Unit 5: Physical Properties of Solutions, Colligative Phenomena & Distribution Law

A solution is a homogeneous mixture of two or more chemical substances forming a single thermodynamic phase down to the sub-nanometer scale. The constituent present in the greatest stoichiometric abundance is typically designated the solvent, while the dispersed species are termed solutes.

Thermodynamic Taxonomy of Solutions

Solutions exist across all physical states of matter:

• Gas-in-Gas: e.g., atmospheric air (completely miscible due to negligible intermolecular forces at low pressure).
• Gas-in-Liquid: e.g., dissolved oxygen ($\text{O}_2$) in aquatic ecosystems or carbon dioxide in carbonated beverages.
• Liquid-in-Liquid: e.g., ethanol in water (completely miscible) vs octane in water (immiscible).
• Solid-in-Liquid: e.g., aqueous sodium chloride ($\text{NaCl}$) or glucose solutions.
• Gas-in-Solid: e.g., hydrogen gas absorbed interstitially within metallic palladium ($\text{PdH}_x$).
• Solid-in-Solid (Solid Solutions): e.g., substitutional alloys (alpha-brass: $\text{Zn}$ in $\text{Cu}$) and interstitial alloys (carbon in iron forming austenite and martensite).

The Three-Step Energetic Model of Dissolution

The formation of a solution from pure components is understood by decomposing the thermodynamic process into a three-step Born-Haber-type energetic cycle:

1. Step 1: Solute-Solute Particle Separation ($\Delta H_1 > 0$):

Solute molecules or ions must be pulled apart from their pure condensed phase, breaking solute-solute intermolecular attractions or ionic crystal lattices:

2. Step 2: Solvent-Solvent Cavity Expansion ($\Delta H_2 > 0$):

Molecules of the solvent must be pushed apart to create microscopic cavities of sufficient volume to accommodate the incoming solute particles, breaking solvent-solvent cohesive bonds (e.g., hydrogen bonds in water):

3. Step 3: Solute-Solvent Solvation / Hydration ($\Delta H_3 < 0$):

Solute particles insert into the solvent cavities, establishing attractive solute-solvent interactions (ion-dipole, dipole-dipole, or dispersion forces):

By Hess's Law, the overall Enthalpy of Solution is:

The Gibbs Free Energy of Dissolution & "Like Dissolves Like"

The spontaneous formation of a solution at constant temperature and pressure is dictated by the change in Gibbs Free Energy of Mixing:

1. Entropy of Dissolution ($\Delta S_{soln}$):

Dispersing solute particles throughout the solvent volume drastically increases the spatial configurational entropy of the system. For an ideal binary solution of components 1 and 2:

Because $x_1, x_2 < 1$, the logarithms are negative, ensuring $\Delta S_{mix} > 0$ unconditionally.

2. The "Like Dissolves Like" Principle:

• Polar Solute in Polar Solvent (e.g., $\text{NaCl}$ in $\text{H}_2\text{O}$): The immense exothermic ion-dipole solvation enthalpy ($\Delta H_3 \ll 0$) is large enough to compensate for the endothermic lattice breaking ($\Delta H_1$) and cavity creation ($\Delta H_2$). Consequently, $\Delta H_{soln}$ is small (slightly positive or negative), and the positive $-T\Delta S_{soln}$ term drives spontaneous dissolution ($\Delta G_{soln} < 0$).
• Nonpolar Solute in Polar Solvent (e.g., Oil in $\text{H}_2\text{O}$): Nonpolar hydrocarbons interact with water only through weak dispersion forces ($\Delta H_3$ is tiny). Furthermore, water molecules around nonpolar solutes are forced into rigid, highly ordered hydrogen-bonded "iceberg" clathrate cages (Hydrophobic Effect), resulting in a net negative entropy change ($\Delta S_{soln} < 0$). Both $\Delta H_{soln} > 0$ and $-T\Delta S_{soln} > 0$, making $\Delta G_{soln} \gg 0$ (completely insoluble).
• Nonpolar Solute in Nonpolar Solvent (e.g., $\text{I}_2$ in $\text{CCl}_4$): Intermolecular forces in both pure components and the mixture are all weak London dispersion forces ($\Delta H_1 + \Delta H_2 \approx -\Delta H_3 \implies \Delta H_{soln} \approx 0$). The process is driven entirely by the favorable entropy of mixing ($\Delta G_{soln} \approx -T\Delta S_{mix} < 0$).

