What is the C₁V₁ = C₂V₂ equation?

It states that the amount of solute is conserved during dilution. The concentration of the stock times its volume equals the concentration of the final solution times its volume, so you can solve for any one unknown if you know the other three.

Which units should I use?

Any units, as long as the two concentrations share a unit and the two volumes share a unit. If C₁ and C₂ are both in molar and V₂ is in millilitres, the V₁ result comes out in millilitres. The tool does not assume units.

How much diluent do I add?

You add diluent equal to the final volume minus the stock volume, V₂ minus V₁. Measure out the stock volume first, then top up to the final volume — do not simply add V₂ of diluent, which would over-dilute.

What is the dilution factor?

The dilution factor is C₁ divided by C₂, the ratio of stock to final concentration. A factor of 5 means a 1-in-5 dilution: one part stock to four parts diluent, giving a fifth of the original concentration.

Can I use this for serial dilutions?

Yes, one step at a time. Compute each step's stock volume against the next target concentration, then use that diluted solution as the stock for the following step. The same equation applies at every stage of a serial dilution.

What is the Dilution Calculator (C₁V₁ = C₂V₂)?

Apply C₁V₁ = C₂V₂ to solve for any missing variable when diluting a stock solution to a target concentration and volume, with the diluent amount and dilution factor. A browser-based lab calculator for scientists, pharmacists, and QC analysts. It runs free in your browser on Gera Tools, with nothing uploaded.

Dilution Calculator (C₁V₁ = C₂V₂)

Name: Dilution Calculator (C₁V₁ = C₂V₂)
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

Get one useful tool a week

Like this tool? Enter your email and we'll send you one genuinely useful Gera tool a week — plus a link to come back to this one. No spam, one-click unsubscribe any time.

The dilution equation C₁V₁ = C₂V₂ is the workhorse of any lab that prepares solutions. It captures the simple fact that diluting a solution does not change the amount of solute present — only the concentration changes because the same quantity of solute now occupies more volume. This calculator solves for whichever variable you leave blank and tells you how much diluent to add, all in your browser.

Why the equation works

When you dilute a solution, you are not removing or adding solute — you are adding solvent (the diluent). The number of moles or mass of solute stays constant. Because concentration is amount per unit volume, and the amount is fixed:

C₁ × V₁ = moles of solute = C₂ × V₂

This holds regardless of the unit system, as long as the units on each side are consistent: if concentration is in mol/L (molar), volume must be in litres. The tool is unit-agnostic — use any matched pair of concentration and volume units and the solved answer comes back in those same units.

Solving for each unknown

Rearranging for each of the four variables:

V₁ = (C₂ × V₂) / C₁    — how much stock to take
C₁ = (C₂ × V₂) / V₁    — what stock concentration you need
C₂ = (C₁ × V₁) / V₂    — what final concentration you get
V₂ = (C₁ × V₁) / C₂    — what final volume to make up

Fill in three values and leave one blank. The tool selects the correct rearrangement automatically.

Worked example

You need 50 mL of a 2 M working solution from a 10 M stock. Leave V₁ blank:

V₁ = (2 × 50) / 10 = 10 mL

Take 10 mL of the 10 M stock and bring the total volume up to 50 mL with diluent — adding 40 mL of diluent. This is a 5× dilution (dilution factor = C₁ / C₂ = 10 / 2 = 5).

The most common mistake

Do not add V₂ of diluent. The correct procedure is: measure out V₁ of stock, then top up the combined solution to V₂ total. If you add 50 mL of diluent to 10 mL of stock, you get 60 mL of solution — more dilute than you intended. Always measure the stock first, then bring the total up to the target volume.

Serial dilutions

For a serial dilution — where you dilute a diluted solution to dilute it further — apply the equation at each step. The output of step one becomes the C₁ and V₁ of step two. The same principles apply at every stage, and the final concentration is the product of each step’s dilution factor applied sequentially.