Dilution Calculator C₁V₁ = C₂V₂: Stock and Final Volumes

The dilution calculator solves C1 V1 = C2 V2: take a volume V1 from a stock solution of concentration C1, add solvent up to a total volume V2, and obtain a concentration C2. Any three of the four quantities determine the fourth—how much water to add to 10 mL of 12 mol/L concentrated hydrochloric acid to reach 1 mol/L, how much stock to pipette to prepare 250 mL of a 0.5 mol/L solution, and how concentrated a sample really was when 0.1 mol/L is measured after a 20-fold dilution.

Dilution Calculator C₁V₁ = C₂V₂: Stock and Final Volumes

About this calculator

The dilution calculator solves $C_1 V_1 = C_2 V_2$ : take a volume $V_1$ from a stock solution of concentration $C_1$ , add solvent up to a total volume $V_2$ , and obtain a concentration $C_2$ . Any three of the four quantities determine the fourth—how much water to add to 10 mL of 12 mol/L concentrated hydrochloric acid to reach 1 mol/L, how much stock to pipette to prepare 250 mL of a 0.5 mol/L solution, and how concentrated a sample really was when 0.1 mol/L is measured after a 20-fold dilution.

The equation says one thing: the amount of solute is unchanged by dilution; only the volume changes. Because it is a proportion, the concentration need not be molar: mass fraction (37% HCl), g/L, mg/mL, ppm and a buffer's “10×” factor can all be entered directly, as long as $C_1$ and $C_2$ use one unit and $V_1$ and $V_2$ use one unit. What a mole is, how the five concentration measures convert, and why the volume is made up to the mark rather than added up are the subject of the chemistry-cluster anchor Molarity calculator; this page mentions them only where they are needed.

What it does not do: it does not compute how much solvent to add ( $V_2 - V_1$ , one hand-calculation, see Example 1); it does not check whether the target concentration is above the stock concentration (dilution can only make a solution weaker, and a $C_2 > C_1$ result has no physical meaning, see Example 4); it does not handle non-additive volumes, heat release or evaporation; it does not convert mL ↔ L; and it cannot be used for logarithmic “concentrations” such as pH.

Formula and variables

C_1 V_1 = C_2 V_2 \qquad\Longleftrightarrow\qquad \text{dilution factor}\ \frac{C_1}{C_2} = \frac{V_2}{V_1}

The four inverse forms (all implemented by the engine):

V_2 = \frac{C_1 V_1}{C_2},\quad C_2 = \frac{C_1 V_1}{V_2},\quad V_1 = \frac{C_2 V_2}{C_1},\quad C_1 = \frac{C_2 V_2}{V_1}

Symbol	Meaning	Calculator field	Unit	Range and notes
$C_1$	stock (concentrated) concentration	C₁ (mol/L)	mol/L (or any concentration unit consistent with $C_2$ )	$C_1 > 0$ ; default 12 (commercial concentrated hydrochloric acid is about 12 mol/L)
$V_1$	volume taken from the stock	V₁ (mL)	mL (or any volume unit consistent with $V_2$ )	$V_1 > 0$ ; default 10
$C_2$	target concentration after dilution	C₂ (mol/L)	same as $C_1$	$0 < C_2 < C_1$ ; no default
$V_2$	total volume after dilution (the volume made up to the mark, not the amount of solvent added)	V₂ (mL)	same as $V_1$	$V_2 > V_1$ ; no default

The panel is labelled in mol/L and mL, but like quantities cancel on both sides of the equation: percentages, g/L and litres all work as long as they are used consistently. The solvent to add $= V_2 - V_1$ (when volumes are additive).

