7.1 Solution Formation, Solubility Factors & Henry's Law

7.1 Solution Formation, Solubility Factors & Henry's Law

Quick Summary: A solution forms when solute particles disperse homogeneously within a solvent. The overall enthalpy of solution ($\Delta H_{\text{soln}}$) represents a balance between endothermic solute separation, endothermic solvent cavity formation, and exothermic solute-solvent attraction (hydration). Dissolution is driven by a favorable increase in entropy of mixing ($\Delta S_{\text{mix}} > 0$). While most solid solutes dissolve more readily at elevated temperatures, gas dissolution in water is typically exothermic, so gas solubility generally decreases as temperature climbs. Henry's law ($C = k_{\text{H}} P_{\text{gas}}$) quantitatively relates gas solubility to partial pressure above the solution.

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1. Solution Terminology & Saturation Equilibria

A solution is a homogeneous mixture existing as a single phase. The component present in the largest molar amount is the solvent, while the dispersed species is the solute:

• Unsaturated Solution: Contains less dissolved solute than the thermodynamic equilibrium limit at a given temperature; added solute dissolves readily.
• Saturated Solution: Solute and dissolved particles exist in dynamic equilibrium: $\text{Undissolved Solute} \rightleftharpoons \text{Dissolved Solute}$ The rate of dissolution equals the rate of crystallization. The concentration of dissolved solute at this equilibrium is the substance's solubility.
• Supersaturated Solution: Contains more dissolved solute than the equilibrium saturation limit, prepared by cooling a saturated solution slowly in a pristine vessel. Supersaturated solutions are metastable; introducing a microscopic seed crystal or mechanical agitation triggers rapid crystallization of excess solute until equilibrium is restored.

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2. Thermodynamics of Solution Formation

Dissolution involves three energetic steps modeled in a thermodynamic cycle:

1. Solute Separation ($\Delta H_{\text{solute}} > 0$): Energy consumed to overcome attractive forces holding solute particles together. For crystalline ionic salts, this endothermic quantity equals the crystal lattice energy ($\Delta H_{\text{solute}} = -\Delta H_{\text{lattice}} > 0$).
2. Solvent Expansion ($\Delta H_{\text{solvent}} > 0$): Energy consumed to disrupt solvent-solvent interactions (e.g., hydrogen bonds in water) to create cavities for solute species.
3. Solvation / Mixing ($\Delta H_{\text{mix}} < 0$): Energy released as attractive forces form between solute and solvent molecules. In water, combining steps 2 and 3 defines the hydration enthalpy ($\Delta H_{\text{hyd}} = \Delta H_{\text{solvent}} + \Delta H_{\text{mix}} < 0$).

By Hess's law: $\Delta H_{\text{soln}} = \Delta H_{\text{solute}} + \Delta H_{\text{solvent}} + \Delta H_{\text{mix}}$

• Exothermic Dissolution ($\Delta H_{\text{soln}} < 0$): Occurs when hydration energy exceeds component separation energies ($|\Delta H_{\text{mix}}| > \Delta H_{\text{solute}} + \Delta H_{\text{solvent}}$), as seen with $\text{CaCl}_2$ (used in some instant hot packs) and $\text{NaOH}$.
• Endothermic Dissolution ($\Delta H_{\text{soln}} > 0$): Occurs when separation energies exceed hydration release, as seen with $\text{NH}_4\text{NO}_3$ (used in instant cold packs) and $\text{KCl}$.

Entropy of Mixing & Gibbs Free Energy

Endothermic salts dissolve spontaneously because dissolution disperses matter and energy: $\Delta G_{\text{soln}} = \Delta H_{\text{soln}} - T\Delta S_{\text{soln}}$ Transitioning a rigid ionic lattice into mobile hydrated ions creates a large positive entropy of mixing ($\Delta S_{\text{soln}} > 0$). At room temperature, $-T\Delta S_{\text{soln}}$ outweighs positive $\Delta H_{\text{soln}}$, yielding $\Delta G_{\text{soln}} < 0$.

