Why a Soda Bottle Fizzes the Moment You Open It

Sugar vanishes into tea, carbon dioxide stays dissolved in a sealed soda bottle, and a mixture of ethanol and water boils below the boiling point of either liquid. Each is a solution, and a few simple laws predict how they behave.

This part covers ways of expressing concentration, Henry's law, Raoult's law, and ideal and non-ideal solutions.

Concentration can be expressed as mass, volume or mass-by-volume percentage, parts per million, mole fraction, molarity or molality; molarity uses the volume of solution and changes with temperature, while mole fraction and molality use amounts and masses that do not.

The units:

- Mass percentage $=\frac{\text{mass of solute}}{\text{mass of solution}}\times 100$
- Volume and mass-by-volume percentage — similar ratios using volumes
- ppm $=\frac{\text{mass of solute}}{\text{mass of solution}}\times 10^6$
- Mole fraction $x_A=\frac{n_A}{n_A+n_B}$
- Molarity $=$ moles of solute per litre of solution
- Molality $=$ moles of solute per kilogram of solvent

Worked example. $4$ g of NaOH (molar mass $40$ g mol${}^{-1}$) is $0.1$ mol.

- Made up to $250$ mL of solution: $M=\frac{0.1}{0.250}=0.4$ mol L${}^{-1}$
- Dissolved in $250$ g of water: $m=\frac{0.1}{0.250}=0.4$ mol kg${}^{-1}$, mass percentage $=\frac{4}{254}\times 100\approx 1.57$
- Mole fraction: water is $\frac{250}{18}\approx 13.9$ mol, so $x_{NaOH}=\frac{0.1}{14.0}\approx 0.0071$

An everyday example. Fluoride in drinking water is reported in ppm, because the amounts are tiny — $1$ ppm is $1$ g in a million grams of water.

The substance. Molality is preferred for colligative properties because it does not change with temperature, while molarity does as the solution expands.

**Henry's law states that the partial pressure of a gas above a solution is proportional to its mole fraction in the solution, $p=K_{H}x$; a larger $K_H$ means a less soluble gas, and gases dissolve less as temperature rises.

Interpreting $K_H$:**

- $K_H$ has units of pressure and depends on the gas and the temperature
- **Higher $K_H$ means lower solubility** at a given pressure
- $K_H$ rises with temperature, so gases are less soluble in warm liquids

Effect of pressure. Gas solubility rises with pressure; solids in liquids are hardly affected, because solids and liquids are nearly incompressible.

Solids and temperature. Solubility rises with temperature if dissolving absorbs heat and falls if it releases heat, following Le Chatelier's principle.

Worked example. $K_H$ for nitrogen in water at $293$ K is $76.48$ kbar. Air supplies nitrogen at a partial pressure of $0.8$ bar:

$$x_{N_2}=\frac{p}{K_H}=\frac{0.8}{76480}\approx 1.05\times 10^{-5}$$

In $1$ L of water ($55.5$ mol), that is $1.05\times 10^{-5}\times 55.5\approx 5.8\times 10^{-4}$ mol of nitrogen.

An everyday example. Opening a bottle of soda lowers the pressure above the liquid, so dissolved carbon dioxide escapes as bubbles — and a warm bottle fizzes even more.

The substance. Henry's law fails for gases that react with water, such as ammonia and hydrogen chloride, which dissolve far more than it predicts.

**Raoult's law states that the partial vapour pressure of each volatile component equals its pure vapour pressure times its mole fraction, $p_A=x_A{p}_A^0$; with a non-volatile solute, only the solvent contributes, so the vapour pressure falls.

Two volatile liquids:**

$$p_{total}=x_1{p}_1^0+x_2{p}_2^0$$

The vapour is richer in the more volatile component, with vapour mole fraction $y_1=\frac{p_1}{p_{total}}$.

Non-volatile solute: $p=x_{solvent}{p}^0$

Worked example. Liquid A ($p_A^0=450$ mm Hg) and liquid B ($p_B^0=700$ mm Hg) are mixed with $x_A=0.4$:

$$p_A=0.4\times 450=180,p_B=0.6\times 700=420,p_{total}=600\text{ mm Hg}$$

$$y_B=\frac{420}{600}=0.7$$

The liquid has $x_B=0.6$, but the vapour has $y_B=0.7$ — enriched in the more volatile B.

An everyday example. Sugar syrup left in an open bowl dries out more slowly than plain water, because the dissolved sugar lowers the vapour pressure.

The substance. Raoult's law is a special case of Henry's law with $K_H=p^0$, since both make partial pressure proportional to mole fraction.

**An ideal solution obeys Raoult's law at all compositions, with ${\Delta }_{mix}H=0$ and ${\Delta }_{mix}V=0$; non-ideal solutions show positive deviation when unlike molecules attract more weakly than like ones, and negative deviation when they attract more strongly.

Ideal solutions. A–B attractions match A–A and B–B attractions, as in benzene and toluene.

Positive deviation:

- A–B attractions are weaker, so molecules escape more easily
- Vapour pressure is higher than Raoult's law predicts, and mixing absorbs heat
- Examples: ethanol and water, acetone and carbon disulphide
- Can form a minimum-boiling azeotrope** — ethanol and water at about $95$ parts ethanol per $100$ by volume

Negative deviation:

- A–B attractions are stronger
- Vapour pressure is lower than predicted, and mixing releases heat
- Examples: chloroform and acetone, nitric acid and water
- Can form a maximum-boiling azeotrope — nitric acid and water at about $68$ parts acid per $100$ by mass

An azeotrope boils at constant temperature with vapour of the same composition as the liquid, so simple distillation cannot separate it.

Worked example. Two liquids with $p^0$ values of $200$ and $350$ mm Hg are mixed with equal mole fractions. Raoult's law predicts $0.5\times 200+0.5\times 350=275$ mm Hg. A measured value of $250$ mm Hg shows negative deviation.

An everyday example. Diluting concentrated sulphuric acid releases a lot of heat — a sign of strong attraction between acid and water, which is why acid is always added slowly to water.

The substance. Azeotropes explain why ordinary distillation cannot give pure ethanol from an ethanol–water mixture.