Atoms and molecules are far too small and too numerous to count one by one, so chemists count them in a fixed-size bundle called the mole. This concept is the arithmetic of that bundle — converting freely between mass, moles, number of particles, and volume of a gas.
1 — The mole and Avogadro's number. One mole is the amount of a substance that contains exactly Nₐ = 6.022×10²³ elementary entities (atoms, molecules, ions, electrons — whatever is specified). So number of particles = moles × Nₐ, and moles = number of particles / Nₐ. One mole of water contains 6.022×10²³ molecules; one mole of Na⁺ contains 6.022×10²³ ions.
2 — Molar mass. The molar mass M (in g/mol) is the mass of one mole. Numerically it equals the relative atomic mass (for an element) or the sum of the atomic masses in the formula (for a compound): H₂O has M = 2(1) + 16 = 18 g/mol; glucose C₆H₁₂O₆ has M = 6(12) + 12(1) + 6(16) = 180 g/mol. The central bridge is moles = mass / molar mass, i.e. n = m / M.
3 — The interconversion web. Everything hangs off moles as the hub:
moles = mass / M = number of particles / Nₐ = gas volume at STP / 22.4 L.
Pick the box you are given, convert to moles, then convert out to the box asked for. For example, mass → number of molecules is a two-step chain: m → n = m/M → N = n·Nₐ.
4 — Gas volume at STP. One mole of any ideal gas occupies a fixed molar volume: 22.4 L at the old STP (0 °C, 1 atm), or 22.7 L at the current IUPAC STP (0 °C, 1 bar). So moles of gas = volume at STP / 22.4 L. Use the molar volume the question specifies; JEE problems usually still take 22.4 L unless "1 bar" is stated. This only applies to gases — never to a solid or liquid.
5 — Counting a specific sub-species. To count the atoms of one particular element in a sample, first get the moles of the compound, then multiply by the number of that atom in one formula unit, then by Nₐ. In 1 mole of H₂SO₄ there are 4 moles of O atoms, i.e. 4 × 6.022×10²³ = 2.409×10²⁴ oxygen atoms. The same logic gives the total atoms (multiply by all atoms per formula unit), or the number of electrons/protons (moles × electrons-per-formula-unit × Nₐ).
6 — Average (relative) atomic mass. An element's tabulated atomic mass is the abundance-weighted average of its isotopes: M̄ = Σ (fractional abundance × isotope mass). For chlorine, 75% of mass-35 and 25% of mass-37 give 0.75(35) + 0.25(37) = 35.5. The reverse question — find an isotope's abundance from the known average — is solved by letting the fraction be x, writing the weighted-average equation, and solving the single linear equation.
7 — Mass of one particle; comparisons. The mass of a single atom or molecule is molar mass / Nₐ (e.g. one water molecule weighs 18 / 6.022×10²³ ≈ 2.99×10⁻²³ g). To decide which sample has the most atoms/molecules, convert each to moles first — equal masses of two substances do not contain equal numbers of particles (the lighter molar mass has more moles), whereas equal moles always contain equal numbers of molecules regardless of mass.
Common traps. Confusing "molecules" with "atoms" (a mole of O₂ has 6.022×10²³ molecules but 1.2×10²⁴ atoms); applying 22.4 L to a non-gas; comparing particle counts by mass instead of by moles; forgetting to multiply by the number of a given atom per formula unit; and mixing the 22.4 L / 22.7 L molar volumes.
How this concept is examined. JEE Main tests: mass ↔ moles; moles ↔ number of molecules/atoms; mass → number of atoms of a specified element; gas volume at STP ↔ moles/molecules; electrons/protons/neutrons in a given mass; average atomic mass from isotopic abundances (and the reverse); mass of a single particle; and "which sample has the most atoms/molecules". The one habit that prevents most errors: always convert to moles first, then out — and read carefully whether the question asks for molecules, atoms of one element, or total atoms.