Solution Dilution Calculator
Calculate dilutions with C₁V₁ = C₂V₂. Solve for any one variable, mix stock + diluent, and get a dilution factor instantly.
Enter any 3 values above to solve for the 4th.
Table of Contents
What is a solution dilution?
A dilution is the process of lowering a solution's concentration by adding more solvent (diluent). In lab work, this usually means taking a small volume of a concentrated stock and adding buffer or water until you reach a target concentration and final volume.
Dilutions are fundamental in chemistry, biology, medicine, and many other scientific disciplines. They allow researchers to work with reagents at optimal concentrations for their specific applications, whether that's running an assay, preparing a buffer, or creating a standard curve.
The key principle behind dilution is conservation of mass: the amount of solute (the dissolved substance) remains constant before and after dilution. Only the volume of the solution changes, which in turn changes the concentration.
Key terms
- Stock solution: The concentrated solution you start with
- Diluent: The solvent added to reduce concentration (e.g., water, buffer, saline)
- Working solution: The final diluted solution at the desired concentration
- Aliquot: A measured portion taken from a larger volume
The C₁V₁ = C₂V₂ formula
The dilution equation is derived from the principle that the amount of solute remains constant during dilution. Since concentration × volume = amount of solute (in moles or mass), we can write:
C₁V₁ = C₂V₂
(Amount of solute before) = (Amount of solute after)
Where:
- C₁ (Stock concentration): The concentration of your starting solution
- V₁ (Stock volume): The volume of stock solution you will use
- C₂ (Final concentration): The target concentration after dilution
- V₂ (Final volume): The total volume of the diluted solution
This equation can be rearranged to solve for any one of the four variables when the other three are known:
Solve for V₁:
V₁ = (C₂ × V₂) / C₁
Solve for V₂:
V₂ = (C₁ × V₁) / C₂
Solve for C₁:
C₁ = (C₂ × V₂) / V₁
Solve for C₂:
C₂ = (C₁ × V₁) / V₂
How to use this calculator
This calculator solves the C₁V₁ = C₂V₂ equation for any variable. Here's how to use it effectively:
Step 1: Select what to solve for
Click the button for the variable you want to calculate. Most commonly, you'll solve for V₁ (how much stock to use) when you know your stock concentration and want to make a specific final volume at a target concentration.
Step 2: Enter the three known values
Fill in the values you know. Select appropriate units from the dropdown menus. The calculator supports various concentration units (M, mM, µM, nM for molar; g/L, mg/mL, mg/L, µg/mL, % w/v for mass/volume) and volume units (L, mL, µL, nL).
Step 3: Review the results
The calculator automatically computes the unknown variable along with the dilution factor, stock fraction, and step-by-step mixing instructions. Use the "Copy Result" button to save all details to your clipboard.
Examples
Example 1: Basic 10× dilution
Problem: Make 100 mL of 0.1 M NaCl from a 1.0 M stock solution.
Solution: Solve for V₁:
Procedure: Measure 10 mL of the 1.0 M stock, then add 90 mL of diluent (water) to reach 100 mL total volume.
Example 2: Converting between unit prefixes
Problem: Prepare 500 µL of a 50 µM working solution from a 10 mM stock.
Solution: The calculator handles unit conversion automatically. Setting C₁ = 10 mM, C₂ = 50 µM, V₂ = 500 µL:
Procedure: Pipette 2.5 µL of the 10 mM stock and add 497.5 µL of diluent.
Example 3: Mass/volume concentration
Problem: How much of a 10 mg/mL protein stock is needed to make 2 mL at 0.5 mg/mL?
Procedure: Add 100 µL of stock to 1900 µL of buffer.
Example 4: Calculating final concentration
Problem: You added 25 µL of a 4 M NaCl stock to reach a final volume of 1 mL. What is the final concentration?
Example 5: Working with percent solutions
Problem: Dilute a 20% (w/v) SDS stock to make 50 mL of 1% SDS.
Procedure: Add 2.5 mL of 20% SDS stock to 47.5 mL of water.
Understanding dilution factor
The dilution factor (DF) describes how much a solution has been diluted. It can be expressed in two equivalent ways:
DF = C₁/C₂ = V₂/V₁
A dilution factor of 10 (often written as "10×" or "1:10") means the final concentration is 10 times lower than the starting concentration. Equivalently, it means the final volume is 10 times the volume of stock used.
| Dilution Factor | Notation | Stock Fraction | Example |
|---|---|---|---|
| 2 | 2× or 1:2 | 50% | 1 mL stock + 1 mL diluent = 2 mL |
| 5 | 5× or 1:5 | 20% | 1 mL stock + 4 mL diluent = 5 mL |
| 10 | 10× or 1:10 | 10% | 1 mL stock + 9 mL diluent = 10 mL |
| 100 | 100× or 1:100 | 1% | 10 µL stock + 990 µL diluent = 1 mL |
| 1000 | 1000× or 1:1000 | 0.1% | 1 µL stock + 999 µL diluent = 1 mL |
The stock fraction is simply 1/DF, representing what proportion of the final volume comes from the stock solution. This is useful for understanding the composition of your diluted solution.
Serial dilutions
A serial dilution is a series of sequential dilutions used to reduce concentration in a controlled, stepwise manner. This technique is essential when you need a range of concentrations (e.g., for standard curves) or when the required dilution factor is too large for a single step.