How to use this calculator

Open the panelthis page corresponds to “Math tools → Formulas → Dilution law” (/#/mathtools/formula?calculator=calculator.chem-dilution). On open, “C₁” = 12 and “V₁” = 10 are already filled in, while both “C₂” and “V₂” carry a “calculate” marker and show 120—those two 120s are not results: when the other unknown is empty the engine falls back to a denominator of 1, and $12 \times 10 / 1 = 120$ . The display only becomes meaningful after you fill in either $C_2$ or $V_2$ .
Find the final volume(the most common use): fill in the target concentration $C_2$ ; the final-volume field shows $C_1 V_1 / C_2$ . The solvent to add is $V_2 - V_1$ .
Find the target concentrationclear $C_2$ , fill in the final volume $V_2$ , and the target-concentration field shows $C_1 V_1 / V_2$ .
Solve for the volume to take(the most common question when preparing a solution): clear “V₁”, fill in $C_1$ , $C_2$ and $V_2$ , and the field shows $C_2 V_2 / C_1$ —how much stock to draw up with a pipette.
Solve for the stock concentrationclear “C₁”, fill in $V_1$ , $C_2$ and $V_2$ , and the field shows $C_2 V_2 / V_1$ —recovering the stock concentration from a diluted measurement.
All four filled inevery field simply shows its input value; the engine stops calculating and does not check the four numbers for consistency. To calculate again, clear the quantity you want to solve for.
Read the resultvalues keep at most 8 decimal places with trailing zeros removed. With $C_2 = 0$ , $V_2$ displays ∞—diluting to zero concentration would need an infinite volume.
Copy the formulathe copy button at the right of the panel title puts the LaTeX C_1 V_1 = C_2 V_2 on the clipboard.

Worked examples

All four examples were recomputed by the engine's chem-dilution compute routine; the display convention is at most 8 decimal places with trailing zeros removed, with $C_1$ and $C_2$ in one unit and $V_1$ and $V_2$ in one unit.

Example 1: making 1 mol/L from 12 mol/L concentrated hydrochloric acid

Inputs: $C_1 = 12$ , $V_1 = 10$ , $C_2 = 1$ . $V_2 = 12 \times 10 / 1 = 120$ . Interface shows: final volume V₂ 120. The 10 mL of stock contains $12 \times 0.010 = 0.12$ mol HCl, and after making up to 120 mL it is still 0.12 mol, so $0.12 / 0.120 = 1$ mol/L. The water to add is $120 - 10 = 110$ mL, not 120 mL— $V_2$ is the total volume.

Checking in reverse: clear $C_2$ and enter $V_2 = 120$ : $C_2 = 12 \times 10 / 120 = 1$ . Interface shows: target concentration C₂ 1. The dilution factor is $C_1/C_2 = V_2/V_1 = 12$ .

Figure 1: Example 1. Both sides contain the same 24 dots of solute, merely spread further apart by more water—this is C₁V₁ = C₂V₂

Example 2: the reverse question when preparing a solution—how much stock to pipette

Solving for the volume taken: to prepare 250 mL of 0.5 mol/L hydrochloric acid from the same 12 mol/L stock, clear $V_1$ and enter $C_1 = 12$ , $C_2 = 0.5$ , $V_2 = 250$ : $V_1 = 0.5 \times 250 / 12 = 10.41666667$ . Interface shows: volume taken V₁ 10.41666667—a 10 mL pipette cannot deliver this, so in the lab you would use a 25 mL graduated pipette to take 10.4 mL, or choose a target that divides exactly into 12 mol/L.

Another case: prepare 500 mL of 0.1 mol/L from a 1 mol/L stock. Clear $V_1$ and enter $C_1 = 1$ , $C_2 = 0.1$ , $V_2 = 500$ : $V_1 = 50$ . Interface shows: 50—pipette 50 mL of stock into a 500 mL volumetric flask and make up to the mark. A 10-fold dilution.

Solving for the stock concentration: a sample of unknown concentration is diluted from 5 mL to 100 mL and measures 0.1 mol/L. Clear $C_1$ and enter $V_1 = 5$ , $C_2 = 0.1$ , $V_2 = 100$ : $C_1 = 0.1 \times 100 / 5 = 2$ . Interface shows: stock concentration C₁ 2. This is the most common use in analytical chemistry: the sample is too concentrated for the instrument, so it is diluted, measured, and multiplied back by the 20-fold dilution factor.

