Toolarithm

Science Calculator

Dilution Calculator

Concentration. Volume. Checked.

Set up a dilution

Enter three known values.

View results

Enter nonnegative decimal or scientific values with up to twenty significant digits. Final volume must be positive. The selected unknown is calculated; its disabled editing buffer is ignored.

Report and serial options

Display rounding does not add measurement certainty or change the underlying arithmetic.

Worked examples

Stock aliquot

V1=1.00000 mLV_1=1.00000\,\mathrm{mL}

Conserved-solute dilution with a unique solution.

Conserved-solute model: same solute before and after, no reaction or solute loss. Concentrations refer to final solution volume. Display uses 6 significant figures; this is reporting precision, not measurement certainty.

Concentration and volume record

Stock concentration

C1=1.00000 mol L−1C_1=1.00000\,\mathrm{mol\,L^{-1}}

Final concentration

C2=10.0000 mmol L−1C_2=10.0000\,\mathrm{mmol\,L^{-1}}

Stock aliquot

V1=1.00000 mLV_1=1.00000\,\mathrm{mL}

Final solution volume

V2=100.000 mLV_2=100.000\,\mathrm{mL}

Interpret the final-volume target

Use the reported stock aliquot and make up to the stated final solution volume. The nominal difference below is a volume bookkeeping estimate. Mixing can contract or expand volumes, so adding exactly this much solvent is not guaranteed to achieve the final volume.

Nominal volume difference

ΔV=V2−V1=99.0000 mL\Delta V=V_2-V_1=99.0000\,\mathrm{mL}

Volume dilution factor

DV=V2/V1=100.000D_V=V_2/V_1=100.000

Concentration ratio

DC=C1/C2=100.000D_C=C_1/C_2=100.000

Stock portion of final volume

100%⋅V1/V2=1.00000 %100\%\cdot V_1/V_2=1.00000\,\%

Conserved solute in the final solution

n=0.00100000 moln=0.00100000\,\mathrm{mol}

For positive concentrations and a positive aliquot, the two dilution factors agree: DV=DCD_V=D_C. When both concentrations are zero, the concentration ratio is undefined even if the volume ratio exists. A zero aliquot has no finite volume dilution factor.

Step-by-step conserved-solute working

Convert concentration bases and volumes before rearranging. The record below uses liters and the common concentration basis; results above use your chosen report units. All arithmetic uses the exact input decimals before display rounding.

  1. Step 1

    C1V1=C2V2C_1V_1=C_2V_2

  2. Step 2

    V1=C2V2C1V_1=\frac{C_2V_2}{C_1}

  3. Step 3

    V1=(1100 mol L−1)(110 L)(1 mol L−1)V_1=\frac{\left(\frac{1}{100}\,\mathrm{mol\,L^{-1}}\right)\left(\frac{1}{10}\,\mathrm L\right)}{\left(1\,\mathrm{mol\,L^{-1}}\right)}

    Long expression: complete values remain in the on-screen mathematics, copied report and CSV.

  4. Step 4

    C1=1.00000 mol L−1C_1=1.00000\,\mathrm{mol\,L^{-1}}

  5. Step 5

    C2=0.0100000 mol L−1C_2=0.0100000\,\mathrm{mol\,L^{-1}}

  6. Step 6

    V1=0.00100000 LV_1=0.00100000\,\mathrm L

  7. Step 7

    V2=0.100000 LV_2=0.100000\,\mathrm L

  8. Step 8

    C1V1=C2V2=11000 molC_1V_1=C_2V_2=\frac{1}{1000}\,\mathrm{mol}
What Can You Create?

Build a concentration and final-volume record

Solve any one dilution quantity

Find the stock aliquot, final concentration, final solution volume or original stock concentration from three known quantities. Independent concentration and volume selectors make conversions explicit. The selected unknown is excluded from input validation so an old editing buffer cannot quietly determine the next answer.

Compare a separate serial plan

Choose a constant factor and final volume for each stage, then inspect how concentration decreases and cumulative dilution grows. Transfer volumes and intermediate retained volumes appear beside each stage. The plan begins from the original stock and explicitly compares its last concentration with the main final target.

Keep complete numerical evidence

Preserve active unit choices, model status, adopted molar mass when needed and every serial row in complete reports. CSV includes exact arithmetic certificates for the entered decimals. A selected screen page and a complete exported plan serve different purposes, with print scope clearly stated.