Quantitative physical chemistry requires exact mathematical precision when expressing the relative abundance of components in a solution phase.

Primary Concentration Scales

Let a solution contain solute $A$ (molar mass $M_A$, mass $m_A$, moles $n_A$) dissolved in solvent $B$ (molar mass $M_B$, mass $m_B$, moles $n_B$), producing a total solution mass $m_{soln} = m_A + m_B$, total solution volume $V$, and macroscopic mass density $\rho = m_{soln} / V$.

1. Mass Percent ($\% w/w$):

2. Mole Fraction ($x_A$):

Mole fraction is dimensionless and strictly temperature-independent.

3. Molarity ($C$ or $M$):

Because liquid volume expands with temperature according to the thermal expansion coefficient $\alpha_P$:

Molarity changes with temperature, introducing systematic errors in precision temperature-dependent experiments.

4. Molality ($m$ or $b$):

Because mass is an invariant scalar under temperature and pressure variations, molality is strictly independent of temperature and pressure, making it the indispensable metric for thermodynamic colligative properties.

Exact Mathematical Interconversion Derivations

Let solution density be $\rho$ in $\text{g}\cdot\text{mL}^{-1} = \text{g}\cdot\text{cm}^{-3} = 10^3\text{ kg}\cdot\text{m}^{-3}$, and solute molar mass be $M_A$ in $\text{g}\cdot\text{mol}^{-1}$.

1. From Molarity ($M$) to Molality ($m$):

Consider exactly $1.000\text{ Liter} = 1000\text{ mL}$ of solution:

• Moles of solute: $n_A = M\text{ mol}$.
• Mass of solute: $m_A = n_A M_A = M \cdot M_A\text{ (grams)}$.
• Total mass of solution: $m_{soln} = V \cdot \rho = 1000 \cdot \rho\text{ (grams)}$.
• Mass of solvent: $m_B = m_{soln} - m_A = 1000\rho - M \cdot M_A\text{ (grams)} = \frac{1000\rho - M \cdot M_A}{1000}\text{ (kg)}$.

Applying the definition of molality:

2. From Molality ($m$) to Molarity ($M$):

Consider an amount of solution containing exactly $1.000\text{ kg} = 1000\text{ g}$ of pure solvent ($m_B = 1000\text{ g}$):

• Moles of solute: $n_A = m\text{ mol}$.
• Mass of solute: $m_A = m \cdot M_A\text{ (grams)}$.
• Total solution mass: $m_{soln} = 1000 + m \cdot M_A\text{ (grams)}$.
• Solution volume: $V_{soln} = \frac{m_{soln}}{\rho} = \frac{1000 + m \cdot M_A}{\rho}\text{ (mL)} = \frac{1000 + m \cdot M_A}{1000\rho}\text{ (Liters)}$.

Applying the definition of molarity:

3. From Molality ($m$) to Mole Fraction ($x_A$):

In a sample containing $1.000\text{ kg} = 1000\text{ g}$ of solvent of molar mass $M_B$ ($\text{g}\cdot\text{mol}^{-1}$):

• Moles of solvent: $n_B = \frac{1000}{M_B}\text{ mol}$.
• Moles of solute: $n_A = m\text{ mol}$.

