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500 ml of a 5 percent solution are to be made from a 30 percent solution. Sounds like a simple calculation. In practice, however, errors often arise not in the math but in measuring out, making up to volume, mixing, or interpreting the concentration specification.

Especially with acids and bases, there is another point to consider: a mathematically correct dilution can still be performed unsafely. Mixing can generate heat, and liquid can boil over or splatter.

This guide shows how to plan dilutions systematically, which formulas really help in everyday laboratory work, and where typical sources of error lie.

Table of Contents

What does diluting actually mean?

When diluting, the concentration of a substance is reduced by adding a suitable diluent.

In aqueous solutions, this is often:

• distilled water,
• demineralized water,
• Ultrapure water.

Depending on the application, however, other solvents can also be used.

The key point is:

The amount of dissolved substance remains the same during a pure dilution – only the total volume of the solution changes.

For this reason, a dilution can be calculated mathematically.

A simple example:

You have 100 ml of a concentrated salt solution. If you add water, the amount of salt it contains does not decrease. It is merely distributed over a larger volume.

The concentration decreases.

What concentration specifications exist?

Before a dilution is calculated, it must be clear which concentration specification is being used.

This is one of the most common sources of error.

Product labels or laboratory instructions contain, for example:

• Percentage values,
• mol/l,
• mmol/l,
• g/l,
• mg/l,
• ppm,
• Mass fractions,
• Volume fractions.

These specifications must not simply be equated with one another.

Mass percent

A value such as:

20 % m/m

means:

20 g of substance are contained in 100 g of solution.

The reference quantity is therefore the mass, not the volume.

Volume percent

A value such as:

70 % V/V

simply means:

70 ml of a liquid component per 100 ml of solution – provided the underlying definition and preparation method are specified accordingly.

This specification is often found, for example, in liquid mixtures.

Masse pro Volumen

A value such as:

5 % m/V

corresponds to:

5 g of substance per 100 ml of prepared solution.

Molar concentration

A value such as:

1 mol/l

means that one mole of the substance in question is present per liter of solution.

This is exactly why the statement “I have a 20% solution” is not always sufficient for a proper calculation. First, it must be clarified which type of percentage is meant.

The most important dilution formula: c₁ × V₁ = c₂ × V₂

For many laboratory dilutions, one formula is sufficient:

c₁ × V₁ = c₂ × V₂

Where:

• c₁ = concentration of the stock solution
• V₁ = required volume of the stock solution
• c₂ = desired final concentration
• V₂ = desired final volume

V₁ is often what is sought.

Then the rearranged formula is:

V₁ = (c₂ × V₂) / c₁

Example: diluting 30% to 5%

You want to prepare 500 ml of a 5% solution from a 30% stock solution.

Calculation:

V₁ = (5 × 500 ml) / 30

V₁ = 83.3 ml

So, by calculation, you need:

83.3 ml stock solution

Then it is made up to a final volume of 500 ml with the appropriate diluent.

This last sentence is important.

This does not automatically mean:

83.3 ml concentrate + 416.7 ml water.

In precise laboratory work, the stock solution is placed in a suitable measuring vessel and then made up to the final volume mark.

The dilution formula c₁V₁ = c₂V₂ is also used by Merck in its current dilution calculator for solutions.

Calculating the dilution factor correctly

A second important quantity is the dilution factor.

It is calculated as:

Dilution factor = final volume / initial volume

Example:

10 ml of sample is made up to a total volume of 100 ml.

Dilution factor:

100 ml / 10 ml = 10

The sample was therefore diluted by a factor of 10.

In a subsequent analysis, the measured value must be taken into account accordingly using the dilution factor.

Merck describes exactly this procedure for analytical dilutions and recommends a multi-step dilution if necessary for larger dilution factors.

Diluting percentage solutions correctly

For percent concentrations, the formula c₁V₁ = c₂V₂ works directly only if the initial and target concentrations are based on the same concentration definition.

Example: 20% to 4%

Desired:

250 ml of a 4% solution.

Available:

20% stock solution.

Calculation:

V₁ = (4 × 250 ml) / 20

V₁ = 50 ml

So you need:

50 ml stock solution

Then it is made up to:

250 ml final volume

made up to volume.

The dilution factor is:

250 / 50 = 5

This is therefore a fivefold dilution.

Caution with concentrated acids.

For concentrated acids, the concentration is often given as mass percent.

Hydrochloric acid at, for example, 37% does not automatically mean 37 mol/l, nor 37 ml of hydrogen chloride per 100 ml of solution.

Merck points out that the density, mass concentration, and molarity of a solution are linked to one another in a temperature-dependent manner.

If a molar concentration is to be calculated from a mass percentage, the density of the solution must also be known.

