Comprehensive University Study Guide on Liquid Solutions, Solubility, and Colligative Properties

Definition and Fundamental Nature of Solutions

A solution is defined as a homogeneous mixture composed of two or more than two components. The term homogeneous implies that the composition and properties of the mixture are uniform throughout its entire volume. In any solution, the component present in the largest quantity is designated as the solvent, which primarily determines the physical state in which the solution exists (solid, liquid, or gas). All other components present in the solution, aside from the solvent, are referred to as solutes. This study focuses primarily on binary solutions, which consist of exactly two components. Solutions are integral to biological and industrial processes; for instance, almost all processes within the human body occur in liquid solutions. The utility of these mixtures often depends on their specific composition. Examples include brass, which is a mixture of copper and zinc; German silver, a mixture of copper, zinc, and nickel; and bronze, a mixture of copper and tin. Furthermore, the concentration of specific ions, such as fluoride, significantly alters properties: $1\,ppm$ of fluoride ions in water prevents tooth decay, while $1.5\,ppm$ causes teeth to become mottled, and high concentrations are used as rat poison (e.g., sodium fluoride).

Classification of Solutions Based on Physical State

Solutions are classified into nine types based on the physical state of the solute and solvent. Gaseous solutions occur when the solvent is a gas; examples include a gas-gas mixture like oxygen and nitrogen, a liquid-gas mixture such as chloroform mixed with nitrogen gas, and a solid-gas mixture such as camphor in nitrogen gas. Liquid solutions occur when the solvent is a liquid; examples include a gas-liquid mixture like oxygen dissolved in water, a liquid-liquid mixture like ethanol dissolved in water, and a solid-liquid mixture such as glucose dissolved in water. Solid solutions involve a solid solvent; examples include a gas-solid solution of hydrogen in palladium, a liquid-solid solution such as an amalgam of mercury with sodium, and a solid-solid solution such as copper dissolved in gold.

Quantitative Methods for Expressing Concentration

The composition of a solution is described using concentration, which can be expressed qualitatively (dilute or concentrated) or, more accurately, quantitatively. Mass percentage $(w/w)$ is defined as the mass of the component in the solution divided by the total mass of the solution, multiplied by $100$. For instance, a $10\%$ glucose solution contains $10\,g$ of glucose in $90\,g$ of water. Volume percentage $(V/V)$ is the volume of the component divided by the total volume of the solution, multiplied by $100$. A $35\%\,(v/v)$ solution of ethylene glycol is used as an antifreeze in cars, lowering the freezing point of water to $255.4\,K$ ($-17.6^{\circ}C$). Mass by volume percentage $(w/V)$ is the mass of solute dissolved in $100\,mL$ of the solution, commonly used in medicine.

Parts per million $(ppm)$ is used when a solute is present in trace quantities, defined as the number of parts of the component divided by the total number of parts of all components, multiplied by $10^6$. For example, a liter of sea water weighing $1030\,g$ containing $6 \times 10^{-3}\,g$ of dissolved oxygen is expressed as $5.8\,ppm$. Mole fraction ($x$) is the ratio of the number of moles of a component to the total number of moles of all components. In a binary mixture of $A$ and $B$, the mole fraction of $A$ is given by $x_A = \frac{n_A}{n_A + n_B}$. The sum of all mole fractions in a solution is always unity: $x_1 + x_2 + ... + x_i = 1$.

Molarity ($M$) is defined as the number of moles of solute dissolved in one liter (or one cubic decimetre) of solution: $Molarity = \frac{\text{moles of solute}}{\text{volume of solution in litre}}$. For example, $0.278\,M$ NaOH indicates $0.278\,mol/L$. Molality ($m$) is the number of moles of the solute per kilogram of the solvent: $m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}}$. Notably, mass percentage, ppm, mole fraction, and molality are temperature-independent because they are based on mass, whereas molarity changes with temperature because volume is temperature-dependent.