For dilute aqueous solutions ($M_B = 18.015\text{ g/mol}, m M_B \ll 1000$):

Solubility is the maximum concentration of a solute that dissolves in a given solvent at a specified temperature and pressure to establish a dynamic thermodynamic equilibrium with undissolved solute:

Effect of Temperature on Solubility

The temperature dependence of solubility is governed by Le Chatelier's Principle and the van 't Hoff equation:

1. Solids in Liquids:

• Endothermic Dissolution ($\Delta H_{soln} > 0$): Ingesting thermal energy shifts equilibrium toward the dissolved state. Solubility increases with temperature. Most salts exhibit this behavior (e.g., $\text{KNO}_3, \text{NH}_4\text{NO}_3, \text{glucose}$).
• Exothermic Dissolution ($\Delta H_{soln} < 0$): Releasing heat means elevated temperature opposes dissolution. Solubility decreases with temperature (e.g., cerium sulfate $\text{Ce}_2(\text{SO}_4)_3$, calcium acetate, lithium carbonate $\text{Li}_2\text{CO}_3$).

2. Gases in Liquids:

Dissolution of a gas involves bringing freely translating gas molecules into a confined liquid phase. Because condensation releases intermolecular attractive energy and restricts translational freedom:

Because $\Delta H_{soln} < 0$, Eq. (5.6) dictates that the solubility of gases in liquids universally decreases as temperature rises.

• Environmental Implication (Thermal Pollution): Industrial discharge of warm cooling water into rivers decreases dissolved $\text{O}_2$ concentrations, causing mass asphyxiation of aquatic organisms.

Effect of Pressure on Gas Solubility: Henry's Law

While pressure has a negligible effect on the solubility of liquids and solids (due to their tiny molar volumes), it exerts a dominant effect on gas solubility.

In 1803, William Henry demonstrated empirically that at constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas above the solution:

Henry's Law:

where $C_g$ is the equilibrium concentration (molarity or molality) of the dissolved gas, $P_g$ is its partial pressure above the liquid, and $k_H$ is the Henry's Law Constant characteristic of the specific gas-solvent pair at temperature $T$. Alternatively, expressing concentration in mole fraction:

Thermodynamic Derivation of Henry's Law

At liquid-gas phase equilibrium, the chemical potential of the gas in the vapor phase must equal its chemical potential in the solution phase:

Assuming the gas behaves ideally:

In an infinitely dilute real solution ($x_g \rightarrow 0$), the solute obeys Henry's limiting law:

Equating expressions:

Since $\Delta G_{transfer}^\circ$ depends only on $T$, the ratio $P_g / x_g$ is an invariant constant at constant temperature, deriving Henry's Law from first thermodynamic principles!

Limitations of Henry's Law

Henry's Law fails when:

1. Gas pressure is very high ($P > 5 - 10\text{ bar}$), where non-ideal gas compressibility and solvent saturation effects manifest.
2. The gas undergoes a chemical reaction with the solvent. For example, carbon dioxide reacts with water:

and ammonia ionizes:

These chemical equilibria buffer and dramatically inflate total apparent gas solubility beyond Henry's linear prediction.

Physiological Application: Decompression Sickness ("The Bends")

According to Henry's Law, when a deep-sea diver breathes compressed air at depth ($P_{total} = 4\text{ bar}$ at $30\text{ m}$ depth), the partial pressure of nitrogen is $P(\text{N}_2) = 0.78 \times 4 = 3.12\text{ bar}$—four times atmospheric level. Substantial volumes of $\text{N}_2$ dissolve into blood and lipid-rich neural tissues. If the diver ascends rapidly to the surface, ambient hydrostatic pressure drops abruptly. The dissolved nitrogen becomes supersaturated, nucleating gas bubbles directly within bloodstream capillaries, joints, and myelin sheaths, causing excruciating pain, paralysis, or fatal embolism. Safe diving requires slow decompression stops, or substituting nitrogen with helium (Heliox), which has a much lower Henry's constant and lipid solubility.

Colligative properties (from Latin colligatus, meaning "bound together") are physical properties of solutions that depend strictly on the ratio of the number of solute particles to solvent particles, completely independent of the chemical identity, size, or structure of the solute.

The four classical colligative properties are:

1. Vapor Pressure Lowering ($\Delta P$)
2. Boiling Point Elevation ($\Delta T_b$)
3. Freezing Point Depression ($\Delta T_f$)
4. Osmotic Pressure ($\Pi$)

All colligative phenomena originate from a single microscopic reality: the addition of a nonvolatile solute dilutes the solvent, lowering its chemical potential $\mu_A(soln)$ relative to pure liquid solvent $\mu_A^*$:

Because $x_B > 0$, $\ln(1 - x_B) < 0$, which depresses the liquid chemical potential curve, expanding the temperature range over which the liquid phase is thermodynamically stable!

