i = 1 + \alpha\left(\frac{1}{n} – 1\right) \implies \alpha = \frac{i – 1}{\frac{1}{n} – 1}$
Important Facts / Formulas
| Colligative Property | Standard Expression | Corrected Expression (with $i$) |
|---|---|---|
| Relative Lowering of Vapour Pressure | $\frac{P_1^\circ – P_1}{P_1^\circ} = x_2$ | $\frac{P_1^\circ – P_1}{P_1^\circ} = i \cdot x_2$ |
| Elevation in Boiling Point | $\Delta T_b = K_b \cdot m$ | $\Delta T_b = i \cdot K_b \cdot m$ |
| Depression in Freezing Point | $\Delta T_f = K_f \cdot m$ | $\Delta T_f = i \cdot K_f \cdot m$ |
| Osmotic Pressure | $\pi = CRT$ | $\pi = i \cdot CRT$ |
Key Points to Remember
- Colligative properties depend exclusively on the number of solute particles, independent of their chemical nature.
- Molality ($m$) is temperature-independent, which is why it is preferred over molarity ($M$) in boiling point and freezing point calculations.
- Osmotic pressure is ideal for large molecular mass determinations because it can be measured accurately at room temperature.
- The van ‘t Hoff factor ($i$) is greater than 1 for dissociation, less than 1 for association, and exactly 1 for nonelectrolytes.
- Raoult’s law is a special case of Henry’s law where the volatile solute behaves like the solvent under high mole fraction limits.
- Solutions exhibiting positive deviations from Raoult’s law show higher vapour pressures than expected, whereas negative deviations show lower vapour pressures.
- Always use molality values expressed in $\text{mol kg}^{-1}$ when working with $K_b$ and $K_f$ values.
Quick Revision Summary
- Raoult’s law relates the partial vapour pressure of a volatile solvent to its mole fraction in the presence of a non-volatile solute.
- Vapour pressure lowering is a direct consequence of surface coverage by non-volatile solute molecules.
- Boiling point elevation ($\Delta T_b = K_b \cdot m$) and freezing point depression ($\Delta T_f = K_f \cdot m$) are proportional to solution molality.
- Osmotic pressure ($\pi = iCRT$) provides the most accurate method for finding high molecular weights.
- Abnormal colligative results occur due to solute association or dissociation in solution.
- The van ‘t Hoff factor ($i$) corrects theoretical formulas to match real experimental observations for electrolytes.
- Degree of dissociation ($\alpha$) relates directly to $i$ and the total number of ions produced per formula unit.
