9.3 Solutions: Raoult's and Henry's Laws, Colligative Properties, and Electrolytes

Solutions: Raoult's and Henry's Laws, Colligative Properties, and Electrolytes

A solution is a homogeneous mixture of two or more substances down to the molecular scale. In physical chemistry and on the PRC Chemist Licensure Examination, solution theory spans ideal solutions, volatile and non-volatile solute behaviors, colligative phenomena, macromolecular characterization by osmometry, and the non-ideality of electrolyte solutions treated by Debye-Hückel theory.

---

1. Ideal Solutions, Raoult's Law, and Thermodynamics of Mixing

An ideal solution is defined at the molecular level as a mixture in which the intermolecular forces between unlike molecules ($A-B$) are identical in magnitude to those between like molecules ($A-A$ and $B-B$).

Raoult's Law

For an ideal solution in equilibrium with its vapor, the partial vapor pressure of each volatile component $i$ is proportional to its mole fraction in the liquid phase ($x_i$):

$P_i = x_i P_i^*$

where $P_i^*$ is the saturation vapor pressure of pure component $i$ at the specified temperature.

For a binary mixture of volatile components $A$ and $B$, the total vapor pressure is:

$P_{tot} = P_A + P_B = x_A P_A^* + x_B P_B^* = x_A P_A^* + (1 - x_A) P_B^* = P_B^* + x_A (P_A^* - P_B^*)$

The mole fraction of component $A$ in the coexisting vapor phase ($y_A$) is given by Dalton's law:

$y_A = \frac{P_A}{P_{tot}} = \frac{x_A P_A^*}{x_A P_A^* + x_B P_B^*}$

If component $A$ is more volatile than $B$ ($P_A^* > P_B^*$), then $y_A > x_A$, meaning the vapor is enriched in the more volatile component.

Thermodynamic Functions of Mixing for Ideal Solutions

When $n_A$ moles of pure $A$ and $n_B$ moles of pure $B$ mix to form an ideal solution at constant $T$ and $P$:

1. Enthalpy of Mixing: No net bond breaking or making energetics occurs: $\Delta_{mix} H = 0$
2. Volume of Mixing: Molecules pack identically to their pure states: $\Delta_{mix} V = 0$
3. Entropy of Mixing: Driven entirely by configurational disorder ($x_i < 1 \implies \ln x_i < 0$): $\Delta_{mix} S = -nR \sum_{i} x_i \ln x_i > 0$
4. Gibbs Free Energy of Mixing: $\Delta_{mix} G = \Delta_{mix} H - T\Delta_{mix} S = nRT \sum_{i} x_i \ln x_i < 0$ Because $\Delta_{mix} G < 0$ for all compositions, ideal mixing is spontaneous at all temperatures.

---

2. Dilute Real Solutions and Henry's Law

In real solutions, $A-A$, $B-B$, and $A-B$ intermolecular potentials differ. However, in the limit of high dilution (where solute mole fraction $x_2 \rightarrow 0$ and solvent mole fraction $x_1 \rightarrow 1$):

• Solvent Behavior (Raoult's Law): Every solvent molecule is surrounded virtually exclusively by other solvent molecules. Thus, the solvent behaves ideally and obeys Raoult's Law: $P_1 = x_1 P_1^* \quad (\text{as } x_1 \rightarrow 1)$

• Solute Behavior (Henry's Law): Every solute molecule is completely surrounded by solvent molecules, experiencing a constant solvated environment. The partial pressure of the solute is directly proportional to its mole fraction, but governed by Henry's Law constant ($K_{H, 2}$): $P_2 = K_{H, 2} x_2 \quad (\text{as } x_2 \rightarrow 0)$

Alternatively, expressing solute concentration in molarity ($c$): $c = k_H P$

Because the dissolution of gases in liquids is generally an exothermic process ($\Delta H_{soln} < 0$), the solubility of gases in water decreases with increasing temperature, as predicted by Le Chatelier's principle.

---

3. Colligative Properties of Non-Electrolyte Solutions

Colligative properties are properties of solutions that depend solely on the number of solute particles dissolved per unit volume or mass of solvent, and are independent of the chemical identity, size, or charge of the solute particles.

Thermodynamically, all colligative properties originate from a single phenomenon: the lowering of the solvent's chemical potential in solution compared to pure liquid solvent:

$\mu_1(soln) = \mu_1^*(l) + RT \ln x_1 < \mu_1^*(l) \quad (\text{since } x_1 < 1)$

This stabilization of the liquid phase expands the liquid temperature range, lowering both the vapor pressure and the freezing point, while elevating the boiling point.

