Dilution Calculator with Steps (Free) | StatsUnlock

A dilution calculator uses C₁V₁ = C₂V₂. For 250.0 mL of 0.1000 M from a 1.000 M stock, V₁ = 0.1000 × 250.0 ÷ 1.000 = 25.00 mL; make up to 250.0 mL (dilution factor 10). With Class A glassware the solution is 0.10000 ± 0.00019 M (k = 2).

Rule of thumb: a 1 in 10 dilution is 1 part stock in 10 parts total (1 + 9). For factors above about 100, dilute in steps so every transfer is at least a few microliters.

A dilution lowers the concentration of a solution by adding solvent; the amount of solute stays the same. That one fact gives the dilution equation, C₁V₁ = C₂V₂, where C₁ and V₁ are the stock concentration and the volume you take, and C₂ and V₂ are the final concentration and final volume. A dilution calculator rearranges it for any unknown, keeps the units consistent, and turns the answer into a recipe: how much stock to pipette and what to make it up to.

A worked one-liner: V₁ = C₂V₂ ÷ C₁ = 0.1000 M × 250.0 mL ÷ 1.000 M = 25.00 mL. With a Class A 25 mL pipette (±0.03 mL, ASTM E969), a Class A 250 mL flask (±0.12 mL, ASTM E288) and a stock known to ±0.1%, the solution made is 0.10000 ± 0.00019 M at k = 2.

Who uses it: AP Chemistry and gen chem students; biology, microbiology and molecular biology labs diluting buffers, antibodies, DNA, primers and siRNA; cell culture labs seeding plates; clinical and QC labs diluting samples into an instrument's range and back-calculating; EPA water labs making ppm standards; and anyone diluting concentrated acid. One page replaces the separate "molarity dilution", "serial dilution", "dilution factor", "cell dilution" and "primer dilution" calculators found on supplier sites.

Quick answers to the questions people ask most

4.1 What the result means

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4.2 How sure you can be

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4.3 Is it practical?

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4.4 The detail that changes the answer

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4.5 What to do next

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4.6 Make up to the mark, do not add a fixed volume

The dilution calculator gives the diluent as V₂ − V₁, but volumes do not always add. Mixing 73.7 mL of 95% ethanol with 26.3 mL of water gives about 97 mL, not 100 mL, because the molecules pack together. Acids shrink and warm on mixing too. Pipette the stock into a volumetric flask and fill to the mark: then V₂ is exact whatever the shrinkage.

4.7 Which input usually dominates the uncertainty

For the headline dilution (25.00 mL into 250.0 mL), the 25 mL pipette (±0.03 mL) and a stock known to ±0.1% share most of the variance, and the flask adds a little. For microliter transfers the pipette dominates completely: a P10 pipette at 2 µL can be off by several percent, which is why steps of at least 5 to 10 µL are safer.

4.8 Units that look the same but are not

mg/mL and g/L are the same number; µg/mL, mg/L and ng/µL are the same number; ppm equals mg/L only in dilute water solutions. M and mg/mL are different kinds of unit and need the molar mass to convert. X units (10X, 100X) are already concentration factors, so 10X to 1X is a 10-fold dilution.

Press Calculate above to fill all five examples with your result.

The same worked problem as Sections 0, 9, 10, 11 and 14: make 250.0 mL of 0.1000 M from a 1.000 M stock, with a Class A 25 mL pipette and a 250 mL flask.

1. Pick the kind of dilution. Choose a mode in Section 2: stock dilution, dilution factor and ratio, serial dilution, cells, primers and DNA, or acids and reagents.
2. Load a sample or start from your numbers. Ten worked problems are in the sample menu. Sample 1 is the headline problem. Nothing is calculated until you press Calculate.
3. Choose what to solve for. In stock dilution mode, tick the unknown: stock volume, final concentration, final volume or stock concentration.
4. Pick units for both sides. Use M, mM, µM, mg/mL, ng/µL, ppm, % v/v, X, U/mL or cells/mL. Both concentrations must be the same kind; add the molar mass to mix M with mg/mL.
5. Set the ratio convention. In ratio mode, say whether 1:N means 1 part in N total (antibodies, biology) or 1 part plus N parts (many chemistry texts).
6. Add the tolerances. Pick your Class A pipette and flask, and give the stock uncertainty. The page combines them by the GUM rule.
7. Press Calculate Dilution. Press the button or Enter. The answer, a Verified badge, the working, every step and a results table appear in Section 3.
8. Read the figure and charts. The figure is drawn from your numbers; the four charts show the relationship, a comparison, the volumes and the uncertainty or counting range.
9. Copy the write-up. Section 5 fills five ready sentences for a lab report, label, method, exam and chat; Section 20 fills a methods sentence.
10. Export or run a batch. Download Doc or PDF, or use the Batch tab for many dilutions at once, typed or from a CSV or Excel file.

