Plan simple and serial dilutions, compute stock and diluent volumes, convert percent solutions, and solve molarity/normality with clear tables and visuals.
Select a dilution mode and enter your values to calculate stock volumes, serial dilutions, or solution concentrations.
C1 = initial concentration
V1 = initial volume
C2 = final concentration
V2 = final volume
Each step: C_n = C_0 / F^n
F = dilution factor
n = step number
M = n / V
n = moles
V = volume (L)
Solution dilution is the process of decreasing the concentration of a solution by adding more solvent (usually water) while keeping the total amount of solute constant. It's one of the most fundamental operations in chemistry, biology, medicine, and environmental science—essential for preparing reagents at the right concentration for experiments, maintaining standards for analytical work, and understanding how solutions behave when you change their composition.
The core principle behind dilution is simple: when you add solvent to a concentrated stock solution, you're spreading the same amount of dissolved substance (solute) across a larger volume. This decreases the concentration—how much solute is present per unit volume—without changing the total quantity of solute. For example, if you have 10 mL of a 1.0 M salt solution (containing a certain number of moles of salt) and add 90 mL of water, you now have 100 mL of 0.1 M solution with the exact same number of moles of salt, just distributed in 10× the volume.
This relationship is captured by the **dilution equation**: **C₁V₁ = C₂V₂**, where C₁ is the initial (stock) concentration, V₁ is the volume of stock solution you use, C₂ is the final (diluted) concentration, and V₂ is the final total volume. This equation says "amount of solute before dilution equals amount of solute after dilution," mathematically expressed as concentration × volume. It's remarkably versatile—whether you're making a buffer for a biology lab, diluting a cleaning solution (conceptually), or preparing calibration standards for analytical chemistry, C₁V₁ = C₂V₂ is your starting point.
Our **Solution Dilution Calculator** automates this equation and extends it to more complex scenarios: serial dilutions (stepwise dilutions by a constant factor to span wide concentration ranges), percent-to-molarity conversions, stock + diluent volume breakdowns, and multi-stock mixing. You input any three of the four variables (C₁, V₁, C₂, V₂), and the calculator solves for the fourth instantly. It also shows how much diluent (V₂ − V₁) to add, presents results in tables and charts, and helps you visualize the dilution process—perfect for chemistry homework, exam prep, lab planning (conceptual), and building intuition about concentration changes.
Dilution is everywhere in real-world chemistry. In teaching labs, students prepare diluted acids or bases from concentrated stocks. In research, scientists make serial dilutions of proteins or dyes to find optimal assay concentrations. In environmental monitoring, samples are diluted to bring analytes within detection ranges. In medicine, drugs are diluted to safe dosing levels. Understanding dilution—and being able to calculate volumes and concentrations accurately—is a critical skill for anyone working with solutions, from high school chemistry students to professional chemists and biologists.
**Important safety and context note:** This calculator is an **educational tool** designed to help you learn and solve dilution problems conceptually. It does NOT replace proper lab training, safety protocols, or chemical handling procedures. Real dilutions involve hazardous substances, precise techniques, and safety equipment (PPE, hoods, etc.). Always consult lab instructors, safety data sheets (SDS), and institutional protocols before performing any actual dilutions in a lab setting. Use this calculator to understand the math, check your homework, and plan experiments—but never substitute it for professional lab guidance when handling chemicals.
This calculator is designed to handle multiple dilution scenarios—from simple C₁V₁ = C₂V₂ problems to serial dilutions and multi-stock mixing. Here's a comprehensive guide to each mode:
The calculator typically offers several modes:
Select the mode that matches your problem type. For most homework and basic lab planning, "Simple Dilution" is the go-to.
For **Simple Dilution (C₁V₁ = C₂V₂)**:
**Critical:** Concentration units for C₁ and C₂ must be compatible (both molarity, both %, etc.), and volume units for V₁ and V₂ must be the same or convertible. The calculator usually handles unit conversions automatically, but double-check consistency.