Solubility and Factors Affecting it

Solubility is the maximum amount of a substance that can be dissolved in a specified amount of solvent at a specific temperature. It depends on the nature of the solute and solvent, temperature, and pressure. The general rule is "like dissolves like," meaning polar solutes (e.g., sodium chloride, sugar) dissolve in polar solvents (water), and non-polar solutes (naphthalene, anthracene) dissolve in non-polar solvents (benzene). Dissolution occurs when a solid solute is added to a solvent and its concentration increases. Simultaneously, crystallization occurs when solute particles collide with solid solute and separate from the solution. When these processes occur at the same rate, a state of dynamic equilibrium is reached: $\text{Solute} + \text{Solvent} \rightleftharpoons \text{Solution}$. A solution containing the maximum amount of solute at a given temperature and pressure is called a saturated solution.

The solubility of solids in liquids is significantly affected by temperature according to Le Chatelier's Principle. If the dissolution process is endothermic ($\Delta_{sol}H > 0$), solubility increases with temperature; if exothermic ($\Delta_{sol}H < 0$), it decreases. Pressure has no significant effect on the solubility of solids and liquids because they are highly incompressible. Conversely, the solubility of gases in liquids is greatly influenced by pressure and temperature. Solubility of gases increases with increasing pressure. As temperature increases, the solubility of gases typically decreases because the dissolution of gases is generally an exothermic process similar to condensation.

Henry's Law

Henry's law provides a quantitative relationship between the pressure and solubility of a gas. It states that at a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas above the surface of the liquid. The most common form states: "the partial pressure of the gas in vapour phase ($p$) is proportional to the mole fraction of the gas ($x$) in the solution," expressed as $p = K_H x$, where $K_H$ is the Henry's law constant. Different gases have different $K_H$ values, and higher $K_H$ values at a given pressure imply lower solubility. For example, at $293\,K$, $K_H$ for $N_2$ is $76.48\,kbar$ and for $O_2$ is $34.86\,kbar$. As temperature increases, $K_H$ increases, leading to lower solubility. This explains why aquatic species are more comfortable in cold water than in warm water.

Applications of Henry's Law include: (1) sealing soft drinks and soda water under high pressure to increase $CO_2$ solubility; (2) management of "bends" in scuba divers. High pressure underwater increases the solubility of atmospheric gases in the blood. When divers surface, pressure decreases, releasing dissolved nitrogen as bubbles that block capillaries and cause pain. To prevent this, divers use air diluted with helium ($11.7\%\,He$, $56.2\%\,N_2$, and $32.1\%\,O_2$). (3) At high altitudes, low partial pressure of oxygen leads to low blood oxygen levels, causing "anoxia," a condition where climbers cannot think clearly and feel weak.

Vapour Pressure and Raoult's Law

Vapour pressure of liquid solutions depends on the volatility of their components. Raoult's Law states that for a solution of volatile liquids, the partial vapour pressure of each component ($p_i$) is directly proportional to its mole fraction ($x_i$): $p_1 = x_1 p_1^0$ and $p_2 = x_2 p_2^0$, where $p^0$ is the vapour pressure of the pure component. According to Dalton’s law, the total pressure is the sum of partial pressures: $p_{total} = p_1 + p_2 = p_1^0 + (p_2^0 - p_1^0) x_2$. The composition of the vapour phase is determined by $p_i = y_i p_{total}$, where $y_i$ is the mole fraction in the vapour phase. Vapour phase is always richer in the more volatile component. Raoult’s law can be viewed as a special case of Henry’s law where $K_H$ becomes equal to $p_1^0$.

When a non-volatile solute is added to a solvent, the vapour pressure of the solution is lower than that of the pure solvent because the solute particles occupy space on the surface, reducing the fraction of solvent molecules that can escape into the vapour phase. This decrease depends on the quantity of the non-volatile solute, not its identity. For such solutions, Raoult's law is expressed as $p_1 = x_1 p_1^0$. A plot of vapour pressure against the mole fraction of the solvent is linear.

Ideal and Non-Ideal Solutions

Ideal solutions are those that obey Raoult’s law over the entire range of concentration. They are characterized by $\Delta_{mix}H = 0$ and $\Delta_{mix}V = 0$. At the molecular level, this occurs when the intermolecular attractive forces between components ($A-B$) are nearly equal to those in pure components ($A-A$ and $B-B$). Examples include mixtures of n-hexane/n-heptane, bromoethane/chloroethane, and benzene/toluene. Non-ideal solutions do not obey Raoult's law and show deviations. Positive deviation occurs when $A-B$ interactions are weaker than $A-A$ or $B-B$ interactions (e.g., ethanol and acetone; carbon disulphide and acetone). This results in a higher vapour pressure than predicted. Negative deviation occurs when $A-B$ interactions are stronger than pure component interactions (e.g., phenol and aniline; chloroform and acetone). Chloroform forms a hydrogen bond with acetone, reducing the escaping tendency of both molecules and lowering vapour pressure.

