Solutions — Notes

A solution is a homogeneous mixture of two or more components. The component in the larger amount is the solvent and decides the physical state; the others are solutes. A binary solution has two components. Nine types are possible (gas, liquid or solid solute in a gas, liquid or solid solvent), e.g. air, soda water, sugar syrup, brass, hydrogen in palladium.

Concentration can be stated in several ways:

• Mass percentage $= \dfrac{\text{mass of component}}{\text{mass of solution}} \times 100$; volume percentage similarly with volumes; mass by volume percentage = grams of solute per 100 mL of solution.
• Parts per million (ppm) for traces: parts of component per $10^{6}$ parts of solution (by mass, volume or mass/volume).
• Mole fraction $x_\text{B} = \dfrac{n_\text{B}}{\sum n}$; all mole fractions add up to 1.
• Molarity $M$ = moles of solute per litre of solution (changes with temperature because volume does).
• Molality $m$ = moles of solute per kilogram of solvent (independent of temperature, like mass % and mole fraction).

When a percentage and a density are given, assume $100\ \text{g}$ of solution; the volume is mass ÷ density.

Solubility is the maximum amount of a substance that dissolves in a specified amount of solvent at a specified temperature. A saturated solution is in dynamic equilibrium with undissolved solute.

Solids in liquids. Like dissolves like: ionic and polar solutes dissolve in polar solvents (water), non-polar solutes in non-polar solvents (benzene). If dissolving is endothermic, solubility increases with temperature; if exothermic, it decreases (Le Chatelier’s principle). Pressure has no significant effect.

Gases in liquids. Solubility increases with pressure and decreases with temperature. Henry’s law: at constant temperature, the partial pressure of a gas above a solution is proportional to its mole fraction in the solution, $p = K_\text{H}\,x$. $K_\text{H}$ depends on the gas, the solvent and the temperature; a higher $K_\text{H}$ means lower solubility, and $K_\text{H}$ rises with temperature.

Applications: carbonated drinks are sealed under high $\ce{CO2}$ pressure; divers breathing compressed air risk the bends (nitrogen bubbles in blood on rapid ascent), so tanks use air diluted with helium; at high altitude the low partial pressure of oxygen causes anoxia; aquatic life suffers in warm water, which holds less oxygen.

Raoult’s law: for a solution of volatile liquids, the partial vapour pressure of each component is proportional to its mole fraction in the liquid: $p_1 = p^\circ_1 x_1$, $p_2 = p^\circ_2 x_2$. By Dalton’s law $p_\text{total} = p_1 + p_2 = p^\circ_1 + (p^\circ_2 - p^\circ_1)x_2$, a straight line against composition. The mole fraction in the vapour is $y_1 = p_1 / p_\text{total}$; the vapour is richer in the more volatile component.

For a non-volatile solute, only the solvent contributes: $p = p^\circ_1 x_1$. Raoult’s law is a special case of Henry’s law with $K_\text{H} = p^\circ$.

Ideal solutions obey Raoult’s law over the whole range; A–B interactions are about the same as A–A and B–B; $\Delta_\text{mix}H = 0$ and $\Delta_\text{mix}V = 0$ (n-hexane + n-heptane, bromoethane + chloroethane, benzene + toluene).

Non-ideal solutions:

• Positive deviation: A–B attractions weaker; vapour pressure higher than predicted; $\Delta_\text{mix}H > 0$, $\Delta_\text{mix}V > 0$ (ethanol + acetone, carbon disulfide + acetone). Large positive deviation gives a minimum-boiling azeotrope (ethanol + water at about 95% ethanol by volume).
• Negative deviation: A–B attractions stronger (often hydrogen bonding); vapour pressure lower; $\Delta_\text{mix}H < 0$, $\Delta_\text{mix}V < 0$ (chloroform + acetone, phenol + aniline). Large negative deviation gives a maximum-boiling azeotrope (nitric acid + water at about 68% nitric acid by mass).

An azeotrope boils at constant temperature with the same composition in liquid and vapour, so fractional distillation cannot separate it.

Colligative properties depend only on the number of solute particles in a given amount of solvent, not on their nature. They are measured for dilute solutions of non-volatile solutes.

Relative lowering of vapour pressure equals the mole fraction of the solute: $\dfrac{p^\circ_1 - p_1}{p^\circ_1} = x_2 \approx \dfrac{w_2 M_1}{M_2 w_1}$ for dilute solutions.

Elevation of boiling point. A solution’s vapour pressure reaches atmospheric pressure at a higher temperature: $\Delta T_\text{b} = K_\text{b}\, m$. $K_\text{b}$ (ebullioscopic constant) for water is $0.52\ \text{K kg mol}^{-1}$.

Depression of freezing point. The solution’s vapour pressure equals that of the solid solvent at a lower temperature: $\Delta T_\text{f} = K_\text{f}\, m$. $K_\text{f}$ (cryoscopic constant) for water is $1.86\ \text{K kg mol}^{-1}$; for benzene $5.12\ \text{K kg mol}^{-1}$.

Molar mass from either: $M_2 = \dfrac{K \times 1000 \times w_2}{\Delta T \times w_1}$ with masses in grams. A temperature difference has the same value in K and °C.

Osmosis is the net flow of solvent through a semipermeable membrane (one that passes solvent but not solute) from pure solvent or a dilute solution into a more concentrated solution. The osmotic pressure $\pi$ is the excess pressure that must be applied to the solution to just stop osmosis.

For dilute solutions $\pi = CRT = \dfrac{n_2 RT}{V}$, so $M_2 = \dfrac{w_2 RT}{\pi V}$. Osmotic pressure is preferred for the molar masses of proteins and polymers: it is measured at room temperature and uses molarity, and its values are large enough to measure even for very dilute solutions.

Isotonic solutions have the same osmotic pressure (0.9% w/v sodium chloride is isotonic with blood cells). In a hypertonic solution cells lose water and shrink; in a hypotonic solution they gain water and swell. Applying a pressure greater than $\pi$ reverses the flow: reverse osmosis, used to desalinate sea water.

If a solute dissociates in solution (electrolytes in water), the number of particles increases and the molar mass calculated from a colligative property is lower than the true value. If it associates (ethanoic acid dimerising in benzene through hydrogen bonds), the number of particles decreases and the calculated molar mass is higher.

The van’t Hoff factor $i = \dfrac{\text{normal molar mass}}{\text{abnormal molar mass}} = \dfrac{\text{observed colligative property}}{\text{calculated colligative property}}$. Corrected equations: $\Delta T_\text{b} = iK_\text{b}m$, $\Delta T_\text{f} = iK_\text{f}m$, $\pi = iCRT$, $\dfrac{\Delta p}{p^\circ_1} = i\,\dfrac{n_2}{n_1}$.

• Dissociation into $n$ ions with degree $\alpha$: $i = 1 + (n - 1)\alpha$, so $\alpha = \dfrac{i - 1}{n - 1}$.
• Association of $n$ molecules into one with degree $\alpha$: $i = 1 - \alpha + \dfrac{\alpha}{n}$; for dimers $i = 1 - \dfrac{\alpha}{2}$.
• Non-electrolytes (glucose, urea, sucrose): $i = 1$. Complete dissociation: $\ce{NaCl}$ $i = 2$, $\ce{CaCl2}$ and $\ce{K2SO4}$ $i = 3$.