Example inputs:
C₁ = 2.0 M, C₂ = 0.5 M, V₂ = 100 mL → Solve for V₁
C₁ = 10% w/v, V₁ = 20 mL, V₂ = 200 mL → Solve for C₂
Indicate which variable is unknown. The calculator will rearrange C₁V₁ = C₂V₂ to solve for it:
For serial dilutions, you typically specify starting concentration, dilution factor, number of steps, and volumes per step. The calculator generates a table showing concentration at each step.
Click "Calculate" or submit the form. The calculator returns:
Example result:
Problem: Dilute 2.0 M stock to 0.5 M in 100 mL final volume.
Solution: V₁ = (C₂ × V₂) / C₁ = (0.5 × 100) / 2.0 = 25 mL
Diluent: 100 mL − 25 mL = 75 mL water
Procedure: Measure 25 mL of 2.0 M stock, transfer to flask, add water to 100 mL mark, mix.
Use the **Copy Result** button to paste the calculation into your lab notebook or homework. For serial dilutions, **Copy as CSV** exports a table you can open in Excel or Google Sheets. Save results for later reference or share with classmates (for conceptual understanding, not to copy answers!).
**Pro tip:** After getting results, always double-check the numbers make sense. C₂ should be lower than C₁ for a dilution (unless you're solving a reverse problem). V₁ should be less than V₂. Diluent volume should be positive. If something looks off, recheck your inputs for unit consistency or typos.
Understanding the math behind dilution helps you use the calculator effectively and solve problems confidently. Here's the conceptual and mathematical foundation:
**Core idea:** Amount of solute before dilution = Amount of solute after dilution.
Amount of solute is concentration × volume. So:
C₁ × V₁ = C₂ × V₂
C₁: Concentration of stock (initial) solution (M, %, mg/mL, etc.)
V₁: Volume of stock solution to use (mL, L, μL)
C₂: Concentration of diluted (final) solution (same units as C₁)
V₂: Total volume of diluted solution (same units as V₁)
**Why it works:** Concentration is amount/volume (e.g., moles/L for molarity). When you multiply concentration by volume, you get amount (moles). In a simple dilution, you don't add or remove solute, so the amount stays constant. You just change the volume (by adding solvent), which changes concentration proportionally.
You can solve for any variable by rearranging C₁V₁ = C₂V₂:
Solve for V₁ (volume of stock needed):
V₁ = (C₂ × V₂) / C₁
Most common: "How much stock do I pipette?"
Solve for C₂ (final concentration):
C₂ = (C₁ × V₁) / V₂
Check what concentration you'll get from given volumes.
Solve for V₂ (final volume):
V₂ = (C₁ × V₁) / C₂
Find total volume needed for target concentration.
Solve for C₁ (stock concentration):
C₁ = (C₂ × V₂) / V₁
Reverse problem: determine stock concentration from dilution result.
Once you know V₁ and V₂, the volume of solvent (diluent) to add is simply:
V_diluent = V₂ − V₁
For example, if V₁ = 25 mL and V₂ = 100 mL, add 75 mL of diluent. The final solution has 25 mL stock + 75 mL diluent = 100 mL total.
Dilution factor (DF) describes how much you've diluted the stock:
DF = V₂ / V₁ = C₁ / C₂
For example, if V₂ = 100 mL and V₁ = 25 mL, DF = 4 (you diluted 4-fold). Equivalently, if C₁ = 2 M and C₂ = 0.5 M, DF = 4. A 10-fold dilution (DF = 10) means final concentration is 1/10 of stock.
In serial dilution, you dilute stepwise by a constant factor F. After n steps:
Cₙ = C₀ / Fⁿ
C₀: Starting stock concentration
F: Dilution factor per step (e.g., 10 for 1:10 dilution)
n: Step number (1, 2, 3, ...)
