Dilution calculator

What is a dilution calculator?

A dilution calculator answers the single question that comes up every time somebody works with a concentrated stock solution: how much of the stock do I take, and how much solvent do I add, to end up with a weaker solution of exactly the concentration and volume I need?

Dilution is the process of lowering the concentration of a solute by adding more solvent. The crucial point is that adding solvent does not change how much solute is present — it only spreads the same amount of solute through a larger volume. That single conservation idea is the whole basis of the calculation.

This tool works in both directions. Pick which of the four quantities is unknown, type in the other three, and the unknown one is returned together with the dilution factor and the volume of solvent you have to add.

The dilution equation

Because the amount of solute is conserved, the number of moles in the aliquot you take from the stock equals the number of moles in the finished solution:

$$C_1 V_1 = C_2 V_2$$

where:

• $C_1$ is the concentration of the stock (starting) solution.
• $V_1$ is the volume taken from that stock.
• $C_2$ is the concentration of the final (diluted) solution.
• $V_2$ is the total volume of the final solution.

The relationship is often attributed to the routine bench practice of nineteenth-century analytical chemistry, but it needs no special theory: it is simply the statement that moles in equals moles out. It holds for molarity, for percentage strength, for parts per million and for any other concentration unit, provided you use the same unit on both sides of the equation.

How does the dilution calculator work?

1. Choose the quantity you want to find in the Calculate selector. The matching input field disappears, because that value is the answer rather than something you supply.
2. Enter the three remaining values. Each concentration field and each volume field has its own unit dropdown, so you can mix units freely — a stock in M with a target in mM and a final volume in mL is perfectly fine.
3. Read off the calculated quantity, the dilution factor and the volume of solvent to add.

Internally every value is converted to a common base unit — molar (M) for concentration and litres (L) for volume — before the equation is applied, and the answer is converted back into whatever unit you selected. That is why mixed units never distort the result.

Formulas

Rearranging the dilution equation for each unknown gives four working formulas.

Stock volume to take:

$$V_1 = \frac{C_2 \times V_2}{C_1}$$

Stock concentration:

$$C_1 = \frac{C_2 \times V_2}{V_1}$$

Final volume:

$$V_2 = \frac{C_1 \times V_1}{C_2}$$

Final concentration:

$$C_2 = \frac{C_1 \times V_1}{V_2}$$

The dilution factor $DF$ tells you how many times the solution has been weakened:

$$DF = \frac{V_2}{V_1} = \frac{C_1}{C_2}$$

It is quoted as a ratio, so a dilution factor of 4 is written 1 : 4 and means one part stock made up to four parts total. Finally, the amount of solvent (diluent) you actually have to add to the aliquot is:

$$V_{solvent} = V_2 - V_1$$

Note that this is the volume added to the aliquot, not the total final volume. In practice you place $V_1$ into the flask first and then bring the contents up to the $V_2$ mark.

Worked examples

Example 1: how much stock do I need?

You have a 2 M stock and need 1 L of a 0.5 M working solution. Solve for the stock volume:

$$V_1 = \frac{0.5 \times 1}{2} = 0.25$$

Take 0.25 L of the stock. The dilution factor is 1 : 4 and the solvent to add is 1 − 0.25 = 0.75 L.

Example 2: how strong was my stock?

You know that 0.25 L of a stock made 1 L of a 0.5 M solution, but the stock bottle has lost its label. Solve for the stock concentration:

$$C_1 = \frac{0.5 \times 1}{0.25} = 2$$

The stock was 2 M.

Example 3: how far can this aliquot be diluted?

You have already measured out 0.25 L of a 2 M stock and want a 0.5 M solution. Solve for the final volume:

$$V_2 = \frac{2 \times 0.25}{0.5} = 1$$

Make the aliquot up to 1 L in total.

Example 4: what concentration will I end up with?

You add 0.25 L of a 2 M stock to a flask and fill it to the 1 L mark. Solve for the final concentration:

$$C_2 = \frac{2 \times 0.25}{1} = 0.5$$

The result is a 0.5 M solution.

Example 5: mixed units

Units do not have to match. Suppose the stock is 2 M, the target is 50 mM and you want 500 mL of it. Converting to the base units gives 0.05 M and 0.5 L, so:

$$V_1 = \frac{0.05 \times 0.5}{2} = 0.0125$$

That is 0.0125 L, or 12.5 mL of stock, made up to 500 mL — a 1 : 40 dilution.

Example 6: a 1 : 20 dilution

To make 0.1 L of a 0.05 M solution from a 1 M stock:

$$V_1 = \frac{0.05 \times 0.1}{1} = 0.005$$

Take 0.005 L (5 mL) of stock and add 0.1 − 0.005 = 0.095 L of solvent. The dilution factor is 1 : 20.

Practical notes

• Add concentrate to solvent, not the other way round. With strong acids this is a safety rule, not a preference: pouring water into concentrated sulfuric acid can boil and spit. Always add the acid to the water.
• Bring to volume, do not add volume. A volumetric flask is calibrated to a total volume. Adding 0.75 L of water to a 0.25 L aliquot is only approximately the same as making 0.25 L up to the 1 L mark, because volumes of mixtures are not perfectly additive.
• Very large dilution factors are better done in steps. Pipetting 5 µL accurately is harder than doing two successive 1 : 100 dilutions. Serial dilutions multiply: two 1 : 100 steps give a 1 : 10 000 overall factor.
• The equation is unit-agnostic but not unit-forgiving. It works for % w/v, ppm, mg/mL and molarity alike, as long as $C_1$ and $C_2$ share a unit and $V_1$ and $V_2$ share a unit. The calculator handles that conversion for you.
• Temperature matters for exact work. Volumetric glassware is usually calibrated at 20 °C; solutions prepared warm will be slightly off once they cool.
• If you need the concentration itself rather than a dilution, start from the Molarity calculator; for acidity of the diluted solution see the pH calculator.

Frequently asked questions

What does C1V1 = C2V2 mean?

It states that the amount of solute taken from the stock equals the amount of solute in the final solution. Concentration multiplied by volume gives an amount, and adding solvent does not create or destroy solute.

Can I use it with percentages or ppm instead of molarity?

Yes. The equation only requires that both concentrations are expressed in the same unit. A 10 % stock diluted to a 2 % solution behaves exactly like a 10 M stock diluted to 2 M.

What is a dilution factor?

The dilution factor is the ratio of the final volume to the stock volume, equivalently the ratio of the starting concentration to the final concentration. A factor of 20 is written 1 : 20 and means the solution is twenty times weaker than the stock.

Why can’t I dilute to a concentration higher than my stock?

Adding solvent can only lower concentration. If you request a final concentration above the stock concentration, the equation returns a stock volume greater than the final volume, which is physically impossible — you would need to remove solvent, that is, to concentrate rather than dilute.

Why does the calculator return nothing sometimes?

A blank result means the equation cannot be solved with the numbers given: either one of the required fields is empty, or the value in the denominator is zero. Dividing by a zero concentration or a zero volume has no meaning, so the field is left empty instead of showing an error value.

What is the difference between the final volume and the solvent to add?

The final volume $V_2$ is the total volume of the finished solution, including the stock aliquot. The solvent to add is only the extra liquid, $V_2 - V_1$. Confusing the two is the most common practical mistake in dilution work.