Core answer: the four formulas — molar mass M = sum of atomic weights in the formula; amount n=m/M (1 mol = 6.022×10²³ particles); ideal gas PV=nRT (R=8.314 J/(mol·K)); dilution C₁V₁=C₂V₂; pH=−lg[H⁺], neutral at 7, each unit a 10× concentration jump.
Computing molar mass
Sum atomic weight × subscript over the formula: H₂O = 1.008×2 + 16.00 = 18.015 g/mol; CO₂ = 12.01 + 32.00 = 44.01 g/mol; Ca(OH)₂ = 40.08 + (16.00+1.008)×2 = 74.09 (expand parentheses); CuSO₄·5H₂O = 63.55+32.06+64 + 5×18.015 = 249.68 (expand hydrate dots). The [molar mass calculator](/c/science/molar-mass) parses nested parentheses and hydrates, with mass percentages.
Common atomic weights
| Element | Weight | Element | Weight |
|---|---|---|---|
| H | 1.008 | O | 16.00 |
| C | 12.01 | N | 14.01 |
| Na | 22.99 | Mg | 24.31 |
| Cl | 35.45 | K | 39.10 |
| Ca | 40.08 | Fe | 55.85 |
| Cu | 63.55 | S | 32.06 |
The ideal gas law
PV = nRT — pick R to match your units:
| R | P unit | V unit |
|---|---|---|
| 8.314 | Pa | m³ |
| 0.08206 | atm | L |
| 62.36 | mmHg | L |
Temperature must be kelvin: K = ℃ + 273.15. At STP (0℃, 1 atm) 1 mol of any gas = 22.4 L — memorize that.
Dilution and solvent volume
Solute is conserved: C₁V₁ = C₂V₂. Solvent to add = V₂ − V₁ — you bring the total volume up to V₂, not add V₂ of solvent.
pH with everyday examples
pH = −lg[H⁺]; at 25℃, pH + pOH = 14.
| Substance | pH | [H⁺] mol/L |
|---|---|---|
| Stomach acid | 1.5-2 | 10⁻¹·⁵ |
| Lemon juice | 2.4 | 10⁻²·⁴ |
| Vinegar | 2.9 | 10⁻²·⁹ |
| Pure water | 7.0 | 10⁻⁷ |
| Blood | 7.35-7.45 | 10⁻⁷·⁴ |
| Soapy water | 9-10 | 10⁻⁹·⁵ |
| Bleach | ~12 | 10⁻¹² |
Two pH units apart = 100× concentration — stomach acid is about 30× more acidic than lemon juice.
Example: making 100mL of 0.1M saline
① moles needed: n = C×V = 0.1 × 0.1 = 0.01 mol; ② mass: m = 0.01 × 58.44 = 0.584 g; ③ dissolve in ~80mL water, transfer to a 100mL flask, bring to the mark.
Example: diluting household bleach
Stock bleach at 5% available chlorine, surface disinfection needs 0.05%: ① dilution factor = 5 ÷ 0.05 = 100×; ② for 10mL of stock, V₂ = C₁V₁/C₂ = 5×10/0.05 = 1000mL — add water up to a 1L total (about 990mL); ③ mix fresh daily — diluted chlorine decays fast. Solve any three variables with the [dilution calculator](/c/science/dilution).
Common mistakes
- "Dilute 10× means add 10× water": wrong — 10× dilution means total volume becomes 10×: 1 part stock brought to 10 parts total. Adding 10 parts water gives roughly 1/11 concentration.
- Celsius in PV=nRT: R is defined against kelvin — writing 25 instead of 298.15 throws results off by 12×.
- "pH 7 is always neutral": only at 25℃. At 100℃ pure water reads pH≈6.1 yet remains neutral ([H⁺]=[OH⁻]).
- Memorizing molar masses: just sum the atomic weights — the calculator even handles hydrate dots like CuSO₄·5H₂O.