Best examples of concentration calculations for serial dilutions

Starter examples of concentration calculations for serial dilutions

Let’s start with simple, realistic numbers you might see in a general chemistry or biology lab. These examples of concentration calculations for serial dilutions all use the same basic relationship:

C₁ × V₁ = C₂ × V₂
where C = concentration, V = volume, and subscripts 1 and 2 are initial and final.

Example 1: 1:10 serial dilution from a 1.0 M stock

Say you have a 1.0 M NaCl stock solution and you want a series of 1:10 dilutions: 0.1 M, 0.01 M, 0.001 M, and 0.0001 M. You decide to use 10 mL total volume for each step.

Step 1: 1.0 M → 0.10 M
Using C₁V₁ = C₂V₂:

• C₁ = 1.0 M
• C₂ = 0.10 M
• V₂ = 10.0 mL

\( V₁ = \frac{C₂ V₂}{C₁} = \frac{0.10 \times 10.0}{1.0} = 1.0 \text{ mL} \)

So you pipette 1.0 mL of 1.0 M stock and add 9.0 mL of solvent. That gives you 10.0 mL of 0.10 M NaCl.

Step 2: 0.10 M → 0.010 M
Same math:

• C₁ = 0.10 M
• C₂ = 0.010 M
• V₂ = 10.0 mL

\( V₁ = \frac{0.010 \times 10.0}{0.10} = 1.0 \text{ mL} \)

Again, 1.0 mL of the 0.10 M solution + 9.0 mL solvent → 0.010 M.

Repeating this pattern gives:

• Tube 1: 0.10 M
• Tube 2: 0.010 M
• Tube 3: 0.0010 M
• Tube 4: 0.00010 M

This is the classic 1:10 serial dilution, and it’s one of the simplest examples of concentration calculations for serial dilutions you’ll see.

Real examples of serial dilution calculations in biology labs

Serial dilutions show up constantly in microbiology, immunology, and molecular biology. These real examples of concentration calculations for serial dilutions mirror what’s done in research and clinical labs.

Example 2: Bacterial plate counts (CFU/mL) with serial dilutions

You’re estimating bacterial concentration in a culture. You prepare a 1:10 serial dilution series and plate 100 µL from each tube. After overnight incubation, you count colonies.

Suppose you get these counts:

• Undiluted: lawn (too many to count)
• 10⁻¹: lawn
• 10⁻²: >300 colonies (too many to count accurately)
• 10⁻³: 145 colonies
• 10⁻⁴: 18 colonies

You choose the plate with 145 colonies at the 10⁻³ dilution. To calculate CFU/mL in the original culture:

\( \text{CFU/mL} = \frac{\text{colony count}}{\text{volume plated (mL)}} \times \text{dilution factor} \)

Here:

• Colony count = 145
• Volume plated = 0.100 mL
• Dilution factor = 10³ (because it’s the 10⁻³ tube)

\( \text{CFU/mL} = \frac{145}{0.100} \times 10³ = 1450 \times 10³ = 1.45 \times 10^6 \text{ CFU/mL} \)

This is a classic example of concentration calculations for serial dilutions used in microbiology labs and in public health work. For context, agencies like the CDC rely on similar methods for microbial quantification in surveillance labs, even though the exact protocols vary by assay (CDC laboratory resources).

Example 3: ELISA standard curve for protein concentration

In an ELISA, you often start with a high-concentration protein standard and create a serial dilution to generate a standard curve. Imagine you have a 1000 ng/mL cytokine standard and want a 2-fold serial dilution series: 1000, 500, 250, 125, 62.5, and 31.25 ng/mL.

You decide each standard will be 1.0 mL.

Tube 1 (1000 ng/mL): Provided as stock.

Tube 2 (500 ng/mL): Mix 0.50 mL of 1000 ng/mL + 0.50 mL buffer → total 1.0 mL at 500 ng/mL.

Tube 3 (250 ng/mL): 0.50 mL of 500 ng/mL + 0.50 mL buffer → 250 ng/mL.

Keep repeating 1:1 dilutions. Each step halves the concentration:

• Tube 1: 1000 ng/mL
• Tube 2: 500 ng/mL
• Tube 3: 250 ng/mL
• Tube 4: 125 ng/mL
• Tube 5: 62.5 ng/mL
• Tube 6: 31.25 ng/mL

Here, the serial dilution factor is 2 at each step. This is one of the best examples of concentration calculations for serial dilutions because it connects directly to data analysis: you’ll plot absorbance vs. concentration, fit a curve, and interpolate unknowns.

Examples of concentration calculations for serial dilutions in chemistry

Chemistry labs lean heavily on serial dilutions when preparing standard solutions for spectrophotometry, titrations, and calibration curves.

Example 4: Preparing a Beer–Lambert law calibration series

You have a 0.200 M colored solution and need standards at 0.160, 0.120, 0.080, 0.040, and 0.020 M to build a Beer–Lambert plot. You want 25.0 mL per standard.

Use C₁V₁ = C₂V₂ for each target concentration.