Excel has no DILUTION function, but you can rebuild this dilution calculator in one row of cells. The sheet repeats the headline problem (1.000 M to 0.1000 M in 250.0 mL) with the uncertainty, so the numbers match Sections 0, 7, 10, 11 and 14.

9.1 The formulas you need

9.2 Step by step

9.3 Awkward cases

• Mixed units: convert first: µM = mM × 1000; or =CONVERT(A2,"ul","ml") for volumes.
• Serial series: put the factor in B1 and fill =$A$2/$B$1^ROW(A1) down six rows.
• Ratios typed as text ("1:10") cannot be multiplied: store the factor (10) in its own cell.

9.4 Why Excel may differ from R or Python

9.5 Charting a serial dilution

Plot tube number against concentration as a Scatter chart, then set the vertical axis to a logarithmic scale (Format Axis, Logarithmic scale). A 10-fold series becomes a straight line falling one decade per tube.

9.6 Errors you may see

• #DIV/0!: the C₁ cell is empty.
• #VALUE!: a unit typed in a number cell ("25 mL").
• A time instead of a ratio: Excel turned 1:10 into 1:10 AM.

A complete, copy-and-run script that reproduces the dilution calculator for the headline problem. It prints the stock volume, the dilution factor, the uncertainty of the solution made, a serial dilution and a plate count back-calculation, and saves one 300 dpi figure.

10.1 The complete script

R, base only, saves serial_dilution.png

# Dilution with GUM uncertainty, a serial dilution and a plate count
# StatsUnlock - https://statsunlock.com/dilution-calculator/   Base R only.

# ---- 1. Inputs (units in the comments) ----
C1 <- 1.000; tol_C1 <- 0.001   # stock, M (+/- 0.1%)
C2 <- 0.1000                   # target, M
V2 <- 250.0; tol_V2 <- 0.12    # final volume, mL (Class A flask, ASTM E288)
tol_V1 <- 0.03                 # Class A 25 mL pipette, mL (ASTM E969)

# ---- 2. The dilution ----
V1 <- C2 * V2 / C1             # mL of stock
DF <- C1 / C2
cat(sprintf("V1 = %.2f mL of stock, make up to %.1f mL (DF = %g, 1 in %g)\n", V1, V2, DF, DF))

# ---- 3. Uncertainty of the solution made: u = a / sqrt(3) ----
rel <- sqrt((tol_C1 / sqrt(3) / C1)^2 + (tol_V1 / sqrt(3) / V1)^2 + (tol_V2 / sqrt(3) / V2)^2)
cat(sprintf("C2 = %.5f +/- %.5f M (k = 2), relative %.3f%%\n", C2, 2 * rel * C2, 200 * rel))
stopifnot(abs(C1 * V1 - C2 * V2) < 1e-12)
cat("Check: C1 x V1 = C2 x V2. Verified.\n")

# ---- 4. Serial dilution: 10-fold, 6 tubes ----
tubes <- 1:6
dil <- 10^tubes
cat("Tube dilutions:", paste0("1e-", tubes), "\n")

# ---- 5. Plate count (FDA BAM Ch. 3: CFU/mL = N / V) ----
N <- 156; tube <- 5; v_plated <- 0.1
V <- v_plated / dil[tube]      # mL of original sample on the plate
cfu <- N / V
cat(sprintf("CFU/mL = %d / %.1e mL = %.2e CFU/mL (counting u = %.1f%%)\n", N, V, cfu, 100 / sqrt(N)))
cat("Countable (15 to 300 colonies):", N >= 15 && N <= 300, "\n")

# ---- 6. One 300 dpi figure ----
png("serial_dilution.png", width = 6, height = 4, units = "in", res = 300)
plot(tubes, 1 / dil, log = "y", type = "b", pch = 19, col = "#2563eb", lwd = 2,
     xlab = "Tube", ylab = "Fraction of original (log scale)", main = "10-fold serial dilution")
points(tube, 1 / dil[tube], pch = 19, cex = 2, col = "#9333ea")
dev.off()

10.2 Output

V1 = 25.00 mL of stock, make up to 250.0 mL (DF = 10, 1 in 10)
C2 = 0.10000 +/- 0.00019 M (k = 2), relative 0.189%
Check: C1 x V1 = C2 x V2. Verified.
Tube dilutions: 1e-1 1e-2 1e-3 1e-4 1e-5 1e-6
CFU/mL = 156 / 1.0e-06 mL = 1.56e+08 CFU/mL (counting u = 8.0%)
Countable (15 to 300 colonies): TRUE

10.3 Traps

• round() rounds half to even; format with sprintf.
• log() is natural; use log10() for decades.
• 10^tubes is a vector; index it with dil[tube], not dil(tube).

10.4 What each part does

10.5 Adapting it

Change the inputs for your stock and target. For a different series, change the factor (10 to 2 for a two-fold series) and the number of tubes. For an antibody, V1 = V2 / N for a 1:N dilution read as 1 in N.

