Dilution

Core idea

Overview

The dilution equation is a mathematical representation of the conservation of solute mass during the process of adding solvent to a solution. It states that the product of the initial concentration and volume is equal to the product of the final concentration and volume, provided no solute is added or removed.

When to use: This formula is applied when a concentrated stock solution is being diluted to a lower concentration by adding more solvent. It assumes that the total amount of solute remains constant and that the volumes of the liquids are additive.

Why it matters: Dilution is a fundamental technique in laboratory science, pharmacology, and industrial chemistry for creating precise working solutions. It allows scientists to store compact, high-concentration reagents safely and prepare specific lower doses or reaction environments as needed.

Symbols

Variables

$C_{2}$ = Final Conc, $V_{2}$ = Final Vol, $C_{1}$ = Initial Conc, $V_{1}$ = Initial Vol

C_{2}

Final Conc

m o l / d m^{3}

V_{2}

Final Vol

d m^{3}

C_{1}

Initial Conc

m o l / d m^{3}

V_{1}

Initial Vol

d m^{3}

Walkthrough

Derivation

Understanding Dilution

Dilution lowers concentration by adding solvent, while the number of moles of solute stays the same.

No solute is lost (no reaction, no evaporation of solute).
Solutions mix completely after dilution.

1

State that Moles Stay Constant:

Moles of solute n equals concentration c times volume V (with V in dm³).

n = c V

2

Set Before and After Equal:

Since the moles are the same before and after dilution, c₁V₁ must equal c₂V₂.

c_{1} V_{1} = c_{2} V_{2}

Note: If volume is in cm³, you can keep it in cm³ as long as both volumes use the same unit.

Result

c_{1} V_{1} = c_{2} V_{2}

Source: Edexcel GCSE Chemistry — Quantitative Chemistry

Free formulas

Rearrangements

Solve for $C_{2}$

Make C2 the subject

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

Rearrange the dilution equation to solve for the final concentration (C₂).

Difficulty: 2/5

Solve for $V_{2}$

Make V2 the subject

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

To make $V_{2}$ (final volume) the subject of the dilution equation, divide both sides by $C_{2}$ (final concentration).

Difficulty: 2/5

Solve for $C_{1}$

Make C1 the subject

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

Rearrange the dilution equation to solve for the initial concentration, $C_{1}$ .

Difficulty: 2/5

Solve for $V_{1}$

Dilution Equation: Solve for Initial Volume (V1)

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

Rearrange the dilution equation to solve for the initial volume ( $V_{1}$ ).

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph forms a hyperbola because the final volume appears in the denominator, resulting in a horizontal asymptote at zero as the final volume increases. For a chemistry student, this shape illustrates that adding more solvent to a fixed amount of solute causes the concentration to drop rapidly at first and then level off as the solution becomes increasingly dilute. The most important feature is that the curve never reaches zero, meaning that no matter how much volume is added, the concentration will always remain a positive value.

Graph type: hyperbolic

Why it behaves this way

Intuition

The total amount of solute particles remains constant, but they are spread out over a larger volume, making the solution less concentrated.

C_{1}

Initial concentration of the solute

Represents the initial 'strength' or amount of solute per unit volume before dilution.

V_{1}

Initial volume of the solution

The starting quantity of the solution containing the solute.

C_{2}

Final concentration of the solute

The 'strength' or amount of solute per unit volume after dilution, which is always lower than

C_{1}

.

V_{2}

Final volume of the diluted solution

The total quantity of the solution after adding solvent, which is always greater than

V_{1}

.

Free study cues

Insight

Canonical usage

The units of concentration (C) must be consistent on both sides of the equation, and similarly, the units of volume (V) must be consistent on both sides.

Common confusion

The most common mistake is using different units for $C_{1}$ and $C_{2}$ , or for $V_{1}$ and $V_{2}$ , without converting them to be consistent. For example, using $V_{1}$ in liters and $V_{2}$ in milliliters without conversion will lead to an

Unit systems

$C_{1}$ mol/L (M) or g/L or % (w/v) - Initial concentration. Must be in the same units as C_2.

$V_{1}$ L or mL - Initial volume. Must be in the same units as V_2.

$C_{2}$ mol/L (M) or g/L or % (w/v) - Final concentration. Must be in the same units as C_1.

$V_{2}$ L or mL - Final volume. Must be in the same units as V_1.

One free problem

Practice Problem

A chemist has 50 mL of a 2.0 M HCl stock solution. If they dilute it with water until the final volume reaches 250 mL, what is the new molar concentration of the solution?

Initial Conc2 mol/dm^3

Initial Vol50 dm^3

Final Vol250 dm^3

Solve for: $C 2$

Hint: Rearrange the formula to isolate the final concentration: C2 = (C1 ×V1) / V2.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In diluting squash concentrate, Dilution is used to calculate Final Conc from Final Vol, Initial Conc, and Initial Vol. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

Ensure concentration units (e.g., Molarity) are identical on both sides of the equation.
Ensure volume units (e.g., mL or L) are consistent throughout the calculation.
V2 represents the total final volume, which is the sum of the initial volume and the volume of solvent added.
Always add concentrated acids to water, not water to acid, even when calculating for dilution.

Avoid these traps

Common Mistakes

Confusing initial/final values.
Using different volume units (mixing cm³ and dm³).
Thinking concentration increases when diluting (it always decreases).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Dilution lowers concentration by adding solvent, while the number of moles of solute stays the same.

This formula is applied when a concentrated stock solution is being diluted to a lower concentration by adding more solvent. It assumes that the total amount of solute remains constant and that the volumes of the liquids are additive.

Dilution is a fundamental technique in laboratory science, pharmacology, and industrial chemistry for creating precise working solutions. It allows scientists to store compact, high-concentration reagents safely and prepare specific lower doses or reaction environments as needed.

Confusing initial/final values. Using different volume units (mixing cm³ and dm³). Thinking concentration increases when diluting (it always decreases).

In diluting squash concentrate, Dilution is used to calculate Final Conc from Final Vol, Initial Conc, and Initial Vol. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Ensure concentration units (e.g., Molarity) are identical on both sides of the equation. Ensure volume units (e.g., mL or L) are consistent throughout the calculation. V2 represents the total final volume, which is the sum of the initial volume and the volume of solvent added. Always add concentrated acids to water, not water to acid, even when calculating for dilution.

References

Sources

Chemistry: The Central Science (14th ed.) by Brown, LeMay, Bursten, Murphy, Woodward, Stoltzfus
Wikipedia: Dilution (chemistry)
Atkins' Physical Chemistry
IUPAC Gold Book
Chemistry: The Central Science by Brown, LeMay, Bursten, Murphy, Woodward, Stoltzfus
IUPAC Gold Book: Dilution
Edexcel GCSE Chemistry — Quantitative Chemistry

Overview

Variables

Derivation

State that Moles Stay Constant:

Set Before and After Equal:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

pH definition

pOH Calculation

Beer-Lambert Law

Frequently Asked Questions

Sources