For 0.160 M:

• C₁ = 0.200 M
• C₂ = 0.160 M
• V₂ = 25.0 mL

\( V₁ = \frac{0.160 \times 25.0}{0.200} = 20.0 \text{ mL} \)

So you pipette 20.0 mL of 0.200 M stock and add solvent to 25.0 mL.

For 0.120 M:

\( V₁ = \frac{0.120 \times 25.0}{0.200} = 15.0 \text{ mL} \)

For 0.080 M:

\( V₁ = \frac{0.080 \times 25.0}{0.200} = 10.0 \text{ mL} \)

For 0.040 M:

\( V₁ = \frac{0.040 \times 25.0}{0.200} = 5.0 \text{ mL} \)

For 0.020 M:

\( V₁ = \frac{0.020 \times 25.0}{0.200} = 2.5 \text{ mL} \)

This is technically a set of independent dilutions rather than a stepwise serial dilution, but many instructors present it alongside examples of concentration calculations for serial dilutions because the math and logic are the same.

Example 5: Serial dilution to reach very low concentrations

Sometimes your target concentration is so low that a single dilution would require an impractically small pipetted volume. Serial dilutions solve that.

Suppose you have a 0.10 M solution and want 1.0 × 10⁻⁵ M (10 µM) in a final volume of 100.0 mL.

If you tried a single dilution:

\( V₁ = \frac{(1.0 \times 10^{-5} \text{ M}) \times (100.0 \text{ mL})}{0.10 \text{ M}} = 0.010 \text{ mL} = 10 \mu\text{L} \)

Pipetting 10 µL accurately is possible with good micropipettes, but in many teaching labs that’s not ideal. Instead, you can do a two-step serial dilution.

Step 1: 0.10 M → 1.0 × 10⁻³ M
Choose V₂ = 100.0 mL.

\( V₁ = \frac{(1.0 \times 10^{-3}) \times 100.0}{0.10} = 1.0 \text{ mL} \)

So 1.0 mL of 0.10 M + 99.0 mL solvent → 1.0 × 10⁻³ M.

Step 2: 1.0 × 10⁻³ M → 1.0 × 10⁻⁵ M
Again V₂ = 100.0 mL.

\( V₁ = \frac{(1.0 \times 10^{-5}) \times 100.0}{1.0 \times 10^{-3}} = 1.0 \text{ mL} \)

Now 1.0 mL of 1.0 × 10⁻³ M + 99.0 mL solvent → 1.0 × 10⁻⁵ M.

This two-step strategy is a very practical example of concentration calculations for serial dilutions used whenever you need low micromolar or nanomolar concentrations.

Applied examples of concentration calculations for serial dilutions in medicine and public health

Modern clinical and public health labs rely heavily on serial dilutions, especially for assays involving antibodies, viral titers, and drug sensitivity.

Example 6: Antibody titer from a serial dilution series

In serology, an antibody titer is often reported as the highest dilution that still gives a positive signal. Let’s say a lab tests a patient’s serum using 2-fold serial dilutions: 1:10, 1:20, 1:40, 1:80, 1:160, 1:320, 1:640.

Imagine the results:

• 1:10 – positive
• 1:20 – positive
• 1:40 – positive
• 1:80 – positive
• 1:160 – positive
• 1:320 – negative
• 1:640 – negative

The titer is reported as 1:160, because that’s the highest dilution with a positive result.

From a concentration perspective, if the undiluted serum is considered 100% (arbitrary units), then 1:160 corresponds to 1/160 of the original antibody concentration. While the exact antibody concentration in mg/mL might be unknown, the serial dilution pattern still lets clinicians track changes over time. For example, a 4-fold rise in titer (say from 1:40 to 1:160) is often interpreted as significant in infectious disease workups (NIH resources on serologic testing).

This is a clinically meaningful example of concentration calculations for serial dilutions, even though the result is expressed as a ratio rather than a molar value.

Example 7: Serial dilutions for drug sensitivity (IC₅₀ determination)

Pharmacology and toxicology labs often prepare serial dilutions of a drug to determine IC₅₀ (the concentration that inhibits a biological response by 50%).

Suppose you start with a 10 mM drug stock in DMSO and want a series for a cell viability assay: 100 µM, 33.3 µM, 11.1 µM, 3.7 µM, and 1.2 µM in cell culture medium.

First, convert 10 mM to µM: 10 mM = 10,000 µM.

For the 100 µM solution in 10.0 mL:

\( V₁ = \frac{100 \text{ µM} \times 10.0 \text{ mL}}{10{,}000 \text{ µM}} = 0.10 \text{ mL} = 100 \mu\text{L} \)

So 100 µL of 10 mM stock + 9.90 mL medium → 100 µM.

Then you can create a 3-fold serial dilution series:

• Tube 1: 100 µM
• Tube 2: 33.3 µM
• Tube 3: 11.1 µM
• Tube 4: 3.7 µM
• Tube 5: 1.2 µM

For each step, transfer 3.3 mL of the higher concentration into 6.7 mL of medium (approximately a 1:3 dilution). This kind of pattern is used in many IC₅₀ studies and high-throughput screening efforts described in modern pharmacology research.

