What is C₁V₁ = C₂V₂?

The dilution equation. C₁ is the concentration of a stock, V₁ is the volume of stock you pipette, and C₂ and V₂ are the concentration and volume of the diluted (working) solution. The identity says the number of moles of solute you take from the stock equals the number of moles that end up in the diluted solution.

When should I use a serial dilution instead of a single dilution?

When V₁ is smaller than the smallest volume you can accurately pipette (usually ≤ 1 µL on a P2). A serial dilution splits a large dilution factor into several small ones, each in a range your pipette can handle. Use the serial dilution calculator for a step plan.

Can I use C₁V₁ = C₂V₂ for percent (w/v) or mg/mL stocks?

Yes — the identity is unit-agnostic as long as C₁ and C₂ are in the same unit and V₁ and V₂ are in the same unit. This calculator only accepts M/mM/µM/nM; for %(w/v) or mg/mL, do the algebra with those units directly (e.g., 30 % H₂O₂ diluted 1:10 gives 3 %).

Do I add water first or stock first?

For most dilutions the order is not chemically important — but for concentrated acids, always add acid to water, never water to acid. Diluting concentrated H₂SO₄ or HCl releases a large amount of heat; adding water on top of the acid can boil it and splash it. The calculator does not enforce this; treat it as a lab safety rule.

Dilution Calculator — C₁V₁ = C₂V₂ Stock to Working

Dilution is the most common calculation in a wet lab: pipette a smaller amount of a concentrated stock, then add solvent to reach the target volume. C₁V₁ = C₂V₂ is all you need — moles conserved on both sides of the pipette.

Dilution Calculator

Enter any three of C₁, V₁, C₂, V₂ and pick which to solve for. Sig figs default to the lowest-precision input.

Equations used

C₁V₁ = C₂V₂, rearranged for each variable:

C_{1} = \frac{C _{2} V _{2}}{V _{1}}

V_{1} = \frac{C _{2} V _{2}}{C _{1}}

C_{2} = \frac{C _{1} V _{1}}{V _{2}}

V_{2} = \frac{C _{1} V _{1}}{C _{2}}

The equation

The number of moles you take out of the stock equals the number of moles that end up in the diluted solution:

$C_1 V_1 = C_2 V_2$

Rearranged for each variable:

Stock volume: V₁ = C₂V₂ / C₁
Stock concentration: C₁ = C₂V₂ / V₁
Final concentration: C₂ = C₁V₁ / V₂
Final volume: V₂ = C₁V₁ / C₂

The relation is unit-agnostic: as long as C₁ and C₂ share a unit and V₁ and V₂ share a unit, the arithmetic works. This calculator internally converts everything to mol/L and L so you can freely mix M with µM and mL with µL.

Worked example

Prepare 500 mL of 0.100 M HCl from a 12.0 M concentrated HCl bottle:

Given: C₁ = 12.0 mol/L, C₂ = 0.100 mol/L, V₂ = 0.500 L
Formula: V₁ = C₂V₂ / C₁
Substitute: V₁ = (0.100 × 0.500) / 12.0
Result: V₁ ≈ 4.17 mL

Add acid to water, never water to acid. Place ~400 mL of water in a 500 mL volumetric flask first, pipette 4.17 mL of 12 M HCl into the water, swirl, then top up to the 500 mL mark. Diluting concentrated acid releases a lot of heat; pouring water onto acid can boil the surface and splash.

For an interactive, step-by-step derivation with any inputs — plus a “Copy as LaTeX” button — use the calculator above.

Dilution factor shorthand

A “1:X” dilution means the stock is diluted so its concentration drops by a factor of X: DF = C₁ / C₂ = V₂ / V₁. A 1:10 dilution takes 1 part stock into 10 parts final volume (100 µL stock + 900 µL diluent = 1.0 mL); a 1:100 takes 10 µL into 1.0 mL. Below 1 µL of stock, single-step dilutions are pipette-limited — see the dilution cheat sheet for a full DF table and switch to a serial dilution for anything under that.

Common pitfalls

Add acid to water, not water to acid. Concentrated H₂SO₄, HCl, HNO₃ release enough heat when diluted to boil the surface if you pour water on top. Always place water first, add acid second.
Volumes are of the final solution. V₂ is the total volume after dilution, not the volume of diluent you added. “10 µL stock brought up to 1.0 mL” means the diluent is 990 µL, not 1.0 mL.
Pipette accuracy floor. A P20 is not reliable below ~1 µL; a P200 is not reliable below ~10 µL. If C₁V₁ = C₂V₂ tells you to pipette 0.3 µL, redesign as a serial dilution.
Same units on both sides. The identity only works if C₁ and C₂ share a unit and V₁ and V₂ share a unit. This calculator normalizes internally, but if you’re doing it by hand and mixing mM with M you will get an answer 1000× off.

