Solutions: NEET notes

1. Types of solutions and concentration

NCERT §1.1; §1.2

• A solution is a homogeneous mixture: its composition and properties are the same in every part. The chapter deals with binary solutions, which have two components.
• The component present in the largest amount is usually called the solvent, and it decides whether the solution is a solid, liquid or gas; every other component is a solute.
• Solute and solvent can each be a gas, a liquid or a solid. Examples: camphor in nitrogen gas (solid in gas), oxygen in water (gas in liquid), glucose in water (solid in liquid), hydrogen in palladium (gas in solid), sodium amalgam (mercury in sodium, liquid in solid) and copper in gold (solid in solid).
• Mass percentage (w/w) = mass of the component × 100 / total mass of the solution. A 10% glucose solution by mass is 10 g of glucose with 90 g of water; commercial bleach carries 3.62% sodium hypochlorite by mass.
• Volume percentage (V/V) is used for liquids in liquids. A 35% (V/V) solution of ethylene glycol, used as antifreeze in car engines, lowers the freezing point of water to 255.4 K (−17.6 °C).
• Mass by volume percentage (w/V), the grams of solute in 100 mL of solution, is the usual unit in medicine and pharmacy.
• Parts per million = parts of the component × 10⁶ / total parts of all components, for trace amounts such as pollutants. Fluoride at 1 ppm in drinking water prevents tooth decay, while 1.5 ppm mottles the teeth.
• Mole fraction: x_A = n_A / (n_A + n_B) in a binary mixture, and the mole fractions of all components add up to 1.
• Molarity (M) is moles of solute per litre of solution; molality (m) is moles of solute per kilogram of solvent.
• Mass percentage, ppm, mole fraction and molality do not change with temperature, because they are built from masses; molarity does, because the volume of a solution changes with temperature.

2. Solubility and Henry's law

NCERT §1.3

• Solubility is the largest amount of a substance that can dissolve in a stated amount of solvent at a stated temperature. It depends on the nature of solute and solvent, on temperature and, for gases, on pressure.
• Like dissolves like: sodium chloride and sugar dissolve readily in water but not in benzene, while naphthalene and anthracene dissolve in benzene but not in water. A solute dissolves when its intermolecular forces resemble the solvent's.
• In a saturated solution, dissolution and crystallisation run at equal rates, so the solution sits in dynamic equilibrium with undissolved solute and holds the most solute possible at that temperature and pressure. An unsaturated solution can still take more.
• Solid in liquid: by Le Chatelier's principle, a nearly saturated solution whose dissolving is endothermic (ΔsolH > 0) dissolves more on heating, and one whose dissolving is exothermic dissolves less. Pressure has no significant effect, because solids and liquids are almost incompressible.
• Henry's law: at a fixed temperature, how much of a gas a liquid holds is directly proportional to the gas's partial pressure over it. The form used most is p = K_H x, where x is the mole fraction of the gas in the solution.
• K_H depends on the gas and on the temperature. At a given pressure, the larger K_H is, the less the gas dissolves: CO₂ (1.67 kbar at 298 K) is far more soluble in water than N₂ (76.48 kbar at 293 K).
• K_H for N₂ and O₂ rises with temperature, so gases dissolve less in warm water. Dissolving a gas releases heat, as condensation does, so by Le Chatelier's principle warming drives gas out; this is why aquatic species are more comfortable in cold water.
• Henry's law at work: soft drinks are bottled under high CO₂ pressure; divers who surface quickly form nitrogen bubbles in the blood (the bends), so their tanks carry air diluted with helium (11.7% He, 56.2% N₂, 32.1% O₂); at high altitude the low partial pressure of oxygen lowers blood oxygen and can cause anoxia.