Dilution formulas and step-by-step examples

Concentration is a solute quantity per final solution volume. For the same conserved solute, C1V1=C2V2C_1V_1=C_2V_2. Subscripts distinguish the stock aliquot from the final solution. Molar concentrations preserve amount of substance; mass concentrations preserve solute mass. Convert their bases before substituting values, and keep the selected unknown out of the three known inputs.

Find a stock aliquot

Choose this mode when you know the available stock strength, desired final strength and total final solution volume. A stronger target cannot be obtained by adding solvent. The stock-volume result is an aliquot of the original solution, not the amount of pure solute.

V1=C2V2C1V_1=\frac{C_2V_2}{C_1}

Find final concentration

Use this mode to check the concentration obtained from an existing aliquot and a measured final volume. Both volumes enter the same common volume basis. The concentration output uses its own selected unit, so a molar stock may be reported as millimolar without changing the conserved amount.

C2=C1V1V2C_2=\frac{C_1V_1}{V_2}

Find final solution volume

A positive target and a known stock aliquot determine a final volume. A zero target with positive conserved solute has no finite solution. If the target equals the original stock concentration, the final volume equals the aliquot and the nominal increase is zero.

V2=C1V1C2V_2=\frac{C_1V_1}{C_2}

Recover original stock concentration

This rearrangement checks an unknown stock concentration from a known dilution record. A zero aliquot cannot identify a unique stock concentration when the target amount is also zero. An inconsistent positive final amount with a zero aliquot is impossible under the model.

C1=C2V2V1C_1=\frac{C_2V_2}{V_1}

A hundredfold dilution with mixed concentration units

Start with C1=1 mol L−1C_1=1\,\mathrm{mol\,L^{-1}}, request C2=10 mmol L−1C_2=10\,\mathrm{mmol\,L^{-1}}, and choose V2=100 mLV_2=100\,\mathrm{mL}. First convert the target to 0.01 mol L−10.01\,\mathrm{mol\,L^{-1}} and the final volume to 0.1 L0.1\,\mathrm L. Substitute only after those conversions, then report the aliquot in the desired volume unit.

V1=(0.01 mol L−1)(0.1 L)1 mol L−1=0.001 L=1 mLV_1=\frac{(0.01\,\mathrm{mol\,L^{-1}})(0.1\,\mathrm L)}{1\,\mathrm{mol\,L^{-1}}}=0.001\,\mathrm L=1\,\mathrm{mL}ΔVnominal=100 mL−1 mL=99 mL,D=100 mL1 mL=100\Delta V_{\mathrm{nominal}}=100\,\mathrm{mL}-1\,\mathrm{mL}=99\,\mathrm{mL},\qquad D=\frac{100\,\mathrm{mL}}{1\,\mathrm{mL}}=100

Make the stock aliquot up to the final solution volume. The nominal solvent difference is an estimate based on the volume record, not a guarantee that mixing component volumes is additive. A factor of one hundred refers to stock versus final solution; it does not mean adding one hundred aliquot volumes of solvent.

Check concentration bases before converting

1 mg mL−1=1 g L−1,1% (w/v)=10 g L−11\,\mathrm{mg\,mL^{-1}}=1\,\mathrm{g\,L^{-1}},\qquad 1\%\,\mathrm{(w/v)}=10\,\mathrm{g\,L^{-1}}

A user-adopted molar mass relates mass concentration ρs\rho_s to amount concentration csc_s through cs=ρs/Msc_s=\rho_s/M_s. For an adopted Ms=100 g mol−1M_s=100\,\mathrm{g\,mol^{-1}}, a mass concentration of 10 g L−110\,\mathrm{g\,L^{-1}} corresponds to 0.1 mol L−10.1\,\mathrm{mol\,L^{-1}}. This arithmetic does not identify the solute or establish sample purity. Volume percentages use a separate ideal model, and mass percent needs additional information rather than a silent percentage conversion.

Follow cumulative dilution and retained volume

Ci=C1Fi,Vtransfer=VstageF,F=10, i=3 ⟹ Fi=1000C_i=\frac{C_1}{F^i},\qquad V_{\mathrm{transfer}}=\frac{V_{\mathrm{stage}}}{F},\qquad F=10,\ i=3\ \Longrightarrow\ F^i=1000

With a 1 mL1\,\mathrm{mL} final volume and a tenfold stage factor, transfer 0.1 mL0.1\,\mathrm{mL} into each next stage and make up to its selected final volume. Intermediate stages retain 0.9 mL0.9\,\mathrm{mL} after supplying the following transfer. The final stage retains 1 mL1\,\mathrm{mL} because the plan has no later withdrawal. A serial factor of one leaves concentrations unchanged and leaves no retained liquid in an intermediate stage when the full volume supplies the next one.