Again, this is a real example of concentration calculations for serial dilutions that directly feeds into data analysis and dose–response curve fitting.

Planning multi-step serial dilutions: working backward from the target

Sometimes you don’t just want a single diluted solution—you want a whole series that spans several orders of magnitude. One of the best examples of concentration calculations for serial dilutions involves planning a qPCR or digital PCR standard curve.

Example 8: qPCR DNA standard curve from a high-concentration stock

Imagine you have a DNA stock at 1.0 × 10⁹ copies/µL. You want standards for qPCR at:

• 1.0 × 10⁸ copies/µL
• 1.0 × 10⁷ copies/µL
• 1.0 × 10⁶ copies/µL
• 1.0 × 10⁵ copies/µL
• 1.0 × 10⁴ copies/µL

You decide each standard will be 100 µL, using 10-fold serial dilutions.

Tube 1: 1.0 × 10⁸ copies/µL from 1.0 × 10⁹ copies/µL

Want a 10-fold dilution (factor 10):

• C₁ = 1.0 × 10⁹
• C₂ = 1.0 × 10⁸
• V₂ = 100 µL

\( V₁ = \frac{(1.0 \times 10^8) \times 100}{1.0 \times 10^9} = 10 \mu\text{L} \)

So 10 µL stock + 90 µL buffer → 1.0 × 10⁸ copies/µL.

Tube 2: 1.0 × 10⁷ copies/µL
Repeat the 1:10 pattern:

• 10 µL of 1.0 × 10⁸ + 90 µL buffer → 1.0 × 10⁷ copies/µL.

Keep going:

• Tube 3: 1.0 × 10⁶ copies/µL
• Tube 4: 1.0 × 10⁵ copies/µL
• Tube 5: 1.0 × 10⁴ copies/µL

This serial dilution approach is standard in molecular diagnostics and research, including assays used in infectious disease monitoring and cancer genomics (NIH’s NCBI resources on qPCR methods). It’s an especially important example of concentration calculations for serial dilutions because small errors in dilution can dramatically affect quantification.

Common mistakes and how to sanity-check your serial dilution math

Even experienced scientists occasionally mislabel a tube or flip a dilution factor. When you’re working through examples of concentration calculations for serial dilutions, it helps to build a few quick checks into your routine.

Check 1: Does the concentration change in the expected direction?
If you dilute, the concentration should go down. If your math says the diluted sample is more concentrated than the stock, you’ve swapped C₁ and C₂ or inverted the dilution factor.

Check 2: Is the dilution factor realistic for your pipettes?
If your calculation demands pipetting 0.5 µL into 50 mL, that’s not realistic. That’s your cue to break it into two or three serial steps, as in Example 5.

Check 3: Track cumulative dilution factors
For serial dilutions, multiply stepwise factors. If you do three 1:10 dilutions, the total factor is 10 × 10 × 10 = 1000. If your starting solution is 1.0 M, the third tube should be 1.0 × 10⁻³ M. If your answer doesn’t line up with this logic, revisit the math.

Check 4: Compare with reference protocols
Many educational and research institutions publish standard dilution protocols. If you’re unsure, compare your plan with a trusted source such as a university lab manual or a public resource like the Harvard University Department of Chemistry & Chemical Biology teaching labs or CDC’s general laboratory quality resources.

Building these checks into your workflow turns serial dilutions from a source of anxiety into a routine calculation.

FAQ: examples of serial dilution concentration calculations

Q: Can you give another quick example of serial dilution from mg/mL to µg/mL?
Yes. Suppose you have a 5 mg/mL antibiotic stock and you want 50 µg/mL (0.050 mg/mL) in 10.0 mL.

• C₁ = 5 mg/mL
• C₂ = 0.050 mg/mL
• V₂ = 10.0 mL

\( V₁ = \frac{0.050 \times 10.0}{5} = 0.10 \text{ mL} = 100 \mu\text{L} \)

So 100 µL of stock + 9.90 mL solvent → 50 µg/mL. If you then take 1.0 mL of this and add 9.0 mL medium, you create a 5 µg/mL solution—an easy example of combining single-step and serial dilutions.

Q: How do I choose between a single dilution and a serial dilution?
Check the volume you’d need to pipette from the stock. If the required V₁ is very small (say under 5–10 µL) or very large relative to your final volume, a serial dilution is usually more accurate and practical. Many lab manuals and method papers, including those indexed by NIH and NCBI, favor serial dilutions when the overall factor is 1000× or greater.

Q: Are there online tools to check examples of concentration calculations for serial dilutions?
Yes. Many universities host dilution calculators, and several reputable medical sites provide dose and dilution calculators for clinical drugs. While you should always understand the underlying math, tools from academic or hospital systems can be helpful as a cross-check.

Q: Do serial dilution principles change across fields like chemistry, microbiology, and medicine?
The math is the same everywhere: C₁V₁ = C₂V₂ and multiplication of dilution factors. What changes is the unit (M, mg/mL, CFU/mL, copies/µL, etc.) and how results are interpreted. That’s why seeing examples of concentration calculations for serial dilutions across different disciplines is so helpful—it shows the same core idea in multiple real-world contexts.