3. Vapour pressure and Raoult's law

NCERT §1.4

• Two volatile liquids in a closed vessel both evaporate until liquid and vapour are in equilibrium. By Dalton's law of partial pressures the total vapour pressure is p_total = p₁ + p₂.
• Raoult's law: in a solution of volatile liquids, the partial vapour pressure of each component is directly proportional to its mole fraction in the liquid: p₁ = p₁° x₁ and p₂ = p₂° x₂, where p° is the vapour pressure of the pure liquid at the same temperature.
• Combining the two, p_total = p₁° + (p₂° − p₁°) x₂: the total vapour pressure is a straight line in the mole fraction of either component, running from p₁° to p₂°.
• The vapour's composition follows from Dalton's law: y_i = p_i / p_total. The vapour is always richer than the liquid in the more volatile component, the one with the larger p°.
• Raoult's law is a special case of Henry's law: both say a volatile component's partial pressure is proportional to its mole fraction, and for Raoult's law the constant K_H equals p°.
• When the solute is non-volatile, only the solvent contributes vapour: p₁ = x₁ p₁°. Solute particles occupy part of the surface, fewer solvent molecules escape, and the solution's vapour pressure falls below the pure solvent's.
• This lowering depends on how much solute is present, not on what it is: 1.0 mol of sucrose and 1.0 mol of urea in 1 kg of water lower the vapour pressure by nearly the same amount.

4. Ideal and non-ideal solutions

NCERT §1.5

• An ideal solution obeys Raoult's law over the whole range of composition. It also has ΔmixH = 0 (no heat taken in or given out on mixing) and ΔmixV = 0 (the volumes simply add).
• Ideal behaviour needs A–B attractions nearly equal to the A–A and B–B attractions. No solution is perfectly ideal, but n-hexane with n-heptane, bromoethane with chloroethane, and benzene with toluene come close.
• A non-ideal solution does not obey Raoult's law over the whole range; its vapour pressure is either higher than predicted (positive deviation) or lower (negative deviation).
• Positive deviation: A–B attractions are weaker than A–A and B–B, so molecules escape more easily. Ethanol with acetone (acetone breaks some of ethanol's hydrogen bonds) and carbon disulphide with acetone behave this way; mixing is endothermic and the volume increases (ΔmixH > 0, ΔmixV > 0).
• Negative deviation: A–B attractions are stronger, so escape is harder. Phenol with aniline (hydrogen bond between the phenolic H and the lone pair on N) and chloroform with acetone (chloroform's H bonds to acetone's O) behave this way; mixing is exothermic and the volume shrinks (ΔmixH < 0, ΔmixV < 0).
• Azeotropes are binary mixtures whose liquid and vapour have the same composition, so they boil at a constant temperature and cannot be separated by fractional distillation.
• A large positive deviation gives a minimum boiling azeotrope: fractional distillation of fermented ethanol-water stops at about 95% ethanol by volume.
• A large negative deviation gives a maximum boiling azeotrope: nitric acid and water, about 68% nitric acid and 32% water by mass, boiling at 393.5 K.

5. Relative lowering of vapour pressure

NCERT §1.6; §1.6.1

• Colligative properties are set by how many solute particles there are relative to all the particles present, whatever those particles are (Latin co, together, and ligare, to bind). There are four: the relative lowering of the solvent's vapour pressure, the rise in its boiling point, the fall in its freezing point, and the osmotic pressure of the solution.
• All four come from the same cause: a non-volatile solute lowers the vapour pressure of the solvent.
• For a non-volatile solute, the lowering is Δp₁ = p₁° − p₁ = x₂ p₁°, and the relative lowering (p₁° − p₁)/p₁° equals x₂, the mole fraction of the solute.
• With several non-volatile solutes, the lowering depends on the sum of their mole fractions.
• Since x₂ = n₂/(n₁ + n₂), and n₂ is much smaller than n₁ in a dilute solution, (p₁° − p₁)/p₁° ≈ n₂/n₁ = (w₂ × M₁)/(M₂ × w₁), which gives the molar mass M₂ of the solute from measured masses and pressures.