The Four Colligative Properties

Exam Application: Macromolecular Molar Mass by Osmometry For polymers and proteins ($M > 10000\text{ g/mol}$), freezing point depression produces temperature changes of less than $0.001^\circ\text{C}$, which cannot be measured accurately. In contrast, osmotic pressure produces hydrostatic column heights of several centimeters, making osmometry the primary colligative method for determining macromolecular weights: $M = \frac{w R T}{\Pi V}$ where $w$ is the mass of dissolved solute in grams.

---

4. Electrolyte Solutions and the van 't Hoff Factor ($i$)

When ionic salts or ionizable acids dissolve, each formula unit dissociates into $\nu$ ions, producing a greater number of colligative particles than expected from stoichiometry.

The van 't Hoff factor ($i$) is defined as:

$i = \frac{\text{Actual number of particles in solution}}{\text{Number of formula units dissolved}} = \frac{\text{Measured colligative property}}{\text{Calculated property for non-electrolyte}}$

Incorporating $i$ into the colligative formulas: $\Delta T_b = i K_b m, \quad \Delta T_f = i K_f m, \quad \Pi = i c R T$

Degree of Dissociation ($\alpha$)

For a weak electrolyte where each formula unit dissociates into $\nu$ ions with a fractional degree of dissociation $\alpha$:

$\begin{matrix} & A_\nu & \rightleftharpoons & \nu \text{ ions} \\ \text{Initial:} & 1 & & 0 \\ \text{Equilibrium:} & 1 - \alpha & & \nu\alpha \end{matrix}$

Total particles at equilibrium $= (1 - \alpha) + \nu\alpha = 1 + \alpha(\nu - 1)$. Therefore:

$i = 1 + \alpha(\nu - 1) \implies \alpha = \frac{i - 1}{\nu - 1}$

• Ideal vs. Real Electrolytes: For strong electrolytes like $NaCl$ ($\nu = 2$), $i$ approaches $2.00$ at infinite dilution. However, in real solutions ($0.05\text{ M}$), $i \approx 1.90$ due to electrostatic attraction and the formation of transient ion pairs ($[Na^+ Cl^-]^0$).

---

5. Non-Ideal Electrolytes: Ionic Strength and Debye-Hückel Theory

In electrolyte solutions, long-range Coulombic forces ($V(r) \propto 1/r$) prevent ions from behaving independently even at concentrations below $0.01\text{ M}$. Peter Debye and Erich Hückel developed the ionic atmosphere model, where every central cation is statistically surrounded by a net excess of anions, and vice versa.

Ionic Strength ($I$)

The electrostatic field strength of an electrolyte solution depends on both ion concentrations and their charges, quantified by the ionic strength:

$I = \frac{1}{2} \sum_{i} c_i z_i^2 \quad \left[ \text{or } I = \frac{1}{2} \sum_{i} m_i z_i^2 \right]$

where $c_i$ is molar concentration and $z_i$ is ionic valence.

• $1:1$ Salt ($NaCl$, $c$): $I = \frac{1}{2} [c(1)^2 + c(-1)^2] = c$
• $2:1$ Salt ($CaCl_2$, $c$): $I = \frac{1}{2} [c(2)^2 + 2c(-1)^2] = \frac{1}{2} [4c + 2c] = 3c$
• $2:2$ Salt ($MgSO_4$, $c$): $I = \frac{1}{2} [c(2)^2 + c(-2)^2] = \frac{1}{2} [4c + 4c] = 4c$
• $3:1$ Salt ($FeCl_3$, $c$): $I = \frac{1}{2} [c(3)^2 + 3c(-1)^2] = \frac{1}{2} [9c + 3c] = 6c$
• $3:2$ Salt ($Al_2(SO_4)_3$, $c$): $I = \frac{1}{2} [2c(3)^2 + 3c(-2)^2] = \frac{1}{2} [18c + 12c] = 15c$

Mean Ionic Activity and the Debye-Hückel Limiting Law (DHLL)

Because individual ion activities cannot be measured independently due to electroneutrality, we use the mean ionic activity coefficient $\gamma_\pm$:

$\gamma_\pm = \left( \gamma_+^{\nu_+} \gamma_-^{\nu_-} \right)^{1/\nu} \quad \text{where } \nu = \nu_+ + \nu_-$

For very dilute solutions ($I < 0.01\text{ M}$), the Debye-Hückel Limiting Law states:

$\log_{10} \gamma_\pm = -A |z_+ z_-| \sqrt{I}$

In aqueous solution at $25.0^\circ\text{C}$ ($298.15\text{ K}$), the solvent constant is $A = 0.509\text{ kg}^{1/2}\text{mol}^{-1/2}$ (or $\text{L}^{1/2}\text{mol}^{-1/2}$).