Azeotropes are binary mixtures that have the same composition in the liquid and vapour phase and boil at a constant temperature, making separation by fractional distillation impossible. Minimum boiling azeotropes form when solutions show large positive deviations (e.g., $95\%$ ethanol and $5\%$ water by volume). Maximum boiling azeotropes form when solutions show large negative deviations (e.g., $68\%$ nitric acid and $32\%$ water by mass, boiling at $393.5\,K$).

Colligative Properties

Colligative properties depend solely on the number of solute particles relative to the total number of particles and are independent of the nature of the solute. There are four primary colligative properties: (1) Relative lowering of vapour pressure, defined as $\frac{p_1^0 - p_1}{p_1^0} = x_2$. For dilute solutions, this relates to molar mass via $\frac{p_1^0 - p_1}{p_1^0} = \frac{w_2 \times M_1}{M_2 \times w_1}$. (2) Elevation of boiling point ($\Delta T_b$). A solution boils at a higher temperature than the pure solvent because its vapour pressure is lower. The elevation is proportional to molality: $\Delta T_b = K_b m$, where $K_b$ is the molal elevation constant (ebullioscopic constant). The molar mass of the solute can be calculated as $M_2 = \frac{1000 \times w_2 \times K_b}{\Delta T_b \times w_1}$.

(3) Depression of freezing point ($\Delta T_f$). Freezing point is the temperature where the liquid and solid phases have the same vapour pressure. Adding a non-volatile solute lowers the freezing point: $\Delta T_f = K_f m$, where $K_f$ is the molal depression constant (cryoscopic constant). The constants $K_b$ and $K_f$ are calculated using $\Delta_{vap}H$ and $\Delta_{fus}H$. (4) Osmosis and Osmotic Pressure ($\Pi$). Osmosis is the flow of solvent molecules through a semipermeable membrane (SPM) from pure solvent to solution. Osmotic pressure is the excess pressure applied to the solution side to stop this flow. It follows the equation $\Pi = CRT$, where $C$ is molarity. This method is preferred for calculating molar masses of macromolecules like proteins and polymers because measurements are taken at room temperature and the magnitude is large even for dilute solutions.

Osmotic Phenomena and Reverse Osmosis

Isotonic solutions have the same osmotic pressure at a given temperature; for example, blood cell fluid is isotonic with $0.9\%\,(w/V)$ sodium chloride (saline). Solutions with higher salt concentration are hypertonic (cells shrink), while lower concentrations are hypotonic (cells swell). Other natural phenomena include raw mangoes shriveling in brine, wilted flowers reviving in fresh water, and edema (swelling due to salt-induced water retention). If pressure greater than the osmotic pressure is applied to the solution side, the direction of osmosis reverses. This "reverse osmosis" is used for the desalination of seawater. Using a porous membrane like cellulose acetate—which is permeable to water but not to ions or impurities—fresh water is squeezed out of saline water.

Abnormal Molar Masses and van't Hoff Factor

When solutes undergo dissociation or association in a solution, the number of particles changes, leading to "abnormal" molar masses. Dissociation (e.g., $KCl \rightarrow K^+ + Cl^-$) increases the number of particles, resulting in higher observed colligative properties and lower calculated molar masses. Association (e.g., ethanoic acid dimerizing in benzene via hydrogen bonding) decreases the number of particles, resulting in lower colligative properties and higher calculated molar masses. To account for this, the van't Hoff factor ($i$) was introduced: $i = \frac{\text{Normal molar mass}}{\text{Abnormal molar mass}} = \frac{\text{Observed colligative property}}{\text{Calculated colligative property}}$. For dissociation, $i > 1$ (e.g., $KCl \approx 2$, $K_2SO_4 \approx 3$); for association, $i < 1$ (e.g., ethanoic acid in benzene $\approx 0.5$). Colligative property equations are modified to include $i$: $\Delta T_b = i K_b m$, $\Delta T_f = i K_f m$, and $\Pi = i CRT$.