Practice problems

Attempt each on paper, then expand the worked solution to check your arithmetic and sig-figs.

Problem 1 Easy You have a 1.0 M NaCl stock. How much do you pipette to make 100 mL of 100 mM NaCl?

Answer: 10 mL of stock

Solve for V₁

C₁ (stock): 1.0 mol/L
C₂ (target): 0.100 mol/L
V₂ (total): 0.100 L

Formula
$V_{1} = \frac{C _{2} V _{2}}{C _{1}}$
Substitute
$V_{1} = 0.100 \cdot 0.100/1.0$
Note
Pipette 10 mL of the 1.0 M stock, then top up with water to 100 mL total in a graduated cylinder or volumetric flask.
Result
$V_{1} = 10 mL$

Problem 2 Easy A protocol calls for 5.0 mL of 50 µM primer. Your stock is 100 µM. How much stock do you use?

Answer: 2.5 mL of stock

Solve for V₁

C₁: 100 µM (1.00e-4 mol/L)
C₂: 50 µM (5.0e-5 mol/L)
V₂: 5.0 mL (5.0e-3 L)

Formula
$V_{1} = \frac{C _{2} V _{2}}{C _{1}}$
Substitute
$V_{1} = 50 \cdot 5.0/100$
Note
Units cancel — you can keep C in µM and V in mL as long as both C values use the same unit and both V values use the same unit.
Result
$V_{1} = 2.5 mL$

Problem 3 Intermediate You need to run a 1:50 dilution. You start with 10 µL of stock. What is the final volume?

Answer: 500 µL

Solve for V₂ (dilution-factor form)

Dilution factor: 1:50 → C₂ / C₁ = 1/50
V₁: 10 µL

Formula
$V_{2} = V_{1} \cdot C_{1} / C_{2}$
Substitute
$V_{2} = 10 \cdot 50$
Note
A dilution factor of 1:50 means the final volume is 50× the stock volume. Add 490 µL of diluent to 10 µL of stock, or bring 10 µL up to 500 µL total in a small tube.
Result
$V_{2} = 500 μ L$

Problem 4 Hard Prepare 500 mL of 0.100 M HCl from a 12.0 M concentrated HCl bottle. What volume of concentrated acid do you pipette?

Answer: 4.17 mL of concentrated HCl

Solve for V₁ — concentrated acid dilution

C₁ (stock): 12.0 mol/L
C₂ (target): 0.100 mol/L
V₂: 0.500 L

Formula
$V_{1} = \frac{C _{2} V _{2}}{C _{1}}$
Substitute
$V_{1} = 0.100 \cdot 0.500/12.0$
Note
Safety: add acid to water, never water to acid. Place ~400 mL of water in the volumetric flask first, then pipette 4.17 mL of 12 M HCl into it, swirl, and top up to 500 mL.
Result
$V_{1} \approx 4.17 mL$

Frequently asked questions

What is C₁V₁ = C₂V₂?: The dilution equation. C₁ is the concentration of a stock, V₁ is the volume of stock you pipette, and C₂ and V₂ are the concentration and volume of the diluted (working) solution. The identity says the number of moles of solute you take from the stock equals the number of moles that end up in the diluted solution.
When should I use a serial dilution instead of a single dilution?: When V₁ is smaller than the smallest volume you can accurately pipette (usually ≤ 1 µL on a P2). A serial dilution splits a large dilution factor into several small ones, each in a range your pipette can handle. Use the serial dilution calculator for a step plan.
Can I use C₁V₁ = C₂V₂ for percent (w/v) or mg/mL stocks?: Yes — the identity is unit-agnostic as long as C₁ and C₂ are in the same unit and V₁ and V₂ are in the same unit. This calculator only accepts M/mM/µM/nM; for %(w/v) or mg/mL, do the algebra with those units directly (e.g., 30 % H₂O₂ diluted 1:10 gives 3 %).
Do I add water first or stock first?: For most dilutions the order is not chemically important — but for concentrated acids, always add acid to water, never water to acid. Diluting concentrated H₂SO₄ or HCl releases a large amount of heat; adding water on top of the acid can boil it and splash it. The calculator does not enforce this; treat it as a lab safety rule.

Sources

Harris, Quantitative Chemical Analysis, 10th ed. (2020), §1-3
IUPAC Gold Book — dilution
PubChem CID 5234 — Sodium chloride (5 M stock reference)

Last updated: 2026-08-13.