Most of introductory chemistry is one idea applied repeatedly: you cannot count atoms, but you can weigh them. Every calculation below is a way of getting from something you can measure — a mass, a volume, a concentration — to the number of particles actually taking part.
The mole is a counting word
A mole is a quantity, in the same way a dozen is. It is 6.02214076 × 1023 of something — Avogadro’s number, which since 2019 has been an exact defined value rather than a measured one.
The number itself is chosen so that the arithmetic disappears: one mole of a substance weighs its atomic or formula mass in grams. Carbon has an atomic mass of 12.011, so a mole of carbon weighs 12.011 g. That correspondence is the whole reason the mole is useful, and it is why the periodic table is the only lookup you need to convert between mass and particle count.
Molar mass from a formula
Add up the atomic masses, multiplied by how many of each atom the formula contains. For H2SO4: two hydrogens at 1.008, one sulfur at 32.06, four oxygens at 15.999, giving 98.07 g/mol.
The arithmetic is trivial and the transcription is where mistakes happen — a miscounted subscript in something like Ca(NO3)2, where the subscript outside the bracket multiplies everything inside it. The molecular weight calculator parses the formula, which removes that particular class of error.
Concentration: molarity and its cousin
Molarity is moles of solute per litre of solution, written M. It is what almost every problem means by concentration, and the thing to watch is that it is per litre of final solution, not per litre of solvent you started with.
Molality is moles per kilogram of solvent, written m. It appears in freezing point and boiling point problems for a specific reason: it is based on mass, so it does not change with temperature, while molarity does as the solution expands.
Dilution is one equation
C₁V₁ = C₂V₂. Concentration times volume gives moles, and diluting adds solvent without adding solute — so the moles on both sides are the same number, and three knowns give you the fourth.
The practical trap is what the answer means. If you solve for V1 and get 25 mL of stock to make 500 mL, you add solvent up to 500 mL — you do not add 500 mL to the 25. Serial dilutions compound this: each step multiplies, so ten tenfold steps is a factor of 1010. The dilution calculator handles both the single step and the series.
Stoichiometry and the limiting reagent
A balanced equation is a recipe in moles. The coefficients say that two moles of hydrogen react with one of oxygen, not two grams with one gram — so every stoichiometry problem is the same three steps: convert what you have to moles, apply the mole ratio, convert back.
The limiting reagent is the one that runs out first, and it caps the product no matter how much of everything else is present. To find it, divide each reactant’s moles by its coefficient in the balanced equation; the smallest result is limiting. Comparing raw masses is the standard mistake, and it gives the wrong answer whenever the molar masses differ much.
What you calculate from it is the theoretical yield, the most you could possibly get. Real reactions produce less, and percent yield — actual divided by theoretical, times 100 — is how that gap is reported. The stoichiometry calculator works through the ratio, the limiting reagent and the yield together.
pH is a logarithm, which changes how to read it
pH is the negative base-10 logarithm of the hydrogen ion concentration: pH = −log₁₀[H⁺]. Because it is logarithmic, each whole unit is a tenfold change. A pH of 3 is not slightly more acidic than 4, it is ten times more; and stomach acid near pH 1 is roughly a million times more acidic than water at 7.
At 25 °C, pH and pOH sum to 14, which is where the familiar scale comes from. That 14 is temperature-dependent — it is derived from the ionisation of water, which changes as water warms. Neutral water at 50 °C has a pH nearer 6.6 and is still neutral, because neutral means equal H⁺ and OH⁻, not pH 7.
The pH calculator converts in both directions, between concentration and pH or pOH.
Gases: PV = nRT
The ideal gas law ties pressure, volume, amount and temperature together, and the two things that go wrong with it are both about units.
Temperature must be absolute. Kelvin, always. Celsius in this equation gives nonsense, and at low temperatures it gives negative volumes.
R has to match the other units. It is 8.314 J/(mol·K) with pressure in pascals and volume in cubic metres, or 0.08206 L·atm/(mol·K) with atmospheres and litres. Mixing them is the most common error in the whole topic.
One more worth flagging: many textbooks give the molar volume of a gas at STP as 22.4 L/mol, which assumes 1 atm. IUPAC’s definition of standard pressure has been 100 kPa since 1982, which gives 22.71 L/mol. Both numbers are correct for their own definition, so check which one your course uses. The ideal gas law calculator solves for whichever variable you leave blank.
Significant figures, where the marks actually go
A calculator will hand you ten digits regardless of how precisely you measured anything, and reporting all of them claims a precision you do not have. Two rules cover nearly every case, and they are different from each other:
Multiplying or dividing: the answer gets the same number of significant figures as the least precise input.
Adding or subtracting: the answer gets the same number of decimal places as the input with the fewest — not significant figures, decimal places. This is the one people merge with the first rule and get wrong.
Exact numbers — counted objects, defined conversions, the 2 in a balanced equation — have unlimited significant figures and never limit an answer. And round once, at the end; rounding at each step accumulates error. The significant figures calculator counts and rounds, and the density calculator is the quickest check on a mass-and-volume result that looks implausible.