Solutions Formula Sheet
All key Solutions formulas for JEE: concentration units, Raoult's law, colligative properties, osmotic pressure, and van't Hoff factor — quick revision for JEE Main & Advanced.
Every must-know formula, constant, and high-yield fact from the Solutions chapter in one scannable sheet. Use it for last-minute revision before JEE Main and Advanced.
Concentration Units
| Quantity | Formula | Notes |
|---|---|---|
| Molarity (M) | $M = \dfrac{n}{V_{\text{(L)}}}$ | mol per L of solution; temperature dependent |
| Molality (m) | $m = \dfrac{n}{W_{\text{solvent (kg)}}}$ | mol per kg of solvent; temperature independent |
| Mole fraction ($\chi$) | $\chi_A = \dfrac{n_A}{n_{\text{total}}}$ | Dimensionless; $\chi_{\text{solute}} + \chi_{\text{solvent}} = 1$ |
| ppm | $\text{ppm} = \dfrac{\text{mass solute}}{\text{mass solution}} \times 10^6$ | Trace concentrations |
| % (w/w) | $\dfrac{\text{mass solute}}{\text{mass solution}} \times 100$ | Temperature independent |
| % (v/v) | $\dfrac{\text{volume solute}}{\text{volume solution}} \times 100$ | Temperature dependent |
| % (w/v) | $\dfrac{\text{mass solute (g)}}{\text{volume solution (mL)}} \times 100$ | Temperature dependent |
Molarity from Mass
$$\boxed{M = \frac{W_{\text{solute}} \times 1000}{M_{\text{solute}} \times V_{\text{(mL)}}}}$$$$\text{Moles} = M \times V_{\text{(L)}} = \frac{M \times V_{\text{(mL)}}}{1000}$$Molality from Mass
$$\boxed{m = \frac{W_{\text{solute}} \times 1000}{M_{\text{solute}} \times W_{\text{solvent (g)}}}}$$M for Mix the Volume, m for Mass of solvent. Molarity uses total solution volume; molality uses solvent mass. Molality is the one used in colligative properties because it doesn’t change with temperature.
Interconversion Formulas
| Conversion | Formula | Notes |
|---|---|---|
| M $\to$ m | $m = \dfrac{1000M}{1000d - M \cdot M_{\text{solute}}}$ | $d$ = density (g/mL) |
| m $\to$ M | $M = \dfrac{1000\,m\,d}{1000 + m \cdot M_{\text{solute}}}$ | Density is the bridge |
| m $\to \chi_{\text{solute}}$ | $\chi_{\text{solute}} = \dfrac{m \cdot M_{\text{solvent}}}{1000 + m \cdot M_{\text{solvent}}}$ | $M_{\text{solvent}}$ in g/mol |
| m $\to \chi_{\text{solvent}}$ | $\chi_{\text{solvent}} = \dfrac{1000}{1000 + m \cdot M_{\text{solvent}}}$ | — |
| M $\to \chi_{\text{solute}}$ | $\chi_{\text{solute}} = \dfrac{M \cdot M_{\text{solvent}}}{1000\,d}$ | For dilute aqueous ($d \approx 1$) |
| Normality $\leftrightarrow$ Molarity | $N = n \times M$ | $n$ = n-factor (valency factor) |
n-factor (valency factor): number of H⁺ for acids, OH⁻ for bases, or change in oxidation state for redox.
Raoult’s Law (Ideal Solutions)
$$\boxed{P_A = P_A^0 \cdot \chi_A}$$$$\boxed{P_{\text{total}} = P_A^0 \chi_A + P_B^0 \chi_B}$$For a non-volatile solute ($P_B^0 = 0$):
$$\boxed{P_{\text{solution}} = P_{\text{solvent}}^0 \cdot \chi_{\text{solvent}} = P_{\text{solvent}}^0 (1 - \chi_{\text{solute}})}$$Vapor-Phase Composition (Dalton’s Law)
$$\boxed{y_A = \frac{P_A}{P_{\text{total}}} = \frac{P_A^0 \chi_A}{P_A^0 \chi_A + P_B^0 \chi_B}}$$The vapor is always richer in the more volatile component: if $P_A^0 > P_B^0$, then $y_A > \chi_A$. This is the basis of fractional distillation.