For higher concentrations ($0.01\text{ M} < I < 0.1\text{ M}$), the Extended Debye-Hückel equation accounts for finite ion diameter ($a^\circ$):

$\log_{10} \gamma_\pm = -\frac{A |z_+ z_-| \sqrt{I}}{1 + B a^\circ \sqrt{I}}$

---

6. Worked Examination Calculations

Problem A: Molar Mass of a Protein via Membrane Osmometry

Task: A $1.250\text{ g}$ sample of a purified enzyme is dissolved in sufficient isotonic buffer to yield $100.0\text{ mL}$ of solution. At $25.0^\circ\text{C}$, the osmotic pressure is measured as $4.350\text{ mmHg}$. Determine the molar mass of the enzyme.

Solution: $\Pi = \frac{4.350\text{ mmHg}}{760.0\text{ mmHg/atm}} = 5.7237 \times 10^{-3}\text{ atm}$ Using the van 't Hoff osmotic pressure equation $\Pi = \frac{n}{V} R T = \frac{w}{M V} R T$: $M = \frac{w R T}{\Pi V} = \frac{(1.250\text{ g})(0.082057\text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K}))(298.15\text{ K})}{(5.7237 \times 10^{-3}\text{ atm})(0.1000\text{ L})}$ $M = \frac{30.582}{5.7237 \times 10^{-4}} = 53430\text{ g/mol} = 5.34 \times 10^4\text{ g/mol}$

Comparison: What would the freezing point depression be for this same solution in water ($K_f = 1.86^\circ\text{C}\cdot\text{kg/mol}$)? $m \approx \frac{n}{0.100\text{ kg}} = \frac{1.250 / 53430}{0.100} = 2.34 \times 10^{-4}\text{ mol/kg}$ $\Delta T_f = (1.86)(2.34 \times 10^{-4}) = 0.000435^\circ\text{C}$ A $\Delta T_f$ of $0.0004^\circ\text{C}$ is indistinguishable from thermal noise on a standard lab thermometer, whereas $\Pi = 4.35\text{ mmHg}$ corresponds to a column height of $5.9\text{ cm}$ of water ($h = \frac{P}{\rho g}$), proving why osmometry is the definitive colligative tool for biochemistry.

Problem B: Degree of Dissociation of a Weak Acid

Task: A $0.0500\text{ m}$ aqueous solution of monochloroacetic acid ($CH_2ClCOOH$, $\nu = 2$) freezes at $-0.1080^\circ\text{C}$. For water, $K_f = 1.860^\circ\text{C}\cdot\text{kg/mol}$. Calculate the van 't Hoff factor $i$ and the percent dissociation $\alpha$.

Solution: $\Delta T_f = 0.1080^\circ\text{C}$ $i = \frac{\Delta T_f}{K_f m} = \frac{0.1080^\circ\text{C}}{(1.860^\circ\text{C}\cdot\text{kg/mol})(0.0500\text{ mol/kg})} = \frac{0.1080}{0.09300} = 1.1613$

For a monoprotic acid ($CH_2ClCOOH \rightleftharpoons H^+ + CH_2ClCOO^-$), $\nu = 2$: $\alpha = \frac{i - 1}{\nu - 1} = \frac{1.1613 - 1}{2 - 1} = 0.1613 \implies 16.13\%$

Problem C: Ionic Strength and Mean Activity Coefficient

Task: Calculate the ionic strength and mean ionic activity coefficient $\gamma_\pm$ of a $0.0050\text{ M}$ aqueous solution of $Ca(NO_3)_2$ at $25^\circ\text{C}$.

Solution: $Ca(NO_3)_2(aq) \rightarrow Ca^{2+}(aq) + 2NO_3^-(aq)$.

• $[Ca^{2+}] = 0.0050\text{ M}$, $z_+ = +2$
• $[NO_3^-] = 2 \times 0.0050 = 0.0100\text{ M}$, $z_- = -1$

$I = \frac{1}{2} \left[ (0.0050)(+2)^2 + (0.0100)(-1)^2 \right] = \frac{1}{2} [ 0.0200 + 0.0100 ] = 0.0150\text{ M}$

Using the Debye-Hückel Limiting Law ($A = 0.509$): $\log_{10} \gamma_\pm = -A |z_+ z_-| \sqrt{I} = -(0.509) |(+2)(-1)| \sqrt{0.0150} = -(0.509)(2)(0.12247)$ $\log_{10} \gamma_\pm = -0.12468$ $\gamma_\pm = 10^{-0.12468} = 0.7504 \approx 0.750$