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3. "Like Dissolves Like" & the Hydrophobic Effect

• Polar/Ionic Solutes in Polar Solvents: Water dissolves salts and small alcohols because ion-dipole and dipole-dipole attractions provide sufficient hydration energy to offset separation enthalpies.
• Nonpolar Solutes in Nonpolar Solvents: Hydrocarbons mix freely in nonpolar solvents (e.g., heptane in hexane) because dispersion forces in solution are comparable to those in pure components ($\Delta H_{\text{soln}} \approx 0$), making positive entropy the driving force.
• The Hydrophobic Effect: Nonpolar molecules cannot form dipole interactions with water. To accommodate them, water molecules must organize into ordered, hydrogen-bonded clathrate cages. This rigid ordering produces an unfavorable decrease in water entropy ($\Delta S < 0$), causing $\Delta G_{\text{soln}} > 0$ and driving phase separation (immiscibility).

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4. Factors Affecting Solubility

Temperature Dependence

• Solids in Liquids: By Le Chatelier's principle:
  • Endothermic dissolution ($\Delta H_{\text{soln}} > 0$): solubility increases with temperature ($\text{KNO}_3$, glucose).
  • Exothermic dissolution ($\Delta H_{\text{soln}} < 0$): solubility decreases with temperature ($\text{Ce}_2(\text{SO}_4)_3$, $\text{Na}_2\text{SO}_4$ above 32 °C).
• Gases in Liquids: Gas molecules already possess translational independence ($\Delta H_{\text{solute}} \approx 0$). Dissolution in water is typically exothermic ($\Delta H_{\text{soln}} < 0$) and decreases entropy ($\Delta S < 0$). Added thermal energy allows dissolved gases to overcome solvent attractions and escape; thus, gas solubility in water generally decreases as temperature increases.
• Thermal Pollution: Industrial power plants returning heated water to lakes deplete dissolved $O_2$, suffocating aquatic fauna.

Pressure Dependence: Henry's Law

Pressure negligibly affects solids and liquids but dictates gas solubility. Henry's Law states that gas solubility ($C$) is directly proportional to its partial pressure ($P_{\text{gas}}$) above the liquid: $C = k_{\text{H}} P_{\text{gas}} \quad \text{or} \quad \frac{S_1}{P_1} = \frac{S_2}{P_2}$ Higher partial pressure increases gas-liquid surface collision frequency, establishing equilibrium at higher dissolved concentration.

Practical Applications

• Carbonated Beverages: Bottled under 3–5 atm $\text{CO}_2$. Opening lowers $\text{CO}_2$ partial pressure to atmospheric levels ($0.0004\text{ atm}$), driving $\text{CO}_2$ out of solution as effervescent bubbles.
• Decompression Sickness ("The Bends"): Divers breathing compressed air at depth dissolve excess $\text{N}_2$ into blood and tissues. Rapid ascent depressurizes blood faster than gas can be exhaled, forming debilitating $\text{N}_2$ gas bubbles in joints and capillaries. Prevented by slow staged ascent or heliox mixtures.

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5. Solubility Factors Summary

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6. Worked Example: Henry's Law Calculation

Problem: At 20.0 °C, the Henry's law constant for $\text{N}_2$ in water is $6.8 \times 10^{-4}\text{ mol/(L}\cdot\text{atm)}$. Air is 78.0% $\text{N}_2$ by volume.

1. Find the solubility of $\text{N}_2$ in water at $1.00\text{ atm}$ air pressure.
2. Find the solubility at an underwater depth where total pressure is $3.50\text{ atm}$.

Step 1: Calculate partial pressures of $\text{N}_2$ $P_{\text{N}_2, 1} = 0.780 \times 1.00\text{ atm} = 0.780\text{ atm}$ $P_{\text{N}_2, 2} = 0.780 \times 3.50\text{ atm} = 2.73\text{ atm}$

Step 2: Apply Henry's Law $C_1 = (6.8 \times 10^{-4}\text{ mol/(L}\cdot\text{atm)}) \times 0.780\text{ atm} = 5.3 \times 10^{-4}\text{ mol/L}$ $C_2 = (6.8 \times 10^{-4}\text{ mol/(L}\cdot\text{atm)}) \times 2.73\text{ atm} = 1.9 \times 10^{-3}\text{ mol/L}$

Dissolved $\text{N}_2$ concentration increases 3.5-fold at depth, quantifying the risk of rapid ascent.