4.4 Concentration Units, Henry's Law & Colligative Properties

4.4 Concentration Units, Henry's Law & Colligative Properties

5. Solution Concentration Units and Henry's Law

Concentration Definitions

• Molarity ($M$): $$M = \frac{\text{moles of solute}}{\text{liters of solution}} \quad (\text{mol/L})$$ Temperature Dependent: Because liquid volume expands with increasing temperature, molarity decreases as temperature rises.
• Molality ($m$): $$m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \quad (\text{mol/kg})$$ Temperature Independent: Mass does not alter with thermal expansion. Essential for colligative property calculations.
• Mole Fraction ($X_A$): $$X_A = \frac{n_A}{n_{\text{total}}} = \frac{n_A}{n_A + n_B + \dots}$$ Dimensionless and Temperature Independent.
• Mass Percent (w/w%): $$\text{Mass }\% = \left(\frac{\text{mass of solute}}{\text{total mass of solution}}\right) \times 100\%$$

Henry's Law of Gas Solubility

Henry's law states that at a constant temperature, the solubility of a gas ($C$) in a liquid is directly proportional to the partial pressure ($P$) of that gas above the liquid:

$$C = k_H P$$

Where $k_H$ is Henry's law constant (specific to the gas, solvent, and temperature).

• Temperature Dependence of Gas Solubility: Dissolving a gas into a liquid brings free gas molecules into an ordered condensed phase, which is an exothermic process ($\Delta H_{\text{soln}} < 0$). By Le Chatelier's principle: $$\text{Gas}_{(g)} + \text{Solvent}_{(l)} \rightleftharpoons \text{Dissolved Gas}_{(aq)} + \text{Heat}$$ Increasing temperature shifts the equilibrium in reverse, decreasing gas solubility. Warm water holds significantly less dissolved oxygen than cold water.
• Clinical Relevance: In hyperbaric oxygen chambers, raising $P_{\text{O}_2}$ forces more dissolved oxygen directly into blood plasma independently of hemoglobin saturation. Conversely, during rapid scuba ascent, dropping ambient hydrostatic pressure decreases nitrogen solubility, causing dissolved $\text{N}_2$ to precipitate out of blood as painful gas bubbles (decompression sickness / "the bends").

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6. Colligative Properties and the van 't Hoff Factor ($i$)

Colligative properties depend exclusively on the total number (concentration) of dissolved solute particles in solution, regardless of their chemical identity, mass, or size.

The van 't Hoff Factor ($i$)

The van 't Hoff factor accounts for the dissociation of ionic electrolytes into multiple particle species:

$$\ni = \frac{\text{moles of particles in solution}}{\text{moles of solute formula units dissolved}}$$

• Nonelectrolytes (glucose, sucrose, urea): $i = 1$.
• Strong Electrolytes (ideal dissociation):
  • $\text{NaCl} \rightarrow \text{Na}^+ + \text{Cl}^- \implies i = 2$
  • $\text{CaCl}_2 \rightarrow \text{Ca}^{2+} + 2\text{Cl}^- \implies i = 3$
  • $\text{Al(NO}_3)_3 \rightarrow \text{Al}^{3+} + 3\text{NO}_3^- \implies i = 4$
• Non-ideal behavior: Real van 't Hoff factors are slightly less than whole integers at high concentrations due to transient electrostatic ion pairing.

The Four Colligative Properties

1. Vapor Pressure Lowering (Raoult's Law): Dissolving a non-volatile solute reduces the mole fraction of volatile solvent at the liquid surface, decreasing the rate of solvent vaporization. For an ideal solution:

   $$P_A = X_A P_A^\circ$$

   Where $P_A$ is the vapor pressure of the solution, $X_A$ is the solvent mole fraction, and $P_A^\circ$ is the vapor pressure of pure solvent. The vapor pressure lowering ($\Delta P$) is:

   $$\Delta P = P_A^\circ - P_A = X_B P_A^\circ = (i \cdot X_{\text{solute}}) P_A^\circ$$
   • Deviations from Raoult's Law:
     • Positive Deviation: Solute-solvent attractions are weaker than solute-solute or solvent-solvent attractions. Molecules escape more readily into the vapor phase, resulting in a higher observed vapor pressure and endothermic mixing ($\Delta H_{\text{mix}} > 0$, e.g., ethanol and hexane).
     • Negative Deviation: Solute-solvent attractions are stronger than interactions in the pure components (e.g., hydrogen bonding between acetone and chloroform). Molecules are held more tightly in the liquid, resulting in a lower observed vapor pressure and exothermic mixing ($\Delta H_{\text{mix}} < 0$).

2. Boiling Point Elevation: A liquid boils when its vapor pressure equals atmospheric pressure. Because solute lowers vapor pressure, a higher temperature is required to reach atmospheric pressure:

   $$\Delta T_b = i K_b m$$

   Where $K_b$ is the ebullioscopic constant (for water, $K_b = 0.512^\circ\text{C}\cdot\text{kg/mol}$) and $m$ is molality. The new boiling point is $T_b = T_b^\circ + \Delta T_b$.

3. Freezing Point Depression: Solute particles disrupt the orderly packing of solvent molecules into a crystalline solid lattice, requiring lower thermal energy to solidify:

   $$\Delta T_f = i K_f m$$

   Where $K_f$ is the cryoscopic constant (for water, $K_f = 1.86^\circ\text{C}\cdot\text{kg/mol}$). The new freezing point is $T_f = T_f^\circ - \Delta T_f$. Practical applications include salting icy roads and ethylene glycol automotive antifreeze.