Cₙ: Concentration after step n
Example: Start with 1.0 M. Do 3 steps of 10-fold dilution. C₃ = 1.0 / 10³ = 0.001 M = 1 mM. Serial dilutions span wide concentration ranges efficiently.
Problem: You have a 5.0 M NaCl stock solution. You need to prepare 250 mL of 0.75 M NaCl. How much stock and water do you need?
Given:
C₁ = 5.0 M (stock)
C₂ = 0.75 M (desired)
V₂ = 250 mL (desired final volume)
V₁ = ? (volume of stock to use)
Solution:
V₁ = (C₂ × V₂) / C₁
V₁ = (0.75 M × 250 mL) / 5.0 M
V₁ = 187.5 / 5.0
V₁ = 37.5 mL
V_diluent = V₂ − V₁ = 250 − 37.5 = 212.5 mL
Answer:
Pipette 37.5 mL of 5.0 M NaCl stock into a container.
Add water to bring total volume to 250 mL.
Mix thoroughly. Final solution: 250 mL of 0.75 M NaCl.
Dilution calculations are essential across chemistry, biology, medicine, and environmental science. Here are detailed scenarios where this calculator proves invaluable:
**Scenario:** Your homework asks: "Prepare 500 mL of 0.25 M H₂SO₄ from a 6.0 M stock. How much stock and water?" Instead of manually calculating and risking arithmetic errors, use the dilution calculator. Input C₁ = 6.0 M, C₂ = 0.25 M, V₂ = 500 mL, solve for V₁. Instantly get V₁ = 20.83 mL, diluent = 479.17 mL. Check your manual work, build confidence, and move on to the next problem. For exams, practice with the calculator beforehand to internalize the C₁V₁ = C₂V₂ relationship so you can solve problems quickly by hand.
**Scenario:** In your chemistry lab, you need 100 mL of 0.1 M HCl for a titration. The lab provides 12 M concentrated HCl stock. Using C₁V₁ = C₂V₂: V₁ = (0.1 × 100) / 12 = 0.833 mL stock. Add 99.17 mL water. The calculator helps you plan this **conceptually**—in the actual lab, you'd follow proper dilution technique: add acid to water slowly, use PPE, work in a hood. The calculator gives you the numbers; lab training teaches safe execution. Never perform dilutions of concentrated acids or bases without proper supervision and safety equipment.
**Scenario:** You're doing a spectrophotometry experiment and need standards at 1.0, 0.1, 0.01, 0.001 M (covering 3 orders of magnitude). Making each from scratch wastes stock. Instead, use serial dilution: start with 1.0 M stock, do 10-fold dilutions. Step 1: dilute 10 mL of 1.0 M to 100 mL → 0.1 M. Step 2: dilute 10 mL of 0.1 M to 100 mL → 0.01 M. Step 3: dilute 10 mL of 0.01 M to 100 mL → 0.001 M. The calculator generates a table showing concentration at each step, volumes to transfer, and checks your dilution factors. This is standard practice in analytical chemistry and biology (enzyme kinetics, antibody titrations, etc.).
**Scenario:** Your biology textbook problem says: "An enzyme works best at 50 mM substrate concentration. You have 200 mM substrate stock. Design an assay using 2 mL total volume at optimal concentration." Solve: C₁ = 200 mM, C₂ = 50 mM, V₂ = 2 mL. V₁ = (50 × 2) / 200 = 0.5 mL. Use 0.5 mL stock + 1.5 mL buffer. The calculator helps you understand how to hit target concentrations precisely, a skill critical in biochemistry, molecular biology, and pharmacology (conceptually).
**Scenario:** A protocol calls for "5% w/v NaCl" but you only know molarity. NaCl molar mass ≈ 58.44 g/mol. 5% w/v = 5 g/100 mL = 50 g/L. Molarity = 50 / 58.44 ≈ 0.856 M. Now you can use 0.856 M in dilution calculations. Or reverse: you have 3 M stock, want to make 10% w/v. The calculator (if it supports percent-molarity conversion) handles this for you, or you do the conversion manually then plug into C₁V₁ = C₂V₂. This interconversion is common when working with literature protocols that use different concentration units.