Example 3: serial dilution—tenfold, tenfold, tenfold again

Microbial counts and enzyme assays need the concentration lowered by several orders of magnitude, and pipetting 1 μL into 10 L in one step is impractical, so the dilution is done step by step. Take 1 mL of a 0.1 mol/L solution and add diluent to 10 mL: $C_1 = 0.1$ , $V_1 = 1$ , $V_2 = 10$ : $C_2 = 0.01$ . Interface shows: target concentration C₂ 0.01. Treat that 0.01 as the new $C_1$ and again take 1 mL up to 10 mL: $C_1 = 0.01$ , $V_1 = 1$ , $V_2 = 10$ : the interface shows 0.001. Each step is a 10-fold dilution, so after $n$ steps the total factor is $10^n$ ; three steps give 1000-fold.

In practice each step is “1 mL of sample + 9 mL of diluent”—the diluent added is $V_2 - V_1 = 9$ , not 10. The pipetting error of one step is propagated unchanged by every later step, so each step needs a calibrated pipette.

Example 4: percentages work too, and two things the formula cannot do

Mass fraction entered directly: a 5% chlorine bleach is to be made into a 0.5% sanitising solution, taking 100 mL. $C_1 = 5$ , $V_1 = 100$ , $C_2 = 0.5$ : $V_2 = 1000$ . Interface shows: final volume V₂ 1000—100 mL of bleach made up with water to 1 L, a 10-fold dilution. The unit is %, and the formula does not care.

Making 500 mL of 70% disinfecting alcohol from 95% alcohol: clear $V_1$ and enter $C_1 = 95$ , $C_2 = 70$ , $V_2 = 500$ : $V_1 = 70 \times 500 / 95 = 368.42105263$ . Interface shows: volume taken V₁ 368.42105263—measure 368 mL of alcohol and make up to 500 mL with water. You cannot replace this with “add 132 mL of water”: ethanol and water contract on mixing, so 368 + 132 gives less than 500 mL and a concentration slightly above 70%.

Cannot concentrate: $C_1 = 1$ , $V_1 = 10$ , $C_2 = 2$ : the engine returns $V_2 = 5$ . Interface shows 5—smaller than the volume taken. The formula holds algebraically, but physically it would require “removing 5 mL of water”, which is evaporation, not dilution. When the target concentration is above the stock concentration the result is meaningless.

Cannot be used for pH: diluting hydrochloric acid at pH 1 tenfold gives pH 2, not “pH 10”. pH is the negative logarithm of $[\text{H}^+]$ , and $C_1 V_1 = C_2 V_2$ only holds for concentrations proportional to the amount of solute; convert pH back to $[\text{H}^+] = 10^{-\text{pH}}$ first, dilute, and then take the logarithm again.

The inputs of Example 1 can be entered directly into the panel to reproduce it; change the target concentration to 0.5, 0.1 and 0.01 and watch the final volume grow in inverse proportion to 240, 1200 and 12000 mL; change the stock concentration to 37 (mass fraction) and the target to 10 to see that percentages work just as well.

Principle and derivation

From “solute unchanged” to $C_1V_1 = C_2V_2$

Concentration is defined as $c = n/V$ (molarity), so the amount of solute in a solution is $n = cV$ . Dilution only adds solvent: not one molecule of solute is added or removed. Before dilution $n = C_1 V_1$ , after dilution $n = C_2 V_2$ , and the two are equal—that is the whole derivation. The 24 dots on each side of Figure 1 are a picture of that sentence.

Because both sides are the same $n$ , the equation holds for any concentration of the form “amount = concentration × volume”: mass concentration (g/L × L = g), mass fraction (approximately, when the solution density changes little), ppm, activity units/mL, colony counts/mL. That is also why the panel is labelled in mol/L and mL yet accepts % and L.

Dilution factor and inverse proportionality

Rewriting the equation as $C_2 / C_1 = V_1 / V_2$ : the factor by which the concentration falls equals the factor by which the volume rises. With $n$ fixed, $C$ and $V$ are inversely proportional. “Diluted 12-fold”, “1 : 11 dilution” and “12× stock” all say $V_2 / V_1 = 12$ —note that “1 : 11” means 1 part stock plus 11 parts solvent, a total of 12 parts, which differs from “1 : 12” (13 parts in total); both notations appear in laboratory records, so confirm which one is meant before use.