Molarity and amount-of-substance concentration

In analytical and chemical laboratories, the amount-of-substance concentration is commonly used.

An example:

1 mol/l NaOH

means that one liter of prepared solution contains an amount of substance of one mole of sodium hydroxide.

If a concentrated solution of known molarity is already available, the formula can again:

c₁ × V₁ = c₂ × V₂

be used.

Example: 2 mol/l to 0.5 mol/l

You need:

500 ml of a 0.5 M solution.

Stock solution:

2 mol/l.

Calculation:

V₁ = (0.5 mol/l × 500 ml) / 2 mol/l

V₁ = 125 ml

So you need:

125 ml of the 2 M stock solution

and then make up to:

500 ml final volume

up.

Understanding dilutions 1:2, 1:5, or 1:10 correctly

Ratios such as 1:2 or 1:10 regularly lead to misunderstandings.

Therefore, it should always be checked how the manufacturer or the work instruction defines the ratio.

In many laboratory applications, a 1:10 dilution means:

1 part sample to 10 parts total volume

So, for example:

1 ml sample + diluent up to a total volume of 10 ml.

This corresponds to:

• 1 ml sample
• 9 ml diluent

For a 1:10 dilution, Merck, for example, uses exactly this ratio and describes a dilution factor of 10.

Why do mix-ups occur?

Outside the lab, "1:10" is sometimes also interpreted as:

1 part concentrate + 10 parts water

Understood.

That gives 11 parts in total.

Mathematically, that is not the same.

Therefore, a work instruction should be worded as unambiguously as possible.

Better:

Make up 10 ml of concentrate to a total volume of 100 ml.

instead of only:

Dilute 1:10

Multi-step dilutions

Very high dilutions should often not be prepared in a single step.

Example:

You need a dilution by a factor of 1000.

In theory, you could:

Make up 0.1 ml of sample to 100 ml.

In practice, this can be problematic because a very small starting volume causes relatively large pipetting errors.

A better solution is a serial dilution.

For example:

Step 1

Make up 10 ml of the stock solution to 100 ml.

Dilution factor:

10

Step 2

Make up 10 ml of this solution to 100 ml once more.

Total factor:

10 × 10 = 100

Step 3

Make up 10 ml of this solution to 100 ml again.

Total factor:

10 × 10 × 10 = 1000

The dilution factors are multiplied together.

Merck also recommends a multi-step approach for analytical work with very high dilution factors in order to obtain more reliable results.

Why you shouldn't simply add volumes

A common mistake in thinking is:

I need 100 ml of solution.

So I mix:

20 ml concentrate + 80 ml water.

For simple technical applications, this approximation may be sufficient depending on the substance.

For precise laboratory work, however, it is problematic.

When mixing different liquids, volumes can change. The total volume therefore does not have to correspond exactly to the sum of the individual starting volumes.

Temperature changes during mixing also affect the volume.

This is why a volumetric flask is used for accurate solutions:

1. Add the calculated amount of the stock solution.
2. Add a portion of the diluent.
3. Allow to cool if necessary.
4. Only then fill exactly to the calibration mark.
5. Mix well.

Merck recommends volumetric flasks and suitable pipettes for reliable analytical dilutions and emphasizes that after filling to the mark, the solution must be mixed thoroughly.

Diluting acids and bases safely

For acids and bases, a correct calculation is not enough.

Dilution can be highly exothermic. This means:

Heat is released during mixing.

The more concentrated the chemical and the larger the amount used, the stronger the heating can be.

Possible consequences are:

• strong heating of the solution,
• Boiling,
• Splashes,
• Release of vapors or mists,
• Damage to unsuitable containers.

BAuA points out that when acids and bases are transferred or decanted, both droplets and vapors can be released, and suitable technical protective measures may be required.

Add concentrated acid to water – not the other way around.

For many classic acid dilution procedures, the well-known safety rule applies:

Add water first, then slowly add the acid.

The reason is heat generation.

If a small amount of water is added to a larger amount of concentrated acid, the upper water layer can heat up very quickly and evaporate suddenly. Acid can splash out as a result.

If, by contrast, the acid is added slowly to a larger quantity of water, the resulting heat can be distributed better.

Nevertheless, the following applies:

This rule never replaces the information in the safety data sheet of the specific product.

Before diluting, the following in particular must be checked:

• Section 2: Hazards,
• Section 7: Handling,
• Section 8: personal protective equipment,
• Section 10: Reactivity.

BAuA describes safety data sheets as a central source of information for users of chemicals.

Work slowly.

Concentrated acids and bases should be added in a controlled manner and in portions.

Do not:

• pour quickly,
• mix in a closed vessel,
• hold your face over the vessel.