**Scenario:** You need three different concentrations for an experiment: 1 M, 0.5 M, 0.2 M, each 50 mL. Option 1: make each from solid solute (tedious, uses lots of solid). Option 2: make 1 M stock, dilute to others. Make 50 mL of 1 M (your stock). Dilute 25 mL of 1 M to 50 mL → 0.5 M. Dilute 10 mL of 1 M to 50 mL → 0.2 M. The calculator helps you plan volumes, minimize waste, and organize your workflow. This strategy—make concentrated stock, dilute as needed—is standard in research and teaching labs.
**Scenario:** A textbook problem on pollution: "A river has 10 ppm pollutant. A discharge adds 1 L of 1000 ppm pollutant to 1000 L of river water. What's the new concentration?" Treat river as C₁ = 10 ppm, V₁ = 1000 L. Discharge is C₂_source = 1000 ppm, V₂_source = 1 L. Total volume = 1001 L. New concentration ≈ (10 × 1000 + 1000 × 1) / 1001 ≈ 11 ppm. This is a mixing problem, related to dilution concepts. The calculator (if it supports mixing) or manual calculation helps you model how concentrations change when solutions combine.
**Scenario:** You diluted a solution and want to verify you did it right. You used 20 mL of 2 M stock, added water to 200 mL. What should final concentration be? C₂ = (C₁ × V₁) / V₂ = (2 × 20) / 200 = 0.2 M. If you measure final concentration and get ~0.2 M (e.g., by titration or spectrometry), dilution was correct. If you get 0.3 M or 0.1 M, something went wrong—recheck volumes, mixing, etc. The calculator provides the expected value for comparison, a quality control step in lab work.
Even simple dilution problems can trip you up if you're not careful about units, logic, and equation setup. Here are the most frequent errors and how to avoid them:
Mistake: C₁ = 2 M, C₂ = 5% w/v. Plugging these directly into C₁V₁ = C₂V₂ gives nonsense because you're comparing apples (molarity) and oranges (percent).
Why it's wrong: C₁ and C₂ must be in the same units for the equation to work. Molarity and percent are different scales.
Solution: Convert both to the same unit before calculating. If C₁ = 2 M and you want C₂ in %, first convert M to % (or vice versa) using molar mass and density. Alternatively, use a calculator feature that handles unit conversion.
Mistake: C₁ = 5 M, C₂ = 1 M, V₂ = 100 mL, solve for V₁. You get V₁ = 20, but accidentally use 20 L instead of 20 mL because you didn't track units.
Why it's wrong: If V₂ is in mL, V₁ must also be in mL (or both in L, μL, etc.). Mixing mL and L without conversion gives wrong answers.
Solution: Always write units next to numbers: V₁ = 20 mL, V₂ = 100 mL. Double-check units match before and after calculation.
Mistake: Problem says "add 80 mL water to 20 mL stock." Student thinks V₂ = 80 mL (wrong). Correct: V₂ = 20 + 80 = 100 mL.
Why it's wrong: V₂ is the **total final volume** (stock + diluent), not just the volume of water added.
Solution: Read the problem carefully. If it says "dilute to 100 mL," V₂ = 100 mL. If it says "add 80 mL water to 20 mL stock," calculate V₂ = 20 + 80 = 100 mL.
Mistake: C₁ = 0.5 M, C₂ = 2 M, V₂ = 100 mL. Solve for V₁ → V₁ = (2 × 100) / 0.5 = 400 mL. Student thinks "use 400 mL of 0.5 M stock to make 100 mL of 2 M."
Why it's wrong: You can't make a **more concentrated** solution by dilution. C₁V₁ = C₂V₂ assumes simple dilution (adding solvent). To increase concentration, you need to add more solute or evaporate solvent—not covered by this equation.