Why Users Love This Tool

Check the model before trusting the number

Explicit physical and algebraic conditions

  • Negative input is rejected rather than clamped to zero. Final volume must be positive. Impossible, underdetermined, unchanged, blank and ordinary dilution cases receive distinct explanations.
  • A stronger final concentration or a smaller final volume is identified as concentration rather than solvent dilution. The algebraic diagnostic stays visible without a misleading solvent instruction.
  • Zero denominators produce an undefined dilution ratio rather than a fabricated zero or infinity. A positive volume ratio and an undefined zero-over-zero concentration ratio can coexist.

Unit-aware, repeatable reporting

  • Mass and molar units share a common basis only through an explicit adopted molar mass. Ideal volume percentages retain their own assumption and cannot silently become mass concentrations.
  • Rational arithmetic preserves the entered decimals, metric scaling and serial-factor powers before display rounding. Small nonzero values remain visible in scientific notation rather than underflowing to zero.
  • Share links retain active choices while omitting unknown and disabled numerical buffers. Complete text and CSV include every serial stage even when the visible report is paged.
Perfect For

For concentration learning and reviewable calculations

Chemistry students

Practice rearranging a conserved-solute equation, canceling units and distinguishing an aliquot from a final solution volume. Compare the four solve modes using the same numerical record, then explain why a mathematically underdetermined equation needs more information.

Educators and laboratory learners

Prepare transparent concentration examples with the unit basis and final-volume assumption attached. Use a serial plan to explain cumulative factors and retained volume. The result is an educational numerical model; equipment suitability and physical preparation conditions remain separate matters.

Readers checking existing records

Review whether a stated stock, aliquot and final-volume record implies the claimed target. Preserve a positive user-adopted molar mass when comparing mass and molar concentration. Retain the complete arithmetic record so another reader can reproduce the unit conversion.

How It Works

How it works in three quick steps.

1

Choose the unknown and define concentration bases

Select stock aliquot, final concentration, final volume or stock concentration. Enter the three known quantities using explicit units. Both concentrations must describe the same solute on compatible bases. A mass-to-molar conversion needs a positive adopted molar mass, while ideal volume percentages cannot be converted into mass or molar concentration without additional physical information. Check the original labels rather than guessing what an unspecified percentage means.

2

Inspect the answer and the final-volume condition

Read the result status before using the numerical answer. A feasible dilution gives a stock aliquot, positive final solution volume and nominal volume difference. Make the aliquot up to the final volume; do not assume mixing volumes are exactly additive. Zero inputs may give a blank, an impossible target or an underdetermined equation. A target stronger than the stock requires concentration and does not receive dilution preparation instructions.

3

Review serial stages and preserve complete working

Optionally define a separate serial plan starting from the original stock, its factor per stage and its final volume per stage. Review stage concentrations, cumulative factors, transfers and retained volumes. Copy complete results or LaTeX, download every stage and exact arithmetic certificate as CSV, share the active model, or print the selected report view. Keep the concentration basis and any user-adopted molar mass with the answer.

Download & Print

Save the concentration basis with the result

Copy complete text or real LaTeX

Copy summary includes the model status, all four quantities and every serial stage, independent of the screen page. Copy LaTeX preserves authored conserved-solute working and stage concentration expressions for a compatible editor. Invalid numeric inputs remove stale results before any report action is available.

Download every serial stage as CSV

The complete report retains exact numerator and denominator certificates for each displayed quantity in its stated unit. Serial rows record concentration, cumulative factor, transfer, nominal volume difference and volume retained after outgoing transfer. Adopted molar mass is recorded when a cross-basis concentration conversion uses it.

Print the selected calculation view

Use Print / Save PDF for the current valid numerical report. The print layout removes editing controls, navigation and supporting guide text. Serial printing follows the visible stage page and states its scope. Long expressions receive a print notice; full values remain in the on-screen mathematics and complete exports.