Ideal vs Non-Ideal Solutions
| Property | Ideal | Positive Deviation | Negative Deviation |
|---|---|---|---|
| $\Delta H_{\text{mix}}$ | 0 | $> 0$ (endothermic) | $< 0$ (exothermic) |
| $\Delta V_{\text{mix}}$ | 0 | $> 0$ | $< 0$ |
| A–B vs A–A, B–B | Equal | Weaker | Stronger |
| $P_{\text{total}}$ | Obeys Raoult | $> P_{\text{ideal}}$ | $< P_{\text{ideal}}$ |
| Example | Benzene + Toluene | Ethanol + Water | Chloroform + Acetone |
| Azeotrope | — | Minimum boiling | Maximum boiling |
Must-remember examples
- Ideal: Benzene + Toluene; n-Hexane + n-Heptane; CCl₄ + SiCl₄
- Positive deviation: Ethanol + Water; Acetone + CS₂; CCl₄ + CHCl₃
- Negative deviation: Chloroform + Acetone (new H-bond); Water + HNO₃; Water + HCl
Azeotropes (cannot be separated by fractional distillation)
| Type | Deviation | Example (composition, BP) |
|---|---|---|
| Minimum boiling | Positive (max VP) | Ethanol (95.6%) + Water → 78.2°C |
| Maximum boiling | Negative (min VP) | HNO₃ (68%) + Water → 120.5°C |
Colligative Properties
The four colligative properties depend only on the number of solute particles, not their nature. Mnemonic: ROBE — RLVP, Osmotic pressure, Boiling-point elevation, freezing-point dEpression.
1. Relative Lowering of Vapor Pressure (RLVP)
$$\boxed{\frac{P^0 - P_s}{P^0} = \chi_{\text{solute}} = \frac{n_{\text{solute}}}{n_{\text{solute}} + n_{\text{solvent}}}}$$For dilute solutions ($n_{\text{solute}} \ll n_{\text{solvent}}$):
$$\frac{P^0 - P_s}{P^0} \approx \frac{n_{\text{solute}}}{n_{\text{solvent}}} = \frac{w/M}{W/M_{\text{solvent}}}$$For dilute aqueous solutions ($M_{\text{water}} = 18$):
$$\frac{\Delta P}{P^0} = \frac{w \times 18}{M \times W}$$2. Elevation of Boiling Point
$$\boxed{\Delta T_b = K_b \, m = K_b \cdot \frac{w \times 1000}{M \times W}}$$3. Depression of Freezing Point
$$\boxed{\Delta T_f = K_f \, m = K_f \cdot \frac{w \times 1000}{M \times W}}$$where $w$ = mass of solute (g), $M$ = molar mass of solute, $W$ = mass of solvent (g).
Theoretical Relations for $K_b$ and $K_f$
$$K_b = \frac{R \, M_{\text{solvent}} \, (T_b^0)^2}{1000 \, \Delta H_{\text{vap}}} \qquad K_f = \frac{R \, M_{\text{solvent}} \, (T_f^0)^2}{1000 \, \Delta H_{\text{fusion}}}$$Both depend on solvent properties only. Since $\Delta H_{\text{fusion}} < \Delta H_{\text{vap}}$, usually $K_f > K_b$.
Ratio of the Two Constants (same molality)
$$\boxed{\frac{\Delta T_b}{\Delta T_f} = \frac{K_b}{K_f}}$$For water: $\dfrac{\Delta T_b}{\Delta T_f} = \dfrac{0.52}{1.86} \approx \dfrac{1}{3.58}$ — FP depression is ~3.5× the BP elevation.
Molal Constants (memorize)
| Solvent | $K_b$ (K·kg/mol) | Normal BP (°C) | $K_f$ (K·kg/mol) | Normal FP (°C) |
|---|---|---|---|---|
| Water | 0.52 | 100 | 1.86 | 0 |
| Benzene | 2.53 | 80.1 | 5.12 | 5.5 |
| Chloroform | 3.63 | 61.2 | 4.68 | −63.5 |
| Ethanol | 1.22 | 78.4 | — | — |
| CCl₄ | 5.03 | 76.7 | — | — |
| Acetic acid | — | — | 3.90 | 16.6 |
| Camphor | — | — | 37.7 | 179.8 |
Water: $K_b = 0.52$, $K_f = 1.86$. Benzene: $K_b = 2.53$, $K_f = 5.12$. Camphor $K_f = 37.7$ (highest) — used for molar-mass determination by Rast method.
4. Osmotic Pressure
$$\boxed{\pi = CRT = \frac{n}{V}RT = \frac{w}{M\,V}RT}$$where $C$ = molarity (mol/L), $R$ = 0.0821 L·atm/(mol·K) or 8.314 J/(mol·K), $T$ in K, $V$ in L.
Isotonic condition: $\;\pi_1 = \pi_2 \implies C_1 = C_2$ (same T)
Reverse osmosis: $\;P_{\text{applied}} > \pi$
| Tonicity | Concentration vs cell | Water flow | Cell response |
|---|---|---|---|
| Hypotonic | Lower (e.g. < 0.9% NaCl) | Into cell | Swells, may burst (hemolysis) |
| Isotonic | Equal (0.9% NaCl) | No net flow | Normal |
| Hypertonic | Higher (e.g. > 0.9% NaCl) | Out of cell | Shrinks (crenation) |
$\pi$ uses molarity (C), not molality. Temperature must be in Kelvin. Pick the right $R$: use 0.0821 with atm, 8.314 with Pa. Water flows from dilute → concentrated (low $\pi$ → high $\pi$).