6. Elevation of boiling point

NCERT §1.6.2

• A liquid boils when its vapour pressure equals the external pressure: water boils at 373.15 K because there its vapour pressure is 1.013 bar (1 atm).
• A non-volatile solute lowers the vapour pressure, so the solution has to be heated past the solvent's boiling point before its vapour pressure reaches 1.013 bar. Such a solution therefore always boils higher than its solvent: a solution of 1 mol of sucrose per 1000 g of water, at 1 atm, boils at 373.52 K.
• The elevation ΔTb = Tb − Tb° is, for dilute solutions, directly proportional to molality: ΔTb = Kb m.
• Kb, the molal elevation (ebullioscopic) constant, has units K kg mol⁻¹ and depends only on the solvent: 0.52 for water, 2.53 for benzene.
• With w₂ g of solute of molar mass M₂ in w₁ g of solvent, m = (w₂ × 1000)/(M₂ × w₁), so M₂ = (Kb × w₂ × 1000)/(ΔTb × w₁).
• Kb = R × M₁ × Tb² / (1000 × ΔvapH), where M₁ is the solvent's molar mass, Tb its boiling point in kelvin and ΔvapH its enthalpy of vaporisation.

7. Depression of freezing point

NCERT §1.6.3

• At the freezing point, solid and liquid are in dynamic equilibrium, which means their vapour pressures are equal. A solution freezes when its vapour pressure equals that of the pure solid solvent.
• A non-volatile solute lowers the liquid's vapour pressure, so it meets the solid's vapour-pressure curve at a lower temperature: the freezing point drops.
• For dilute solutions the depression ΔTf = Tf° − Tf is directly proportional to molality: ΔTf = Kf m.
• Kf, the molal depression (cryoscopic) constant, has units K kg mol⁻¹ and depends only on the solvent: 1.86 for water, 5.12 for benzene, 20.00 for cyclohexane and 31.8 for carbon tetrachloride.
• Molar mass from freezing point: M₂ = (Kf × w₂ × 1000)/(ΔTf × w₁).
• Kf = R × M₁ × Tf² / (1000 × ΔfusH), with Tf the solvent's freezing point in kelvin and ΔfusH its enthalpy of fusion.
• For water Kf (1.86) is much larger than Kb (0.52), so the same solution lowers the freezing point more than it raises the boiling point. Ethylene glycol in car radiators uses this as antifreeze.

8. Osmosis and osmotic pressure

NCERT §1.6.4; §1.6.5

• Semipermeable membranes, natural (pig's bladder, parchment) or synthetic (cellophane), have submicroscopic pores that let small solvent molecules such as water through but hinder larger solute particles.
• Osmosis is the net flow of solvent through a semipermeable membrane from pure solvent, or a dilute solution, into a more concentrated solution.
• Osmotic pressure is the excess pressure that must be applied to the solution side to just stop osmosis. It is a colligative property.
• For dilute solutions π = CRT, where C is molarity; with n₂ mol of solute in V litres, π = (n₂/V)RT, and with w₂ g of solute, M₂ = w₂RT/(πV).
• Osmotic pressure is the method of choice for molar masses of proteins, polymers and other macromolecules: it is measured near room temperature, uses molarity, and gives a sizeable reading even for very dilute solutions. Biomolecules are often unstable when heated, and polymers are poorly soluble.
• Solutions with equal osmotic pressure at a temperature are isotonic, and no osmosis occurs between them. The fluid in blood cells is isotonic with 0.9% (mass/volume) sodium chloride, normal saline, which is safe for intravenous injection.
• Above 0.9% NaCl (hypertonic) water leaves blood cells and they shrink; below 0.9% (hypotonic) water enters and they swell.
• Everyday osmosis: raw mangoes shrivel in brine, wilted flowers and limp carrots firm up in fresh water, salty diets cause water retention (edema), water rises into plants partly by osmosis, and salted meat or sugared fruit kills bacteria by drawing water out of them.
• Reverse osmosis: a pressure larger than the osmotic pressure applied to the solution side pushes pure solvent out through the membrane. It is used to desalinate sea water, often with a cellulose acetate film that passes water but not ions and impurities.