1. Vapor Pressure Lowering: Raoult's Law (1886)

For an ideal solution, Francois-Marie Raoult established that the partial vapor pressure of solvent $A$ above a solution is proportional to its mole fraction in the liquid phase:

where $P_A^\circ$ is the vapor pressure of pure solvent $A$ at that temperature. For a solution containing an involatile solute $B$ ($x_A = 1 - x_B$):

The relative vapor pressure lowering is exactly equal to the mole fraction of the solute:

2. Boiling Point Elevation ($\Delta T_b$)

Because the nonvolatile solute depresses the vapor pressure curve, the solution must be heated to a higher temperature before its vapor pressure equals the external atmospheric pressure ($P = 1\text{ bar}$).

At the elevated boiling point $T_b = T_b^* + \Delta T_b$, phase equilibrium requires $\mu_A(soln) = \mu_A(vap)$. Using Eq. (5.9) with pure vapor at standard pressure ($\mu_A(vap) = \mu_A^*(vap)$):

Differentiating with respect to temperature using the Gibbs-Helmholtz relation $\frac{d(\Delta G/T)}{dT} = -\frac{\Delta H}{T^2}$:

Integrating from pure solvent ($x_A = 1, T = T_b^*$) to solution ($x_A = 1 - x_B, T = T_b$):

For dilute solutions, $\ln(1 - x_B) \approx -x_B$, $\Delta T_b = T_b - T_b^ \ll T_b^$, so $T_b T_b^ \approx (T_b^)^2$:

Substituting $x_B \approx \frac{m M_A}{1000}$ (where $m$ is molality and $M_A$ is solvent molar mass in $\text{g/mol}$):

where $K_b \equiv \frac{R (T_b^*)^2 M_A}{1000 \Delta H_{vap}^\circ}$ is the Ebullioscopic Constant characteristic of the pure solvent (for water, $K_b = 0.512\text{ K}\cdot\text{kg}\cdot\text{mol}^{-1}$).

3. Freezing Point Depression ($\Delta T_f$)

When a solution freezes, pure solid solvent typically crystallizes out first (e.g., pure ice from seawater). Phase equilibrium between solid solvent and solution requires $\mu_A(soln) = \mu_A^*(solid)$. Following an identical thermodynamic derivation with $\Delta H_{fus}^\circ$:

where $K_f \equiv \frac{R (T_f^*)^2 M_A}{1000 \Delta H_{fus}^\circ}$ is the Cryoscopic Constant of the solvent (for water, $K_f = 1.86\text{ K}\cdot\text{kg}\cdot\text{mol}^{-1}$; for camphor, $K_f = 39.7\text{ K}\cdot\text{kg}\cdot\text{mol}^{-1}$).

4. Osmotic Pressure ($\Pi$): The van 't Hoff Equation

Osmosis is the spontaneous net transport of solvent molecules across a semi-permeable membrane (permeable to solvent but impermeable to solute particles) from a region of lower solute concentration (higher solvent chemical potential) to higher solute concentration.

To halt this osmotic flow, an external mechanical pressure $\Pi$ must be applied to the solution compartment. At osmotic equilibrium:

Since $\left(\frac{\partial \mu}{\partial P}\right)_T = V_{m,A}$:

Therefore:

For a dilute solution, $x_B \approx n_B / n_A$ and $n_A V_{m,A} \approx V_{total}$:

This is the celebrated van 't Hoff Equation for Osmotic Pressure (1887), bearing an identical mathematical form to the ideal gas law!