FAQ

Frequently Asked Questions

What does the dilution equation assume?
It tracks one conserved solute while the final solution volume changes. The stock and final concentrations must describe that same solute, expressed on a consistent basis. Reactions, degradation, adsorption, precipitation and material loss are outside the model. A valid equation is therefore a numerical concentration check rather than a guarantee about a real preparation, its composition, stability or physical measurement uncertainty.
Does nominal solvent difference mean I should add exactly that much solvent?
No. The result is the final solution volume minus the stock aliquot volume in the selected report unit. This is useful bookkeeping, but real mixtures may contract or expand. Interpret the target as making the aliquot up to the stated final solution volume under the relevant preparation conditions. The tool intentionally labels this value nominal and does not certify that two measured component volumes will add exactly.
Can I mix molar concentration and milligrams per milliliter?
Yes, when both describe the same solute and you supply a positive adopted molar mass in grams per mole. The calculator converts the mass concentration into an amount concentration before solving. It records the adopted value rather than choosing a chemical identity or guessing purity. If both concentration units are mass-based, no molar mass is needed; metric scaling alone relates their numerical values.
Which percentage concentrations are supported?
Mass per final solution volume is supported as grams of solute per one hundred milliliters of solution. Volume percent and volume fraction are separate options using an explicit ideal additive-volume assumption. They cannot be mixed automatically with mass or molar units. Mass percent, an unspecified percentage and parts per million are intentionally excluded because their basis or density assumptions are not supplied by a bare numerical input.
Why can zero concentration produce no unique answer?
A zero coefficient changes the algebra. If both sides reduce to zero, the equation alone may leave a range of possible aliquots, stock concentrations or final volumes. If a positive conserved amount would have to equal zero, there is no finite solution. A zero final target from a positive stock can instead require a zero aliquot when final volume is fixed. These cases must not all be replaced with zero.
Is a hundredfold dilution the same as adding one hundred parts solvent?
A hundredfold dilution means the stock aliquot is one hundredth of the final solution volume when both concentrations and the aliquot are positive. The nominal difference is consequently ninety-nine aliquot volumes, subject to the final-volume condition. Ratio notation can be ambiguous about whether it refers to stock versus solvent or stock versus final solution. This calculator uses a named volume dilution factor to remove that ambiguity.
How does the serial dilution plan handle transfers and remaining volume?
The serial plan starts independently from the original stock concentration. Each stage uses the specified final volume and factor, with its transfer taken from the stock or the preceding stage. Intermediate retained volumes exclude the next outgoing transfer; the last stage keeps its full final volume because no following transfer is modeled. The cumulative factor multiplies across stages, and the last concentration is compared with the main target.
Does selecting more significant figures improve preparation accuracy?
It changes the display, not the certainty of the inputs, glassware or adopted molar mass. Exact rational arithmetic preserves the stated decimal inputs and unit conversions before a single reporting-rounding step. It does not model calibration, temperature, mixing, purity or measurement uncertainty. Complete CSV exports retain exact numerator and denominator certificates for reproducing the numerical model, while printed serial reports show only the currently selected stages.
About This Tool

About this Dilution Calculator

The Dilution Calculator is a four-direction concentration tool built around a conserved-solute record. Instead of assuming that every problem asks for stock volume, it lets the reader choose one unknown and enter three known quantities. Each concentration and volume has its own unit choice, and the report identifies a common basis before presenting the answer. A numerical result therefore comes with an explanation of what was conserved and which final solution volume gives the reported concentration.

The distinction between a solvable equation and a feasible dilution is central to the design. A target stronger than the stock cannot be reached by adding solvent, while a zero target can describe a blank, an unreachable finite-volume target or an underdetermined record depending on the unknown. These cases remain separate. Nominal volume difference is labeled as bookkeeping rather than an exact physical instruction, because real mixing volumes can differ from their simple sum. Mass-to-molar conversion requires an explicitly adopted molar mass for the same solute and retains that value in complete reporting.

A separate serial planner extends the concentration explanation across a bounded sequence of stages. It keeps cumulative factor, transfer and retained-volume meanings distinct and compares the last stage with the main target. Exact rational arithmetic avoids accumulating display rounding through the series; it does not create measurement certainty or infer equipment accuracy. Accessible mathematical rendering, readable cards on narrow screens, active-settings links and complete exports help another reader inspect the calculation. The tool supports learning and numerical review rather than identifying a material, selecting a biological procedure or certifying a physical preparation.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

The method identifies the governing relationship, units, significant assumptions, and educational scope. SI terminology is checked against the BIPM reference.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.

Keep building

Explore more calculators

Calculators hub