Molar Mass Determination
| From | Formula |
|---|---|
| Boiling-point elevation | $M = \dfrac{K_b \, w \times 1000}{\Delta T_b \times W}$ |
| Freezing-point depression | $M = \dfrac{K_f \, w \times 1000}{\Delta T_f \times W}$ |
| Osmotic pressure | $M = \dfrac{w\,R\,T}{\pi\,V}$ |
Osmotic pressure is best for macromolecules (polymers, proteins) — large, easily measured $\pi$ at room temperature.
van’t Hoff Factor (i)
$$\boxed{i = \frac{\text{Actual no. of particles in solution}}{\text{No. of formula units dissolved}} = \frac{\text{Observed colligative property}}{\text{Calculated (assuming } i = 1)}}$$Modified Colligative Properties
$$\frac{P^0 - P_s}{P^0} = i\,\chi_{\text{solute}} \qquad \Delta T_b = i\,K_b\,m \qquad \Delta T_f = i\,K_f\,m \qquad \pi = i\,CRT$$Dissociation (i > 1)
$$\boxed{i = 1 + \alpha(n - 1)} \qquad \boxed{\alpha = \frac{i - 1}{n - 1}}$$where $n$ = number of ions on complete dissociation, $\alpha$ = degree of dissociation.
| Compound | Dissociation | Theoretical i |
|---|---|---|
| NaCl, KBr | $\to$ 2 ions | 2 |
| CaCl₂, Na₂SO₄, K₂SO₄ | $\to$ 3 ions | 3 |
| AlCl₃, K₃[Fe(CN)₆] | $\to$ 4 ions | 4 |
| K₄[Fe(CN)₆], Al₂(SO₄)₃ | $\to$ 5 ions | 5 |
Association (i < 1)
$$\boxed{i = 1 - \alpha\left(1 - \frac{1}{n}\right)} \qquad \boxed{\alpha = \frac{(1-i)\,n}{n-1}}$$For dimerization ($n = 2$):
$$\boxed{i = 1 - \frac{\alpha}{2}} \qquad \boxed{\alpha = 2(1 - i)}$$For trimerization ($n = 3$): $\;i = 1 - \dfrac{2\alpha}{3}$
Abnormal Molar Mass
$$\boxed{i = \frac{M_{\text{theoretical}}}{M_{\text{observed}}}} \qquad M_{\text{observed}} = \frac{M_{\text{theoretical}}}{i}$$| Process | i | $M_{\text{observed}}$ |
|---|---|---|
| Dissociation | $> 1$ | $< M_{\text{theoretical}}$ |
| Association | $< 1$ | $> M_{\text{theoretical}}$ |
| Neither | $= 1$ | $= M_{\text{theoretical}}$ |
$i > 1 \Rightarrow$ dissociation (electrolyte). $i = 1 \Rightarrow$ non-electrolyte (glucose, urea, sucrose). $i < 1 \Rightarrow$ association (e.g. acetic acid / benzoic acid in benzene, $i \approx 0.5$). Don’t mix up the dissociation and association formulas.
Real-World i Values
| Substance | Behavior | i |
|---|---|---|
| Glucose / urea / sucrose (water) | Non-electrolyte | 1 |
| NaCl (dilute) | Strong electrolyte | ~1.9–2.0 |
| NaCl (1 M) | Strong electrolyte | ~1.7 (ion-pairing) |
| CaCl₂ | Strong electrolyte | ~2.5–2.7 |
| CH₃COOH (water) | Weak acid | 1.01–1.10 |
| CH₃COOH / benzoic acid (benzene) | Associated (dimer) | ~0.5 |
Henry’s Law (Gas in Liquid)
$$\boxed{P_{\text{gas}} = K_H \cdot \chi_{\text{gas}}}$$Higher $K_H$ $\to$ lower solubility. Applications: carbonated drinks (CO₂), scuba diving (the bends / N₂ narcosis), O₂ solubility in blood. Example: $K_H$ for CO₂ in water at 25°C $= 1.67 \times 10^8$ torr.
Key Constants & Reference Facts
| Item | Value |
|---|---|
| $R$ | 0.0821 L·atm/(mol·K) = 8.314 J/(mol·K) |
| Water: $M$, density | 18 g/mol, 1 g/mL (≈ 55.56 mol per L) |
| Normal saline | 0.9% NaCl — isotonic with blood ($\pi \approx 7.8$ atm) |
| Isotonic glucose | ~5% (w/v) |
| Seawater | ~3.5% salt → $\pi \approx 30$ atm |
| Common molar masses | NaCl 58.5, H₂SO₄ 98, NaOH 40, glucose 180, urea 60, sucrose 342 |
Highest-weightage items across this chapter: M↔m interconversion, RLVP with non-volatile solute, molar mass from $\Delta T_f$, $\pi = iCRT$ for electrolytes, and computing $i$ then $\alpha$. Always check: molality (not molarity) for $\Delta T_b/\Delta T_f$, molarity (not molality) for $\pi$, the ×1000 factor, $\Delta T_f = T_f^0 - T_f$, and the van’t Hoff factor for any electrolyte.