For larger quantities, additional external cooling may be required.

Monitor the temperature.

After diluting, the solution should not be made up to the final volume immediately if it has warmed up significantly.

Why?

Warm liquids have a different volume than cooled ones.

For an accurate solution, therefore:

1. premixed,
2. allowed to cool,
3. only then made up exactly to the final volume.

Which laboratory instruments are suitable for accurate dilutions?

The accuracy of a dilution depends not only on the calculation.

At least as important is the measuring instrument used.

Volumetric flask

A volumetric flask is intended for preparing a defined final volume.

Typical sizes are:

• 50 ml,
• 100 ml,
• 250 ml,
• 500 ml,
• 1000 ml.

For precise dilutions, it is clearly superior to a beaker.

Volumetric pipette

A volumetric pipette is used for the accurate transfer of a fixed defined volume.

For example:

10.00 ml or 25.00 ml.

Graduated pipette

It allows different volumes, but depending on the design and application, it may be less accurate than a corresponding volumetric pipette.

Piston pipette

It is particularly suitable for smaller volumes.

What is crucial is that the desired volume is within the appropriate working range of the pipette.

Graduated cylinder

A graduated cylinder is suitable for many technical and approximate tasks.

For high-quality quantitative analysis, however, it is usually less accurate than volumetric flasks and volumetric pipettes.

Beaker

The scale of a beaker serves mainly for orientation.

A beaker is therefore not a suitable measuring instrument when the concentration must be set exactly.

Which water is suitable for diluting?

The diluent can also affect the result.

Depending on the application, the following may be suitable:

• distilled water,
• demineralized water,
• fully demineralized water,
• Ultrapure water.

For simple technical applications, demineralized water may suffice.

For sensitive analyses, however, even small amounts of:

• ions,
• organic compounds,
• metals,
• microorganisms

can be problematic.

For analytical dilutions, Merck, for example, recommends distilled or fully demineralized water.

The required water quality should therefore depend on the measurement method.

Typical errors when diluting

Not properly checking percentage specifications

20 % m/m and 20 % V/V are not the same.

Before any calculation, the concentration definition must be known.

confusing the final volume with the water volume

“Make up to 500 ml” does not mean:

Add 500 ml of water.

What is meant is:

add enough diluent until the total volume is 500 ml.

misinterpreting 1:10

1 part sample + 10 parts water does not yield the same dilution as 1 part sample made up to 10 parts total volume.

Work instructions should therefore be worded unambiguously.

Measuring exactly with a beaker

Beakers are suitable for mixing, but not for precisely adjusting a volume.

Pipetting very small volumes

For example, if 0.05 ml is needed, the relative measurement error can become large.

A serial dilution is often more accurate.

Not mixing after filling

If the volumetric flask is simply set aside, the concentration within the solution may initially be unevenly distributed.

After making up to volume, the solution should be thoroughly homogenized.

Filling the hot solution directly to the mark

After a strongly exothermic dilution, the volume can be inaccurate because of the increased temperature.

For precise work, the solution should be brought to the intended temperature before being made up to the final volume.

Mixing acid and water in the wrong order

With concentrated acids, an unfavorable order of addition can lead to severe splashing.

Always follow the safety data sheet and the specific work instruction.

Forgetting personal protective equipment

A mathematically simple dilution can be chemically dangerous.

Depending on the product, the following may be required:

• safety goggles,
• face shield,
• chemical-resistant gloves,
• Lab coat or protective clothing,
• Exhaust ventilation.

Take the concentration from the product name.

The same chemical can be sold in different concentrations.

Therefore, before calculating, the label, specification, and safety data sheet for the actual batch or product variant used must be checked.

Practical checklist before dilution

Before you begin, check:

• What starting concentration does the product have?
• Is it m/m, V/V, m/V, or mol/l?
• What final concentration is needed?
• What final volume should be prepared?
• Is c₁ × V₁ = c₂ × V₂ applicable to these concentration specifications?
• Does the density also need to be considered for a conversion?
• Which diluent is suitable?
• What water quality does the application require?
• Which measuring vessel provides the required accuracy?
• Is a single-step dilution appropriate?
• Would a serial dilution be more accurate?
• Is heat released during mixing?
• What order of addition does the safety data sheet specify?
• What personal protective equipment is required?
• Does the work need to be carried out under exhaust ventilation?
• Does the solution need to cool before being made up to the final volume?
• Was the finished solution mixed thoroughly?
• Does the dilution need to be documented or labeled?

A reliable dilution therefore consists of three steps:

calculate correctly, measure correctly, and mix correctly.

The formula is often the simplest part. In practice, the correct concentration specification, suitable volumetric measuring instruments, temperature, and working technique determine whether the desired solution is actually obtained in the end.