Dilution Calculations Made Simple: C1V1 = C2V2 With Worked Examples
Making a dilution is one of the most common tasks in any lab, from teaching labs to research benches. It's also one of the most common sources of quiet errors: a misplaced decimal, mixed-up units or a stock that wasn't what you thought it was. The good news is that nearly every simple dilution comes down to one equation, and once you set it up carefully, it's hard to get wrong.
The equation
C₁V₁ = C₂V₂
- C₁ — concentration of your stock (the concentrated solution you're starting with)
- V₁ — volume of stock you need to take
- C₂ — concentration you want in the final solution
- V₂ — total final volume you want
The idea is that the amount of solute doesn't change when you add solvent — it just spreads out into a larger volume. Concentration × volume gives the amount of solute, so the amount before equals the amount after.
Usually you know C₁, C₂ and V₂ and need V₁:
V₁ = (C₂ × V₂) ÷ C₁
Worked example 1: a simple dilution
You have a 1 M stock of NaCl and need 100 mL of 0.1 M NaCl.
- C₁ = 1 M
- C₂ = 0.1 M
- V₂ = 100 mL
V₁ = (0.1 M × 100 mL) ÷ 1 M = 10 mL
Take 10 mL of the stock and add solvent until the total volume is 100 mL. Note the wording: you bring the total to 100 mL — you don't add 100 mL of water. Adding 90 mL is usually close, but for accurate work, use a volumetric flask or graduated cylinder and fill to the final volume.
Worked example 2: working in microliters
You have a 10 mM stock and need 500 µL at 200 µM.
First, put both concentrations in the same unit. 10 mM = 10,000 µM.
V₁ = (200 µM × 500 µL) ÷ 10,000 µM = 10 µL
So: 10 µL of stock + 490 µL of diluent = 500 µL total.
The key move is the unit conversion. C₁ and C₂ must be in the same units, and V₁ will come out in whatever unit you used for V₂.
Worked example 3: a "10X" stock
Many buffers are sold or made as concentrated stocks labeled 10X, meaning ten times the working concentration. To make 1 L of 1X buffer from a 10X stock:
V₁ = (1X × 1000 mL) ÷ 10X = 100 mL
100 mL of 10X stock, brought to 1 L total. The "X" units cancel just like molarity.
Dilution factors
A dilution factor is the ratio of final volume to the volume of stock: V₂ ÷ V₁ (equivalently C₁ ÷ C₂). In example 1, the dilution factor is 100 ÷ 10 = 10, often written as a 1:10 dilution.
A word of caution: "1:10" is used in two ways. Most labs mean 1 part stock in 10 parts total, but some people mean 1 part stock plus 10 parts diluent (1 in 11 total). If a protocol is ambiguous, ask or check how your lab writes it, and record exactly what you did in your notebook.
Serial dilutions
When you need a very large dilution or a series of concentrations (for a standard curve, for example), a single step can require impractically tiny volumes. A serial dilution repeats a smaller dilution several times.
Example: a 1:10 serial dilution, four steps, starting from 1 mg/mL.
- Tube 1: 100 µL stock + 900 µL diluent → 0.1 mg/mL
- Tube 2: 100 µL of tube 1 + 900 µL diluent → 0.01 mg/mL
- Tube 3: 100 µL of tube 2 + 900 µL diluent → 0.001 mg/mL
- Tube 4: 100 µL of tube 3 + 900 µL diluent → 0.0001 mg/mL
The total dilution is the dilution factors multiplied: 10 × 10 × 10 × 10 = 10,000. Mix each tube thoroughly before moving on, and use a fresh tip each transfer — carryover is the classic serial-dilution error.
When C₁V₁ = C₂V₂ doesn't apply
The equation is for diluting a solution with solvent. It isn't the right tool when:
- You're making a solution from a solid — use mass = molarity × volume × molar mass instead.
- You're mixing two different solutions that both contain the solute at different concentrations.
- A chemical reaction occurs on mixing, as in a titration.
Common mistakes
- Mixed units. mM on one side and µM on the other gives an answer off by 1,000.
- Swapping C₁ and C₂. If V₁ comes out larger than V₂, you've almost certainly flipped them.
- Adding V₂ of diluent instead of bringing the total to V₂.
- Volumes too small to measure accurately. If V₁ is below what your pipette handles reliably, make an intermediate dilution first.
- Not checking the stock. Confirm the label, concentration and expiry date before you start.
Record the calculation, not just the result
In your lab notebook, write the full equation with the numbers filled in: "V₁ = (0.1 M × 100 mL) ÷ 1 M = 10 mL." A bare "10 mL" can't be checked by anyone, including you next month. Also record the stock's lot number and the final volume. If you made a buffer that others will use, log it in a solution prep log with the date, who made it and its expiry. (For more on this, see our guide on how to keep a lab notebook.)
A reference on the bench
Our Research Lab Notebook Pack includes a one-page Common Lab Calculations reference — molarity, C₁V₁ = C₂V₂, percent solutions, serial dilutions, ppm, centrifuge RCF and SI prefixes — with a blank dilution worksheet, alongside experiment entry pages and a solution and buffer prep log for recording what you made.
Set up every dilution the same way — write the equation, convert the units, check that V₁ is smaller than V₂ — and this everyday calculation becomes one you never have to second-guess.