Electrolyte Solutions & The van 't Hoff Factor ($i$)

When an electrolyte dissolves in water, it dissociates into ions, yielding more particles in solution than stoichiometric formula units:

All colligative property equations for electrolytes incorporate $i$:

Degree of Dissociation ($\alpha$)

Consider a weak electrolyte $A_{\nu_+} B_{\nu_-}$ that dissociates into $\nu = \nu_+ + \nu_-$ total ions:

• Initial moles: $1$
• Moles dissociated: $\alpha$
• Remaining undissociated: $1 - \alpha$
• Ions produced: $\nu \alpha$
• Total moles of particles at equilibrium: $n_{tot} = (1 - \alpha) + \nu \alpha = 1 + (\nu - 1)\alpha$.

By definition, $i = n_{tot} / 1 = 1 + (\nu - 1)\alpha$. Rearranging isolates the degree of dissociation:

In real solutions at finite concentrations, $i$ is slightly lower than ideal integer values ($\nu$) due to ion pairing and electrostatic screening described by the Debye-Hückel Theory.

When a solute is added to a system containing two mutually immiscible (or partially miscible) liquid solvents, the solute distributes itself between the two liquid phases until thermodynamic partition equilibrium is achieved.

The Nernst Distribution Law (1891)

Let solute $A$ distribute between two liquid phases 1 and 2 at constant temperature $T$:

At thermodynamic equilibrium, the chemical potential of the solute in phase 1 must equal its chemical potential in phase 2:

Expressing chemical potentials in dilute solutions:

Since $\Delta G_{partition}^\circ$ is a function purely of temperature, the concentration ratio is constant at fixed temperature:

The Nernst Distribution Law:

where $K_D$ is the Partition Coefficient (Distribution Coefficient).

Conditions for Validity:

1. The temperature must be constant throughout the extraction.
2. Both liquid phases must be mutually immiscible (or mutually saturated).
3. The solute must exist in the identical molecular state (same molecular weight and chemical form) in both phases, with neither dissociation nor association occurring in either solvent.

Deviations: Association and Dissociation

When the solute undergoes chemical changes in one of the phases:

1. Association in One Phase:

Suppose the solute exists as normal monomeric molecules $A$ in phase 1 (concentration $C_1$), but associates into $n$-mers ($A_n$) in phase 2 (e.g., benzoic acid dimerizing in benzene via dual hydrogen bonds: $2\text{C}_6\text{H}_5\text{COOH} \rightleftharpoons (\text{C}_6\text{H}_5\text{COOH})_2$):

The Nernst partition applies strictly between monomeric species:

2. Dissociation in One Phase:

Suppose the solute undergoes electrolytic dissociation in aqueous phase 1 into ions with degree of dissociation $\alpha$:

The concentration of undissociated neutral molecules is $C_1(1 - \alpha)$. Nernst distribution applies only to the un-ionized neutral species:

Theory of Multi-Stage Solvent Extraction

Solvent extraction is an indispensable analytical and industrial technique used to extract organic compounds, pharmaceuticals, and metal complexes from aqueous media into organic solvents.

Let $W_0$ be the initial mass of solute dissolved in volume $V$ of water (solvent 1). Suppose we extract this solution using volume $v$ of an immiscible organic solvent (solvent 2) in which the distribution coefficient is:

Single Extraction:

After the first extraction, let $W_1$ be the mass of solute remaining unextracted in the aqueous phase.

• Mass extracted into organic phase: $W_0 - W_1$.
• Concentration in aqueous phase: $C_1 = \frac{W_1}{V}$.
• Concentration in organic phase: $C_2 = \frac{W_0 - W_1}{v}$.

From the distribution law:

If we define partition coefficient as $P = \frac{C_{org}}{C_{aq}} = \frac{1}{K_D}$:

Multiple ($n$) Successive Extractions:

If the aqueous solution is extracted $n$ successive times, each time using a fresh volume $v$ of organic solvent, mathematical induction yields the general Extraction Formula:

Fundamental Extraction Principle: For a fixed total volume of extracting solvent $V_{total} = n \cdot v$, performing multiple extractions with small portions of solvent is vastly more efficient than performing a single extraction with the entire solvent volume. Mathematically, since $\left(\frac{V}{V + P(V_{total}/n)}\right)^n < \frac{V}{V + P V_{total}}$ for all $n > 1$, multi-stage extraction minimizes the unextracted solute $W_n$ exponentially.