Why use serial dilutions?
- Accuracy: Large dilutions (e.g., 1:10,000) are error-prone in a single step due to pipetting very small volumes
- Standard curves: Create a series of known concentrations for calibration
- Titrations: Systematically test a range of concentrations
- Microbiology: Achieve countable colony numbers from concentrated samples
For a serial dilution with n steps, each using the same dilution factor (DF), the total dilution is:
Total dilution = DFn
Example: 10-fold serial dilution
Starting with a 1 M solution and performing three 10× dilutions:
- Tube 1: 1 M → 0.1 M (10× dilution)
- Tube 2: 0.1 M → 0.01 M (100× total)
- Tube 3: 0.01 M → 0.001 M (1000× total)
Concentration units explained
This calculator supports two families of concentration units. Units within a family can be converted directly; converting between families requires knowing the molecular weight of the solute.
Molar concentration (amount per volume)
Molar units express concentration as moles of solute per liter of solution. This is preferred for most biochemical and molecular biology applications because it directly relates to the number of molecules.
| Unit | Full Name | Equivalent |
|---|---|---|
| M | Molar | 1 mol/L |
| mM | Millimolar | 10⁻³ M = 1 mmol/L |
| µM | Micromolar | 10⁻⁶ M = 1 µmol/L |
| nM | Nanomolar | 10⁻⁹ M = 1 nmol/L |
Mass/volume concentration
Mass/volume units express concentration as mass of solute per volume of solution. These are common for proteins, polymers, and situations where molecular weight is unknown or variable.
| Unit | Equivalent | Notes |
|---|---|---|
| g/L | 1 g per liter | Base unit |
| mg/mL | 1 g/L | Same as g/L |
| mg/L | 10⁻³ g/L | Also called ppm (parts per million) in water |
| µg/mL | 10⁻³ g/L | Same as mg/L |
| % (w/v) | 10 g/L | Grams per 100 mL |
Converting between molar and mass/volume
To convert between these families, you need the molecular weight (MW) of the solute:
Molarity (M) = Mass concentration (g/L) / MW (g/mol)
Mass concentration (g/L) = Molarity (M) × MW (g/mol)
Common mistakes to avoid
Mistake 1: Confusing V₂ with diluent volume
V₂ is the total final volume, not the volume of diluent added. The diluent volume is V₂ - V₁. If you want 100 mL final volume and use 10 mL stock, add 90 mL diluent (not 100 mL).
Mistake 2: Mixing incompatible concentration units
You cannot directly use C₁V₁ = C₂V₂ when C₁ is in molar units and C₂ is in mass/volume units (or vice versa). First convert both to the same unit family using the molecular weight.
Mistake 3: Ignoring significant figures
Your calculated result is only as accurate as your least precise input. If your stock concentration is known to ±5%, your final concentration will have at least that much uncertainty.
Mistake 4: Pipetting errors at extreme volumes
Very small volumes (<1 µL) and very large dilution factors (>100×) in a single step are prone to significant error. Consider serial dilutions for better accuracy when the calculated V₁ is extremely small.
Mistake 5: Assuming ideal mixing
C₁V₁ = C₂V₂ assumes volumes are additive (V_stock + V_diluent = V_final). This is approximately true for dilute aqueous solutions but may not hold for concentrated solutions or organic solvents where volume contraction or expansion can occur.
Frequently Asked Questions
They must be in the same unit family (e.g., both molar units like M/mM/µM, or both mass/volume like g/L and mg/L). You can mix different prefixes (mM and µM), but not molar with mass units without first converting using the molecular weight.
Dilution factor (DF) is how many times the solution is diluted: DF = C₁/C₂ = V₂/V₁. A 10× dilution means the final concentration is 10 times lower than the stock. It's the reciprocal of the stock fraction (1/DF = V₁/V₂).
The C₁V₁ = C₂V₂ equation is mathematically valid for concentration (C₂ > C₁), but you cannot concentrate a solution by adding diluent. Concentration requires evaporation, lyophilization, ultrafiltration, or other techniques. The calculator will warn you if your inputs describe a concentration rather than a dilution.
If V₁ > V₂, the calculation implies you need more stock than the final volume you want. This is physically impossible for a dilution. It means either (a) your target concentration is higher than your stock (not achievable by dilution), or (b) there's an error in your inputs.
The calculator uses standard floating-point arithmetic and displays results to 6 significant figures. However, your real-world accuracy depends on the precision of your input values, pipetting technique, and the assumption that volumes are additive. For critical applications, always verify your dilutions.
Percent weight/volume (% w/v) means grams of solute per 100 mL of solution. A 10% (w/v) solution contains 10 g of solute in 100 mL total volume, which equals 100 g/L. This notation is common for buffers, detergents, and many biochemical reagents.
For most applications, the order doesn't matter as long as you mix thoroughly. However, for certain reagents (like acids or highly concentrated solutions), it's safer to add the concentrated solution to the diluent to avoid localized high concentrations. Always follow specific protocols for hazardous materials.
Yes. The calculator automatically saves your inputs to the URL. Simply copy the URL from your browser's address bar to share your exact calculation with colleagues. They'll see the same values and results when they open the link.