C1*V1 = C2*V2 - solve any one of the four dilution variables.
Conservation of moles: C1 · V1 = C2 · V2. The diluent volume is V2 − V1 (assuming additive volumes — fine for aqueous solutions, less so for high-concentration ethanol or sulfuric acid).
The dilution equation C₁ · V₁ = C₂ · V₂ is the most-used formula in laboratory chemistry, biology, pharmacy, and any domain that handles solutions. It's a statement of conservation: moles of solute don't change when you add solvent (only the volume and thus the concentration change). The arithmetic is trivial — but mistakes are common because there are four variables and only one equation; you need to know which three to fix and which to solve. A calculator that lets you pick which variable to solve, validates the others, and visualizes the stock + diluent split removes the per-experiment friction. The same equation is used to prepare PCR master mixes, cell-culture media, working stocks of antibiotics, and (in a different domain) cooking dilutions of stock concentrate.
Conservation of solute: the moles of solute n = C · V are conserved across a dilution (no solute added or removed). Therefore:
C₁ · V₁ = C₂ · V₂
Solving for any one variable: - V₁ (volume of stock to take) = (C₂ · V₂) / C₁. - C₁ (stock concentration, as a sanity-check on a working stock) = (C₂ · V₂) / V₁. - C₂ (final concentration after dilution) = (C₁ · V₁) / V₂. - V₂ (final volume after adding diluent) = (C₁ · V₁) / C₂.
Diluent volume to add = V₂ − V₁.
Dilution factor = C₁ / C₂ — common shorthand: "1:10 dilution" means dilution factor 10, i.e., one part stock plus 9 parts diluent.
The calc assumes additive volumes (V_total = V_stock + V_diluent). This is exact for ideal mixing of dilute aqueous solutions and a good approximation for most practical purposes. For high-concentration ethanol or sulfuric acid, real volumes contract by a few percent (mixing exotherm); use the result as a starting point and top up to V₂ at the meniscus.
Pick which variable to solve for. Enter the three known values (the calc will use C₁, V₁, C₂, V₂ as labeled). Concentrations in mol/L, volumes in mL. The result panel echoes all four values (with the solved one filled in), shows the diluent volume (V₂ − V₁), the dilution factor (C₁ / C₂), and the stock fraction (V₁ / V₂ as %), plus a stacked-bar visualization of stock + diluent.
You have 1.0 mol/L NaCl stock; you need 250 mL of 0.1 mol/L NaCl. Solve V₁.
PCR setup: you have a 10× buffer, you need 100 µL of 1× working solution. Solve V₁.
You have 50 mL of 0.5 mol/L stock and you want to know what concentration you'd get if you topped it up to 500 mL. Solve C₂.
Units must match. C₁ and C₂ in the same unit (mol/L); V₁ and V₂ in the same unit (mL). Mixing M (mol/L) and mM (mmol/L), or mL and µL, gives wrong results by a factor of 1 000.
Additive-volume assumption breaks at high concentration. Mixing 100 mL ethanol + 100 mL water yields ~196 mL, not 200 — there's a ~2 % volume contraction. For exact targeting at high concentrations, dilute by mass (g) rather than volume.
Stock concentration uncertainty. A "1.0 M" stock that's been on the shelf 6 months may be 0.95 M (evaporation through the cap, hygroscopic salt loss). Recalibrate stocks for sensitive applications.
Pipetting accuracy at low V₁. If V₁ < 5 µL on a 200 µL system, pipette accuracy drops below 5 %. Use a serial dilution (intermediate stock) instead.
Serial dilutions. Going from 1.0 mol/L to 1 µmol/L is a ×10⁶ dilution; in a single step, V₁ would be 1 µL out of 1 mL (4-decimal-place pipetting). Better: 6 serial 1:10 dilutions, each from the prior one.
Diluent identity matters. Diluting an HCl stock with water → still HCl (pH set by C₂). Diluting with a buffered solution → buffered acid. Pick the diluent that matches the chemistry you need, not just "water".
Mass / volume vs molar concentration. % w/v and % w/w are not directly compatible with M; convert via molecular weight first.
Limit of solubility. Diluting a saturated solution with the same solvent always reduces concentration; diluting with a different solvent might precipitate the solute (e.g., diluting an aqueous protein stock into pure ethanol). Watch for cloudiness.
Negative diluent volume. If V₁ > V₂ (you accidentally enter a working concentration C₂ > C₁), the diluent volume is negative — meaning you can't dilute up; you need a higher-concentration stock or to evaporate.
Polyprotic dilution. The dilution equation conserves moles of solute (e.g., moles of H₂SO₄). pH and ionic strength change non-trivially because dissociation equilibria shift. C₁V₁ = C₂V₂ is correct for total moles, not for [H⁺] specifically.