Comprehensive Study Notes — Solutions Summary & Study Notes

🧪 Overview

After studying this unit you should be able to describe how solutions form, compute their concentrations in common units, state and apply Henry’s law and Raoult’s law, distinguish ideal and non-ideal behavior, explain deviations from Raoult’s law, and use colligative properties to determine molar masses and understand osmotic phenomena.

🔬 What is a solution?

A solution is a homogeneous mixture of two or more components. The component in largest amount is the solvent and the others are solutes. Most real-world substances are mixtures, not pure substances. We focus on binary solutions (two components) that may be solid, liquid or gas.

📐 Concentration units (definitions & formulas)

• Mass percentage (w/w): expresses mass fraction as a percent. $\mathrm{\%}w\mathrm{/}w=\frac{m_{\text{solute}}}{m_{\text{solution}}}\times 100$.

• Volume percentage (v/v): used for liquid–liquid mixtures. $\mathrm{\%}v\mathrm{/}v=\frac{V_{\text{solute}}}{V_{\text{solution}}}\times 100$.

• Mass by volume percentage (w/v): grams of solute per 100 mL solution. $\mathrm{\%}w\mathrm{/}v=\frac{\text{g solute}}{\text{mL solution}}\times 100$.

• Parts per million (ppm): $\text{ppm} = \frac{\text{mass of solute}}{\text{mass of solution}} \times 10^{6}$.

• Mole fraction ($x$): ratio of moles of a component to total moles. $x_i = \frac{n_i}{\sum_j n_j}$.

• Molarity ($M$): moles solute per litre of solution. $M = \frac{\text{mol solute}}{\text{L solution}}$.

• Molality ($m$): moles solute per kilogram of solvent. $m = \frac{\text{mol solute}}{\text{kg solvent}}$.

Note: Molarity depends on solution volume (temperature-sensitive); molality depends on solvent mass (temperature-independent).

🌡️ Solubility: factors & trends

Solubility is the maximum amount of solute that dissolves in a solvent at a given temperature and pressure. Key influences:

• Nature of solute and solvent ("like dissolves like" — polarity, hydrogen bonding).
• Temperature: solubility of most solids increases with temperature; solubility of gases usually decreases with temperature.
• Pressure: solubility of gases increases with pressure (important for carbonation and diving physiology).

Example applications: CO$_2$ solubility in soft drinks (pressure ↑ increases solubility); decompression sickness (gas solubility changes with pressure).

🧯 Henry’s law (gases in liquids)

Henry’s law: the solubility of a gas in a liquid is proportional to its partial pressure above the liquid. A common form:

$p = k_H x_{\text{gas}}$

where $p$ is partial pressure, $x_{\text{gas}}$ is mole fraction of the gas in solution, and $k_H$ is Henry’s constant (units depend on form used). Another common form relates concentration $c$ to pressure: $c = k'_{H} p$.

Practical notes: $k_H$ depends on gas, solvent and temperature.

☁️ Raoult’s law (volatile components)

Raoult’s law for an ideal binary solution:

$p_i = x_i p_i^{*}$

where $p_i$ is the partial vapor pressure of component $i$ above the solution, $x_i$ its mole fraction in the liquid, and $p_i^{*}$ the vapor pressure of pure component $i$.

• For a non-volatile solute, the solvent vapour pressure becomes $p_{\text{solvent}} = x_{\text{solvent}} p^{*}_{\text{solvent}}$, so the lowering of vapour pressure is

$\Delta p = p^{$}{\text{solvent}} - p{\text{solvent}} = x_{\text{solute}} p^{}_{\text{solvent}}.

• Relative lowering: $\frac{\Delta p}{p^{*}${\text{solvent}}} = x{\text{solute}}.

Raoult’s law is a special case of Henry’s law when the proportionality constant equals the vapor pressure of the pure solvent.

⚖️ Ideal vs non-ideal solutions; deviations from Raoult’s law

• Ideal solutions: obey Raoult’s law at all compositions. Mixing causes no enthalpy or volume change (ΔHmix = 0, ΔVmix = 0). Intermolecular interactions A–A, B–B and A–B are similar.

• Non-ideal solutions: show positive or negative deviations from Raoult’s law.

  • Positive deviation: measured vapour pressure is higher than Raoult’s prediction. Occurs when A–B interactions are weaker than A–A and B–B; molecules escape more easily.

  • Negative deviation: measured vapour pressure is lower. Occurs when A–B interactions are stronger (e.g., hydrogen bonding) so molecules are held more tightly.

Consequences: deviations affect boiling points, azeotrope formation and separation methods.