Why “make up to volume” rather than “add this much water”

$V_2$ is the total volume of the diluted solution. The most reliable procedure uses a volumetric flask: add the stock, then add solvent to the graduation mark, and mix. That way $V_2$ is the nominal capacity of the flask, regardless of whether volumes are additive. “Add $V_2 - V_1$ of water” is equivalent only when volumes are additive—dilute aqueous solutions approximately are, but ethanol and water, and concentrated acids and water, are not (the alcohol in Example 4). Volumetric flasks are calibrated at 20 °C (ISO 1042); a class A 250 mL flask has a tolerance of ±0.15 mL, and a temperature deviation from 20 °C produces a volume drift of about 0.2% per 10 °C.

The mathematics of serial dilution

If $n$ steps dilute by factors $d_i$ , the total factor is the product $\prod d_i$ ; when every step is 10-fold it is $10^n$ . Errors propagate multiplicatively too: a 1% relative error per step becomes about 3% after three steps. That is why microbial counting often runs two dilution series in parallel as a cross-check. Serial dilution was established in bacteriology by the solid-medium counting methods of Koch's era (the 1880s) and remains the standard preparation for colony counts today.

A little history

Volumetric analysis—measuring one solution by volume against another of known concentration—was systematically established by Gay-Lussac around 1824, and he also named the “pipette” and the “burette”. Only once accurate pipettes and volumetric flasks existed did “take $V_1$ and make up to $V_2$ ” become a reproducible operation, and only then did $C_1 V_1 = C_2 V_2$ turn from an algebraic identity into an everyday laboratory tool.

Assumptions

Solute is conserved: during dilution the solute does not evaporate, react, precipitate or adsorb onto the container.
$V_2$ is the total volume after making up to the mark: not the volume of solvent added.
Concentration is proportional to the amount of solute: mol/L, g/L, %, ppm are fine; logarithmic measures such as pH and pKa are not.
Like units are consistent: $C_1$ and $C_2$ in one unit, $V_1$ and $V_2$ in one unit; the mol/L and mL shown on the panel are only labels.
Temperature is fixed: volumes refer to the same temperature basis.
Interface conventions: when both $C_2$ and $V_2$ are empty they display 120 (the denominator falls back to 1); $C_2 = 0$ displays ∞; when all four are filled in the engine neither calculates nor validates.

Scope and limitations

Can calculate: any one of the four quantities from the other three; any positive numbers; any linear concentration unit; serial dilution by entering each step in turn.
Approximate only: replacing make-up-to-volume by “add $V_2 - V_1$ of solvent” gives a small error in dilute aqueous solutions but is not valid for organic solvents or concentrated acids and bases; entering mass fractions implicitly assumes the solution density changes negligibly.
Cannot calculate: the amount of solvent to add (hand-calculate $V_2 - V_1$ ); the result of mixing two solutions of different concentrations (that is a weighted average, $C = (C_1 V_1 + C_2 V_2)/(V_1 + V_2)$ ); concentration (a $C_2 > C_1$ result is meaningless); dilution of pH directly (convert back to $[\text{H}^+]$ first); solute consumed by a reaction.
Does not: convert units; make safety judgements.
Numerics: 8 decimal places, trailing zeros removed.

Common pitfalls

Treating $V_2$ as the water to add: Example 1 needs 110 mL, not 120 (Figure 1).
Using different units on the two sides: $V_1$ in mL with $V_2$ in L is off by a factor of 1000; $C_1$ in mol/L with $C_2$ in % is meaningless. Like quantities just need the same unit.
Assuming the panel only accepts mol/L: % , g/L and ppm all work because the formula is a proportion (Example 4).
Using the formula to “concentrate”: $C_2 > C_1$ gives $V_2 < V_1$ , which has no physical meaning (Example 4).
Applying $C_1V_1 = C_2V_2$ to pH: pH is logarithmic; a 10-fold dilution changes it by 1 unit, not by a factor of 10.
Adding organic solvent volumes arithmetically: ethanol and water contract, so make up to the mark rather than adding a fixed amount of water (Example 4).
Mistaking the 120 shown on opening the panel for a result: it comes from the fallback denominator of 1 and only becomes valid once $C_2$ or $V_2$ is entered.
The two readings of “1 : 10”: 1 + 9 or 1 + 10—confirm which is meant before calculating.

