Engineering · Chemical Engineering · Reaction Engineering
Dilution Calculator
Calculates the final concentration or required solvent volume when diluting a solution using the C1V1 = C2V2 dilution equation.
Calculator
Formula
C₁ is the initial (stock) concentration of the solution; V₁ is the initial volume of stock solution taken; C₂ is the final (desired) concentration after dilution; V₂ is the final total volume of the diluted solution. The volume of solvent (diluent) to add is V_{\text{add}} = V_2 - V_1. This relationship holds because the amount of solute (moles or mass) is conserved during dilution — only the volume changes.
Source: Atkins, P. & de Paula, J. — Physical Chemistry, 10th Edition; IUPAC Green Book (Quantities, Units and Symbols in Physical Chemistry).
How it works
When a solution is diluted, solvent is added to lower its concentration. The total number of moles (or mass) of solute remains constant throughout this process — only the volume increases. This conservation of solute is the physical basis of the dilution law. The relationship C₁V₁ = C₂V₂ expresses that the product of concentration and volume before dilution equals the product after dilution. This equation applies to any unit of concentration (molarity, percent by volume, parts per million, grams per litre) as long as both C₁ and C₂ use the same units, and both V₁ and V₂ use the same units.
The formula C₁V₁ = C₂V₂ contains four variables: initial concentration (C₁), initial volume of stock taken (V₁), final concentration (C₂), and final total volume (V₂). Knowing any three allows calculation of the fourth. The dilution factor — defined as V₂ divided by V₁ — describes how many times the solution has been diluted and is especially useful in serial dilution protocols. The volume of pure solvent to add equals the final volume minus the initial volume taken from stock: V_add = V₂ − V₁.
In industrial chemical engineering, dilution calculations underpin reactor feed preparation, wastewater treatment dosing, acid/base neutralisation processes, and pharmaceutical manufacturing. In analytical laboratories, serial dilutions are used to construct calibration curves for spectrophotometry and chromatography. Environmental engineers use dilution factors to assess pollutant concentrations in receiving water bodies. The calculator supports all four solve modes — for C₂, V₂, V₁, or C₁ — making it flexible for any dilution scenario.
Worked example
Example: Preparing 500 mL of a 0.1 mol/L HCl solution from a 10 mol/L stock.
Given values: C₁ = 10 mol/L, C₂ = 0.1 mol/L, V₂ = 500 mL. Solve for V₁ (volume of stock to use).
Step 1 — Rearrange C₁V₁ = C₂V₂ for V₁:
V₁ = (C₂ × V₂) / C₁ = (0.1 × 500) / 10 = 5 mL
Step 2 — Calculate the dilution factor:
Dilution Factor = V₂ / V₁ = 500 / 5 = 100×
Step 3 — Calculate solvent to add:
V_add = V₂ − V₁ = 500 − 5 = 495 mL of distilled water
Procedure: Carefully pipette 5 mL of the 10 mol/L HCl stock into a 500 mL volumetric flask. Slowly add approximately 400 mL of distilled water while swirling, then make up to the 500 mL mark. Always add acid to water — never water to acid — for safety.
Limitations & notes
The C₁V₁ = C₂V₂ equation assumes ideal behaviour and perfect mixing, with no volume contraction or expansion upon mixing (which can occur with concentrated solutions such as strong acids or ethanol-water mixtures). It does not account for temperature effects on density and therefore concentration. The formula is valid only when the same concentration units are used on both sides of the equation; mixing units (e.g., mol/L and mg/L) will produce incorrect results. For highly concentrated or viscous solutions, non-ideal volumetric behaviour should be corrected using density data. Serial dilution calculations involving multiple successive steps require repeated application of the formula rather than a single-step calculation. The calculator does not account for solute activity, ionic strength effects, or equilibrium chemistry that may alter effective concentrations in complex solutions.
Frequently asked questions
What does C1V1 = C2V2 actually mean?
The equation expresses conservation of solute during dilution. C₁V₁ is the number of moles (or mass) of solute in the stock solution taken, and C₂V₂ is the number of moles in the final diluted solution. Since no solute is added or removed — only solvent is added — both quantities are equal.
Can I use different units for concentration on both sides?
No. C₁ and C₂ must be expressed in identical units (e.g., both in mol/L, both in mg/mL, or both in % w/v) for the equation to hold. Similarly, V₁ and V₂ must use the same volume units. Mixing units will give a mathematically incorrect result.
What is the dilution factor and why does it matter?
The dilution factor is the ratio of the final volume to the initial volume taken (V₂/V₁). A dilution factor of 10× means the solution has been made ten times less concentrated. It is widely used in microbiology, analytical chemistry, and serial dilution protocols to express and communicate how much a sample has been diluted.
How do I perform a serial dilution?
A serial dilution involves a sequence of equal dilution steps. For example, a 1:10 serial dilution repeated three times gives a total dilution factor of 10³ = 1000×, and a final concentration of C₁/1000. Apply the C₁V₁ = C₂V₂ formula at each individual step, using the output concentration of one step as the C₁ for the next step.
Does this calculator work for percent concentrations (% w/v or % v/v)?
Yes. The C₁V₁ = C₂V₂ formula works for any consistent concentration unit, including percent weight per volume (% w/v), percent volume per volume (% v/v), ppm, mg/L, or molarity. Simply ensure both C₁ and C₂ are in the same percentage units and both volumes are in the same unit.
Last updated: 2025-01-15 · Formula verified against primary sources.