❄️ Colligative properties (depend on particle number)

Colligative properties depend on the number of solute particles, not their identity. Main ones:

• Relative lowering of vapour pressure: $\frac{\Delta p}{p^{*}} = x_{\text{solute}}$ (ideal, dilute solutions).

• Boiling point elevation: $\Delta T_b = i K_b m$.

• Freezing point depression: $\Delta T_f = i K_f m$.

• Osmotic pressure: $\pi = i M R T$ (for dilute solutions), or more generally $\pi = \frac{nRT}{V}$.

In these formulas: $i$ is the van’t Hoff factor, $K_b$ and $K_f$ are molal boiling/freezing point constants of the solvent, $m$ is molality, $M$ is molarity of solute, $R$ is gas constant and $T$ is absolute temperature.

Practical use: these relations let you determine molar mass of solutes (including macromolecules) by measuring colligative effects.

🧮 Van’t Hoff factor and dissociation/association

The van’t Hoff factor $i$ accounts for dissociation or association of solute molecules in solution.

• For a solute that produces $\nu$ particles on complete dissociation and with degree of dissociation $\alpha$:

$i = 1 + (\nu - 1)\alpha$.

• For association (e.g., dimerization), $i = 1 - (\nu - 1)\alpha$ where $\nu$ is the number of monomers in the associated particle.

Example: for $\text{NaCl} \rightleftharpoons \text{Na}^+ + \text{Cl}^-$, $\nu = 2$ and $i = 1 + \alpha$.

Observed $i$ is often less than the ideal value due to incomplete dissociation and interionic interactions in solution.

🧾 Using colligative properties to find molar mass

Common approaches:

• From freezing point depression: measure $\Delta T_f$, use $\Delta T_f = i K_f m$ and $m = \frac{\text{mol solute}}{\text{kg solvent}} = \frac{\frac{w}{M}}{\text{kg solvent}}$ to solve for molar mass $M$.

• From boiling point elevation: analogous with $K_b$.

• From osmotic pressure: if mass $w$ of solute is dissolved in volume $V$, then

$\pi = i \left(\frac{w}{M V}\right) R T$ so

$M = i \frac{w R T}{\pi V}$.

Example note from the source: calculation of a protein’s molar mass using osmotic pressure gave about $61{,}022\ \text{g mol}^{-1}$ (illustrates use for macromolecules).

🌊 Osmosis, osmotic pressure & biological relevance

Osmosis: flow of solvent through a semipermeable membrane from dilute to concentrated solution. Osmotic pressure is the external pressure required to stop this flow.

• Classification: isotonic (no net flow), hypertonic (cell loses water — shrinks), hypotonic (cell gains water — swells).

• Biological examples: wilting and revival of plant tissues, blood cell responses to saline, edema due to osmotic imbalances.

• Reverse osmosis: applying pressure greater than osmotic pressure to force solvent from concentrated to dilute side — key for desalination. Common membranes: cellulose acetate and other semipermeable polymers.

🧪 Worked examples & problem-solving tips

• Always identify whether concentration is expressed per solution volume (molarity) or solvent mass (molality). Use molality for colligative problems because $m$ is temperature-independent.

• For vapour-pressure problems use mole fractions and Raoult’s law: compute $x_i$, then $p_i = x_i p_i^{*}$.

• For gas solubility use Henry’s law and check which form of the constant is given ($p = k_H x$ vs $c = k'_H p$).

• When using colligative formulas, include the van’t Hoff factor $i$ for electrolytes and check whether observed data imply incomplete dissociation or association.

✅ Summary of key formulas (compact)

• $\mathrm{\%}w\mathrm{/}w=\frac{m_{\text{solute}}}{m_{\text{solution}}}\times 100$
• $x_i = \frac{n_i}{\sum n_j}$
• $M = \frac{\text{mol solute}}{\text{L solution}}$ ; $m = \frac{\text{mol solute}}{\text{kg solvent}}$
• Henry: $p = k_H x_{\text{gas}}$ (or $c = k'_H p$)
• Raoult: $p_i = x_i p_i^{$}pi​=xi​pi∗​ ; $\Delta p = x_{\text{solute}} p^{$}_{\text{solvent}}
• Colligative: $\Delta T_f = i K_f m$, $\Delta T_b = i K_b m$, $\pi = i M R T$
• Van’t Hoff (dissociation): $i = 1 + (\nu - 1)\alpha$

These notes collect the central concepts and formulas you need to analyze solution behavior, compute concentrations, apply gas solubility laws, predict vapor pressure changes, and use colligative properties to find molar masses and interpret osmotic phenomena.