Typical use cases

Teaching

The derivation from conservation of solute to $C_1V_1 = C_2V_2$ ; dilution factors and inverse proportionality; making up to volume and non-additive volumes; why pH cannot simply be diluted.

Laboratory solution preparation

Preparing working solutions from commercial concentrated acids and bases or from stock solutions: solving for the volume to take; back-calculating a stock concentration from a pre-diluted sample; preparing 1× buffer from a 10× stock.

Biology and microbiology

Tenfold serial dilution followed by colony counting; gradient dilutions of enzymes, antibodies and DNA; adjusting a cell suspension to a target density (counts/mL work the same way).

Everyday use and disinfection

Diluting bleach, disinfectant, pesticide and cleaning products by the percentages on the label; making 70%–75% alcohol from 95% alcohol.

FAQ

Why do both the target concentration and the final volume show 120 when the panel opens?

When both unknowns are empty, the engine falls back to 1 for the other unknown, and $12 \times 10 / 1 = 120$ . Fill in either $C_2$ or $V_2$ and the other becomes a real result.

How much water do I add?

$V_2 - V_1$ . In Example 1 that is $120 - 10 = 110$ mL. The panel gives only the total volume.

Must the concentration be in mol/L and the volume in mL?

No. $C_1$ and $C_2$ just need one unit and $V_1$ and $V_2$ one unit; %, g/L, ppm and L all work.

What if the target concentration is higher than the stock?

Dilution cannot achieve it, and the $V_2 < V_1$ the formula returns is meaningless. Concentrating requires evaporating solvent or making up a fresh solution.

An acid at pH 2 is diluted 100-fold—what is the pH?

About pH 4. $[\text{H}^+]$ goes from $10^{-2}$ to $10^{-4}$ . The pH value must not be substituted into $C_1V_1 = C_2V_2$ .

Why the emphasis on “making up to volume”?

Volumes are not necessarily additive: ethanol and water contract, and concentrated acid and water release heat and expand. Adding solvent to the mark in a volumetric flask is what makes $V_2$ accurate.

How do I calculate mixing two solutions of different concentrations?

That is not dilution but mixing: $C = (C_1V_1 + C_2V_2)/(V_1 + V_2)$ , by hand.

References and further reading

The chem-dilution definition in the CalcX engine source src/data/formulasScience.ts (compute(known, solveFor) solves for V₂ / C₂ / V₁ / C₁, falling back to a denominator of 1 when empty); every figure on this page was recomputed by that engine.
Wikipedia, Dilution (equation) （访问日期：2026-09-09）—derivation of $C_1V_1 = C_2V_2$ , the dilution factor and applicability.
Wikipedia, Serial dilution （访问日期：2026-09-09）—serial factors, error propagation and use in microbial counting.
Wikipedia, Volumetric flask （访问日期：2026-09-09）—making up to volume, class A/B tolerances and 20 °C calibration.
Wikipedia, Titration （访问日期：2026-09-09）—Gay-Lussac's establishment of volumetric analysis in 1824 and the naming of the pipette and burette.
Wikipedia, Sulfuric acid （访问日期：2026-09-09）—the heat of dilution of concentrated acids and the basis of “acid into water”.
ISO 1042:1998, Laboratory glassware — One-mark volumetric flasks—capacity tolerances and the 20 °C reference temperature for one-mark volumetric flasks.

Privacy

All inputs and calculations run inside your browser and are never uploaded to a server. The text on this page is static content; it neither contains nor records any user input.

Sources & review

Reviewed by: CalcX 编辑组
Updated: 2026-09-09

Wikipedia: Dilution (equation) (accessed 2026-09-09)
Wikipedia: Serial dilution (accessed 2026-09-09)
Wikipedia: Volumetric flask (accessed 2026-09-09)
Wikipedia: Titration — History (Gay-Lussac's volumetric analysis, 1824) (accessed 2026-09-09)
Wikipedia: Sulfuric acid — Safety (heat of dilution; “acid into water”) (accessed 2026-09-09)
ISO 1042:1998 Laboratory glassware — One-mark volumetric flasks (capacity tolerances and the 20 °C calibration temperature)

Related calculators

Open Dilution Calculator C₁V₁ = C₂V₂: Stock and Final Volumes in CalcX