Solution: For dilution, C₂ must be ≤ C₁ and V₁ ≤ V₂. If C₂ > C₁, you need a different approach (e.g., dissolve more solid, or start with a more concentrated stock).
Mistake: Mixing acid and base and assuming you can use dilution equation to find final concentration. For example, mixing HCl and NaOH—they react, so the solute (H⁺ or OH⁻) changes.
Why it's wrong: C₁V₁ = C₂V₂ assumes **no chemical reaction**—the solute is the same before and after, just spread in different volume. If a reaction occurs, use stoichiometry instead.
Solution: Only use dilution equation for **inert dilution** (same solute, same chemical identity). For reactions, balance equations and use mole calculations.
Mistake: You calculate correctly, pipette stock and water, but don't mix. Concentration is non-uniform—some parts are still stock concentration, others pure water.
Why it's wrong: The math assumes a **homogeneous** solution. If you don't mix, you don't have a dilution—you have layered liquids.
Solution: Always mix (shake, stir, or invert repeatedly) after adding diluent. This is a practical lab step the calculator can't do for you—but it's essential for the dilution to be real.
Mistake: V₁ = 12.345678 mL. Student rounds to 12 mL, uses that for next step in serial dilution. Errors compound.
Why it's wrong: Premature rounding introduces error. In serial dilutions, small errors propagate exponentially.
Solution: Keep full precision during intermediate calculations (use calculator's full decimal display). Round only for the final answer or when matching sig figs from input data. Follow significant figure rules.
Mistake: Get V₁ = 150 mL, V₂ = 100 mL. Don't notice that V₁ > V₂ is impossible for dilution.
Why it's wrong: You can't use more stock volume than final total volume—that's not dilution, it's an error in setup or input.
Solution: Sanity check: V₁ < V₂, C₂ < C₁, diluent volume > 0. If any fail, recheck inputs for typos, unit errors, or conceptual mistakes.
Mistake: Student thinks C₁V₁ = C₂V₂ applies when heating a solution (volume expands, concentration changes).
Why it's wrong: The equation assumes isothermal, isobaric conditions and no physical volume change from temperature or pressure. Thermal expansion changes V without changing amount, but that's not "dilution" in the chemical sense.
Solution: C₁V₁ = C₂V₂ is for dilution by adding solvent, not thermal/physical effects. For temperature-dependent volume changes, use density corrections or ideal gas law (for gases).
Mistake: Doing 3 steps of 10-fold dilution, but calculating each as if starting fresh from stock, instead of realizing concentration decreases exponentially.
Why it's wrong: Each serial step uses the previous tube's concentration, not the original stock. After 3 steps of 10-fold dilution, concentration is C₀ / 10³ = C₀ / 1000, not C₀ / 10.
Solution: Track cumulative dilution factor: DF_total = F^n. Use the calculator's serial dilution mode to auto-generate step-by-step tables and avoid confusion.
Once you've mastered basic C₁V₁ = C₂V₂, these advanced techniques will make you a dilution expert and help you tackle complex scenarios confidently:
**Why:** The calculator is powerful, but **understanding** comes from doing. Work through 5-10 dilution problems by hand using C₁V₁ = C₂V₂. Make mistakes, catch them, learn. Then use the calculator to check your answers. This feedback loop—manual → verify → internalize—builds deep intuition. You'll eventually be able to estimate answers mentally ("doubling volume halves concentration"), speeding up exams and lab planning.
**Strategy:** Use the calculator experimentally. "What if I double V₂? How does C₂ change?" (Answer: C₂ halves if V₁ stays constant.) "What if I use half as much stock (V₁ / 2) at same final volume?" (Answer: C₂ halves.) By playing with variables, you internalize the inverse relationships: C₂ ∝ V₁, C₂ ∝ 1/V₂, etc. This makes dilution **intuitive**, not just mechanical.