9. Abnormal molar masses and van't Hoff factor

NCERT §1.7

• Ionic solutes dissociate in water and give more particles than formula units: 1 mol of KCl gives 1 mol of K⁺ and 1 mol of Cl⁻, so, ignoring interionic attraction, 1 mol of KCl in 1 kg of water would raise the boiling point by 2 × 0.52 = 1.04 K.
• Dissociation therefore makes the molar mass found from a colligative property lower than the true value.
• Association does the reverse: ethanoic acid forms hydrogen-bonded dimers in benzene, a solvent of low dielectric constant. If every molecule paired up, ΔTb and ΔTf would be half the normal value and the apparent molar mass twice the true one.
• A molar mass found this way that is lower or higher than expected is called an abnormal molar mass.
• van't Hoff factor: i = normal molar mass / abnormal molar mass = observed colligative property / calculated colligative property = total moles of particles after association or dissociation / moles before.
• i > 1 for dissociation and i < 1 for association: i is close to 2 for aqueous KCl and nearly 0.5 for ethanoic acid in benzene.
• With the factor included: relative lowering = i n₂/n₁, ΔTb = i Kb m, ΔTf = i Kf m and π = i n₂RT/V.
• For strong electrolytes i approaches the full value only on dilution because ions attract each other: NaCl gives i = 1.87 at 0.1 m, 1.94 at 0.01 m and 1.97 at 0.001 m, and K₂SO₄ approaches 3.
• For a weak electrolyte splitting into two ions with degree of dissociation α, i = 1 + α; for dimerisation with degree of association x, i = 1 − x/2.

Must-know facts

1. Molality, mole fraction, mass % and ppm are temperature-independent; molarity is not.
2. 35% (V/V) ethylene glycol lowers water's freezing point to 255.4 K (−17.6 °C).
3. Fluoride: 1 ppm prevents tooth decay; 1.5 ppm mottles teeth.
4. Henry's law: p = K_H x; larger K_H means lower solubility.
5. Gas solubility falls as temperature rises (K_H rises).
6. Scuba tanks: 11.7% He, 56.2% N₂, 32.1% O₂, to avoid the bends.
7. Raoult's law: p_i = p_i° x_i; p_total = p₁° + (p₂° − p₁°) x₂.
8. Vapour is richer in the more volatile component: y_i = p_i / p_total.
9. Ideal solution: ΔmixH = 0, ΔmixV = 0; e.g. benzene + toluene, n-hexane + n-heptane.
10. Positive deviation: ethanol + acetone, CS₂ + acetone; negative: phenol + aniline, chloroform + acetone.
11. Minimum boiling azeotrope: ethanol-water at about 95% ethanol (V/V). Maximum boiling: 68% HNO₃, 393.5 K.
12. Relative lowering of vapour pressure = mole fraction of solute.
13. Kb (water) = 0.52 K kg mol⁻¹; Kf (water) = 1.86 K kg mol⁻¹.
14. 1 mol sucrose in 1000 g water boils at 373.52 K.
15. π = CRT; osmotic pressure is used for molar masses of proteins and polymers.
16. Normal saline is 0.9% (m/V) NaCl, isotonic with blood cells.
17. Reverse osmosis uses a cellulose acetate membrane to desalinate sea water.
18. i > 1 for dissociation, i < 1 for association; KCl ≈ 2, ethanoic acid in benzene ≈ 0.5.
19. Weak electrolyte AB: i = 1 + α; dimerisation: i = 1 − x/2.

Common traps

Formulas

Key terms

Binary solution

A solution made of exactly two components.

Molality

Moles of solute per kilogram of solvent; unchanged by temperature.

Saturated solution

A solution in dynamic equilibrium with undissolved solute, holding the most solute it can at that temperature and pressure.

Henry's law constant (K_H)

The ratio of a gas's partial pressure to its mole fraction in solution; large K_H means low solubility.

Ideal solution

A solution that obeys Raoult's law at every composition, with no heat or volume change on mixing.

Azeotrope

A mixture that boils at a fixed temperature with vapour of the same composition as the liquid.

Colligative property

A property that depends on the number of solute particles, not on their identity.

Ebullioscopic constant (Kb)

The boiling point rise produced by a 1 molal solution of a non-electrolyte in that solvent.

Cryoscopic constant (Kf)

The freezing point drop produced by a 1 molal solution of a non-electrolyte in that solvent.

Semipermeable membrane

A film whose pores let solvent molecules through but hold back solute particles.

Osmotic pressure

The excess pressure on the solution side that just stops osmosis.

Isotonic solutions

Solutions with the same osmotic pressure, between which no osmosis occurs.

Reverse osmosis

Forcing solvent out of a solution through a membrane by applying more than the osmotic pressure.

van't Hoff factor (i)

The ratio of particles actually present to formula units dissolved, found from colligative data.