How to Use the Dilution Equation

What dilution actually is

When you dilute a solution, the moles of solute don’t change — only the volume around them does. That single conservation statement (n_before = n_after) is the entire reason C1V1 = C2V2 works. Since concentration times volume equals moles, the moles on each side must match.

C1 × V1 = C2 × V2

• C1, V1: concentration and volume of the stock you start with
• C2, V2: concentration and total volume of the diluted solution you finish with

Notice what V2 is not: it’s not the volume of water you add. It’s the final total volume after dilution. Mixing those two up is the single most common mistake students make with this equation.

How to use it

Find the three values you have. Pick the matching algebraic rearrangement:

• V1 = (C2 × V2) / C1 — “how much stock do I need?”
• V2 = (C1 × V1) / C2 — “how far can I dilute this?”
• C2 = (C1 × V1) / V2 — “what concentration did I end up with?”
• C1 = (C2 × V2) / V1 — “how concentrated was that mystery stock?”

Units have to match within each pair. Both concentrations in the same units, both volumes in the same units. The equation doesn’t care whether you use M and mL or mM and L — only that you stay consistent.

Worked examples

1. Stock volume needed. You want 500 mL of 0.100 M NaCl from a 1.00 M stock. V1 = (0.100 × 500) / 1.00 = 50.0 mL. Pipette 50.0 mL of stock into a 500 mL volumetric flask, fill to the mark with water.

2. Final concentration. You take 25.0 mL of 6.00 M HCl and dilute to 150.0 mL total. C2 = (6.00 × 25.0) / 150.0 = 1.00 M.

3. Final volume. You have 10.0 mL of 12.0 M HCl and want a 0.600 M working solution. V2 = (12.0 × 10.0) / 0.600 = 200 mL total volume. The water you add is V2 − V1 = 190 mL.

4. Serial 1:10 dilution, three rounds, starting from 5.00 M. Each round divides by 10. After three: 5.00 × (1/10)³ = 5.00 × 10⁻³ M. Serial dilutions are how you get to nanomolar concentrations without trying to pipette 0.5 µL of stock — accuracy compounds when you stay in the 10–1000 µL range each step.

5. Diluting concentrated H₂SO₄. You need 1.00 L of 1.00 M from 18.0 M concentrated. V1 = (1.00 × 1000) / 18.0 = 55.6 mL of concentrated acid.

This is where the equation meets lab safety: always add acid to water, never water to acid. Pour ~800 mL of water into a beaker first, then slowly add the 55.6 mL of concentrated H₂SO₄ with stirring. The dilution is highly exothermic — adding water to the acid concentrates the heat at the surface and can boil off droplets of concentrated acid into your face. Transfer to a volumetric flask and bring to volume only after it has cooled.

Traps to watch for

V2 is not the water you add. It’s the final total volume. If you start with 50 mL of stock and your equation gives V2 = 500 mL, you add 450 mL of water — not 500.

Concentration units have to match. A C1 in M and a C2 in mM differs by a factor of 1000. Convert before substituting.

The equation assumes no chemistry happens. C1V1 = C2V2 is pure conservation of moles. The moment you mix two reactive species, or the dilution itself protonates/deprotonates something (e.g., diluting a weak acid changes its degree of dissociation), you’re outside the regime where this equation holds.

Sig figs. Volumetric glassware delivers to 3-4 sig figs at best. Reporting V1 = 50.000 mL when your stock concentration is known to 1.0 M is fiction.

Volumes don’t always add ideally. Mixing 50 mL of ethanol with 50 mL of water gives ~96 mL, not 100. For aqueous chemistry at lab concentrations the error is small enough to ignore, but it’s why volumetric flasks are calibrated to a final mark rather than relying on additive volumes.

Use the Dilution Calculator to solve for any of the four variables and see the rearrangement step-by-step.