4. Osmotic Pressure ($\Pi$): Osmotic pressure is the exact hydrostatic pressure that must be applied to a solution to prevent the net inward osmotic flow of pure water across a semipermeable membrane:

   $$\Pi = i M R T$$

   Where $M$ is molarity, $R = 0.0821\text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K})$, and $T$ is absolute temperature in Kelvin.

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7. Physiological Applications: Intravenous Fluids, Osmolarity & Erythrocytes

Normal human blood plasma has a physiological osmolarity of approximately $280 - 300\text{ mOsm/L}$ (primarily generated by dissolved $\text{Na}^+, \text{Cl}^-$, bicarbonate, glucose, and serum albumin).

Tonicity and Erythrocyte Morphologies

• Isotonic Solutions: Possess an effective osmolarity equal to intracellular fluid (e.g., $0.9\%\text{ w/v NaCl}$ normal saline $\approx 308\text{ mOsm/L}$, or $5\%\text{ w/v}$ dextrose in water [D5W]). Zero net water flux occurs across the erythrocyte membrane; red blood cells maintain normal biconcave disc morphology.
• Hypertonic Solutions ($> 300\text{ mOsm/L}$, e.g., $3.0\%\text{ NaCl}$): External solute concentration exceeds intracellular concentration. Water rapidly diffuses down its chemical potential gradient out of the cell into the hypertonic extracellular environment, causing erythrocyte cellular shrinkage and wrinkling (crenation).
• Hypotonic Solutions ($< 280\text{ mOsm/L}$, e.g., $0.45\%\text{ NaCl}$ or sterile pure water): External solute concentration is lower than intracellular fluid. Water rushes osmotically into the erythrocyte, causing cellular swelling, membrane disruption, and cell rupture (osmotic hemolysis). Intravenous administration of pure water is fatal.

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8. Worked Numerical Examples

Example 1: Colligative Shift for an Electrolyte Solution

Calculate the freezing point ($T_f$) and boiling point ($T_b$) of an aqueous solution prepared by dissolving $111\text{ g}$ of calcium chloride ($\text{CaCl}_2$, molar mass $= 111.0\text{ g/mol}$) into $1000\text{ g}$ ($1.00\text{ kg}$) of pure distilled water, assuming ideal electrolyte dissociation. ($K_f = 1.86^\circ\text{C}\cdot\text{kg/mol}$, $K_b = 0.512^\circ\text{C}\cdot\text{kg/mol}$).

Solution:

1. Calculate moles of solute: $$n = \frac{111\text{ g}}{111.0\text{ g/mol}} = 1.00\text{ mol } \text{CaCl}_2$$
2. Calculate molality ($m$): $$m = \frac{1.00\text{ mol}}{1.00\text{ kg solvent}} = 1.00\text{ m}$$
3. Determine van 't Hoff factor ($i$): $$\text{CaCl}_2(s) \rightarrow \text{Ca}^{2+}(aq) + 2\text{Cl}^-(aq) \implies i = 1 + 2 = 3$$
4. Calculate freezing point depression ($\Delta T_f$): $$\Delta T_f = i K_f m = (3)(1.86^\circ\text{C}\cdot\text{kg/mol})(1.00\text{ mol/kg}) = 5.58^\circ\text{C}$$ $$T_f = 0.00^\circ\text{C} - 5.58^\circ\text{C} = -5.58^\circ\text{C}$$
5. Calculate boiling point elevation ($\Delta T_b$): $$\Delta T_b = i K_b m = (3)(0.512^\circ\text{C}\cdot\text{kg/mol})(1.00\text{ mol/kg}) = 1.536^\circ\text{C}$$ $$T_b = 100.00^\circ\text{C} + 1.54^\circ\text{C} = 101.54^\circ\text{C}$$

Example 2: Osmotic Pressure of a Protein Solution

A physician analyzes an artificial colloidal plasma expander containing human serum albumin (molar mass $\approx 66,500\text{ g/mol}$) at a concentration of $40.0\text{ g/L}$ at normal body temperature ($37.0^\circ\text{C} = 310.15\text{ K}$). Assuming albumin behaves as a monomeric nonelectrolyte ($i = 1$), calculate the osmotic pressure (oncotic pressure) in atmospheres and in $\text{mmHg}$ ($1\text{ atm} = 760\text{ mmHg}$).

Solution:

1. Calculate the molar concentration ($M$): $$M = \frac{40.0\text{ g/L}}{66,500\text{ g/mol}} = 6.015 \times 10^{-4}\text{ M}$$
2. Apply the van 't Hoff osmotic pressure equation ($\Pi = i M R T$): $$\Pi = (1)(6.015 \times 10^{-4}\text{ mol/L})(0.0821\text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K}))(310.15\text{ K})$$ $$\Pi = 1.532 \times 10^{-2}\text{ atm}$$
3. Convert to millimeters of mercury ($\text{mmHg}$): $$\Pi = (1.532 \times 10^{-2}\text{ atm}) \times 760\text{ mmHg/atm} \approx 11.6\text{ mmHg}$$ This oncotic pressure acts across capillary walls to pull fluid back from the interstitial space into the vascular lumen.