**Technique:** Need concentrations from 1 M to 1 μM (6 orders of magnitude)? Serial dilution is the answer. Start with 1 M stock. Do 6 steps of 10-fold dilution → 1 M, 0.1 M, 0.01 M, 0.001 M, 0.0001 M, 0.00001 M = 10 μM, then one more step → 1 μM. Each step uses small volumes (e.g., 1 mL into 9 mL). Total stock used: ~6 mL vs. making each concentration from scratch (requires grams of solute and hundreds of mL). Serial dilution saves time, materials, and enables ultra-low concentrations impossible to weigh accurately.
**Advanced application:** Problem: "React 50 mL of 0.2 M AgNO₃ with excess NaCl. How much 1 M NaCl stock is needed?" First, find moles AgNO₃: 0.2 M × 0.05 L = 0.01 mol. Stoichiometry: AgNO₃ + NaCl → AgCl + NaNO₃ (1:1). Need 0.01 mol NaCl. From 1 M stock: V = n/M = 0.01 / 1 = 0.01 L = 10 mL. This combines molarity (from dilution background) with stoichiometry. The dilution calculator helps with the "how much stock" step; stoichiometry tells you how much solute you need.
**Lab optimization:** Pipettes are most accurate in the middle of their range (e.g., 50-200 μL for a 200 μL pipette, not 5 μL or 199 μL). Design dilutions so volumes fall in accurate ranges. Need 0.01 M from 1 M stock in 10 mL final? V₁ = 0.1 mL = 100 μL (doable, but on the edge). Alternative: dilute 1 M to 0.1 M (1 mL into 10 mL), then dilute 0.1 M to 0.01 M (1 mL into 10 mL). Both steps use 1 mL pipettes (more accurate). Two-step dilution can be more precise than one-step with tiny volume.
**Concept mastery:** Dilution factor DF = C₁ / C₂ = V₂ / V₁. If DF = 5, final concentration is 1/5 of stock. Conversely, if you measure absorbance of a diluted sample, to get original concentration, multiply by DF. Example: dilute unknown sample 10-fold, measure 0.5 M. Original = 0.5 × 10 = 5 M. This "dilution factor correction" is standard in analytical chemistry when samples are too concentrated for direct measurement.
**Analytical strategy:** Your spectrophotometer measures absorbance accurately from 0.1–1.0 AU. Sample gives 2.5 AU (off-scale high). Dilute 1:5 (DF = 5): measure ~0.5 AU. Calculate original concentration = measured × DF. Or sample gives 0.05 AU (too low, noisy). Don't dilute; concentrate (evaporate, or use a larger cuvette). The calculator helps you plan dilution to fit within instrument limits—essential for quantitative analysis.
**Practical skill:** Many commercial reagents are labeled in % (e.g., "37% HCl by mass," "70% ethanol v/v"). To use C₁V₁ = C₂V₂, convert to molarity. Formula: M = (% × density × 10) / molar mass (for % w/v). The calculator (if it has conversion features) automates this, or you do it manually. Knowing how to switch between %, molarity, normality, ppm, etc., makes you fluent in solution language across disciplines (chemistry, biology, medicine, environmental).
**Quality control:** After diluting, check concentration by titration, spectrometry, or pH meter (for acids/bases). If measured concentration matches calculated (within error), dilution was successful. If not, troubleshoot: wrong pipette, evaporation, mixing error, stock concentration wrong. This experimental verification loop—calculate → prepare → measure → compare—is how professionals ensure accuracy. Use the calculator as predictor, measurement as validator.
**Best practice:** Make dilution planning step 1 of every experiment involving solutions. Before starting, list all concentrations needed, design dilution scheme (stocks, final volumes, DF), calculate volumes with this tool, prepare a checklist. This systematic approach prevents mid-experiment crises ("I need 0.05 M but only have 0.1 M and no time to make more"). Plan ahead, dilute in advance, label clearly, and you'll save hours and avoid errors. Dilution mastery = lab efficiency mastery.
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