How to Do Dilution Calculations (C1V1 = C2V2)

C1 V1 = C2 V2

because n = C V, and n₁ = n₂ if you did not spill or react the solute away.

The dilution calculator solves that product for the blank field. No account. It will not notice if C1 is 6.0 M and C2 is 0.15 mM unless you convert.

What each symbol is on the bench

• C1 - concentration of the stock (the bottle).
• V1 - how much stock you transfer (pipette, buret, graduated cylinder).
• C2 - concentration you want (the working solution).
• V2 - final volume after dilution, usually a volumetric flask filled to the mark.

V2 is not “how much water you added” unless you are using the approximation V2 ≈ V1 + V_water and the volumes really add. For a 6.25 mL aliquot in a 250 mL flask, you add water to the mark, not 250 mL of water on top of 6.25 mL (that would be 256 mL).

C can be molarity, or g/L, or any unit proportional to amount per volume, as long as C1 and C2 match. You cannot set C1 in M and C2 in % w/w without converting.

Stock 6.0 M down to 0.15 M in 250 mL

You need 250 mL of 0.15 M HCl. The cabinet has 6.0 M.

V1 = C2 V2 / C1 = (0.15 mol/L × 250 mL) / 6.0 mol/L = 6.25 mL

Pipette 6.25 mL of 6.0 M into a 250 mL volumetric flask, add water toward the mark, mix, fill to the line. Moles transferred: 6.0 mol/L × 0.00625 L = 0.0375 mol. Final M: 0.0375 mol / 0.250 L = 0.150 M. That is the check.

If you only have a 10 mL pipette that delivers 10.00 mL, you can make a larger batch: V2 = C1 V1 / C2 = (6.0 × 10.00) / 0.15 = 400 mL, and use a 500 mL flask only if you actually fill to 400 mL or change the target.

mL vs L. (0.15 × 0.250 L) / 6.0 = 0.00625 L = 6.25 mL. Same answer. (0.15 × 250) / 6.0 with C in mol/L and V in mL works because the milliliters cancel. (0.15 × 250 L) / 6.0 does not.

Serial dilutions are repeated C1V1 = C2V2

Going from 1.0 M to 1.0 × 10⁻³ M in one step is a 1:1000 dilution: 1.0 mL into a 1.0 L flask, or 0.100 mL into 100 mL. Pipetting 0.100 mL well is harder than pipetting 1.00 mL twice.

Two 1:100 steps:

1. 1.00 mL of 1.0 M → 100 mL → 0.010 M
2. 1.00 mL of 0.010 M → 100 mL → 1.0 × 10⁻⁴ M

Each step: C2 = C1 × (1.00/100). Errors multiply, so one sloppy flask ruins the chain. Label every bottle with the new C.

For pH work, dilute first, then compute pH from the new [H+]. 6.0 M HCl is not something you stick a probe into casually, and −log(6.0) is also a bad description of “concentrated HCl” (activity ≠ 6 M). See pH from concentration.

When the mole-conservation story breaks

Reaction. Diluting bleach into an acid, or adding a reagent that precipitates the analyte, means n is not constant. C1V1 = C2V2 is the wrong equation.

Two solutions mixed. If both beakers already contain the solute, n_total = C_a V_a + C_b V_b, and C_mix = n_total / V_mix. That is not C1V1 = C2V2 with a single stock.

Non-additive volumes. 50 mL of ethanol plus 50 mL of water is less than 100 mL. If V2 must be exact, measure V2 (or use a flask). Using V1 + V_water as V2 is the approximation.

Heat. Concentrated H2SO4 into water: add acid to water, in small portions, because the dilution is exothermic. The formula still gives V1; the procedure is a safety issue, not a new algebra.

Molality. C1V1 = C2V2 is a volume relation. Molality uses kilograms of solvent. Do not dilute “2.00 m” with this equation unless you have converted to a volume-based concentration.

Unit and technique traps

C2 larger than C1: you cannot dilute to a higher concentration. You would need to evaporate solvent or add solid. The calculator may still divide; the chemistry will not.

Using the molarity calculator to “find V” from n and M when you meant a dilution from stock. If you do not have n yet, you want V1 from C1V1 = C2V2, not V = n/M with a guessed n.

A 1:10 dilution means V2 = 10 × V1, so C2 = C1 / 10. People write “1:10” and then pipette 10 mL into 10 mL of water (that is 1:2 if V2 = 20 mL). Agree on whether the ratio is parts stock : parts final or parts stock : parts solvent.

Solve for the missing piece, then make it

Leave one of C1, V1, C2, V2 empty in the dilution calculator. Keep both C’s in the same unit and both V’s in the same unit. If you still need to prepare the stock from grams, that is M = n/V on the molarity calculator before you dilute. After a strong-acid dilution, pH uses the working C, not the stock C.