A complete, copy-and-run script that reproduces the dilution calculator for the headline problem. It prints the stock volume, the dilution factor, the uncertainty of the solution made, a serial dilution and a plate count back-calculation, and saves one 300 dpi figure.

Install once: pip install numpy matplotlib

11.1 The complete script

Python with NumPy and Matplotlib, saves serial_dilution.png

# Dilution with GUM uncertainty, a Monte Carlo check, a serial dilution and a plate count
# StatsUnlock - https://statsunlock.com/dilution-calculator/
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt

# ---- 1. Inputs (units in the comments) ----
C1, tol_C1 = 1.000, 0.001        # stock, M (known to +/-0.1%)
C2 = 0.1000                      # target, M
V2, tol_V2 = 250.0, 0.12         # final volume, mL (Class A flask, ASTM E288)
tol_V1 = 0.03                    # Class A 25 mL pipette, mL (ASTM E969)

# ---- 2. The dilution ----
V1 = C2 * V2 / C1                # mL of stock
DF = C1 / C2
print(f"V1 = {V1:.2f} mL of stock, make up to {V2:.1f} mL (DF = {DF:g}, 1 in {DF:g})")

# ---- 3. Uncertainty of the solution made: rectangular tolerances, u = a/sqrt(3) ----
rel = np.sqrt((tol_C1 / np.sqrt(3) / C1) ** 2 + (tol_V1 / np.sqrt(3) / V1) ** 2 + (tol_V2 / np.sqrt(3) / V2) ** 2)
print(f"C2 = {C2:.5f} +/- {2 * rel * C2:.5f} M (k = 2), relative {200 * rel:.3f}%")

# ---- 4. Monte Carlo check (JCGM 101), 100 000 draws ----
rng = np.random.default_rng(1)
sim = (rng.uniform(C1 - tol_C1, C1 + tol_C1, 100_000) * rng.uniform(V1 - tol_V1, V1 + tol_V1, 100_000)
       / rng.uniform(V2 - tol_V2, V2 + tol_V2, 100_000))
print(f"Monte Carlo: SD = {sim.std(ddof=1):.2e} M, 95% interval {np.percentile(sim, 2.5):.5f} to {np.percentile(sim, 97.5):.5f} M")
assert abs(C1 * V1 - C2 * V2) < 1e-12
print("Check: C1 x V1 = C2 x V2. Verified.")

# ---- 5. Serial dilution: 10-fold, 6 tubes, 1.0 mL into 9.0 mL ----
tubes = np.arange(1, 7)
dil = 10.0 ** tubes
print("Tube dilutions:", ", ".join(f"1e-{k}" for k in tubes))

# ---- 6. Plate count back to the original (FDA BAM Ch. 3: CFU/mL = N / V) ----
N, tube, v_plated = 156, 5, 0.1  # colonies, tube number, mL plated
V = v_plated / dil[tube - 1]     # mL of original sample on the plate
cfu = N / V
print(f"CFU/mL = {N} / {V:.1e} mL = {cfu:.2e} CFU/mL (counting u = {100 / np.sqrt(N):.1f}%)")
print("Countable (15 to 300 colonies):", 15 <= N <= 300)

# ---- 7. One 300 dpi figure ----
fig, ax = plt.subplots(figsize=(6, 4))
ax.semilogy(tubes, 1.0 / dil, "o-", color="#2563eb", lw=2, label="fraction of original")
ax.semilogy(tube, 1.0 / dil[tube - 1], "o", color="#9333ea", ms=11, label=f"plated tube: {cfu:.2e} CFU/mL")
ax.set_xlabel("Tube"); ax.set_ylabel("Fraction of original (log scale)"); ax.legend(frameon=False)
fig.tight_layout(); fig.savefig("serial_dilution.png", dpi=300)

11.2 Output

V1 = 25.00 mL of stock, make up to 250.0 mL (DF = 10, 1 in 10)
C2 = 0.10000 +/- 0.00019 M (k = 2), relative 0.189%
Monte Carlo: SD = 9.41e-05 M, 95% interval 0.09982 to 0.10018 M
Check: C1 x V1 = C2 x V2. Verified.
Tube dilutions: 1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6
CFU/mL = 156 / 1.0e-06 mL = 1.56e+08 CFU/mL (counting u = 8.0%)
Countable (15 to 300 colonies): True

11.3 Traps

• round() rounds half to even; format with f-strings.
• Integer powers: 10 ** tubes needs a float base (10.0) to avoid integer overflow on long series.
• numpy.std() defaults to ddof=0; use ddof=1 for replicates.

11.4 What each part does

11.5 Adapting it

Change the inputs for your stock and target. For a different series, change the factor (10 to 2 for a two-fold series) and the number of tubes. For an antibody, V1 = V2 / N for a 1:N dilution read as 1 in N.

12.1 Common dilution factors: stock and diluent for 1 mL and 10 mL

Exact arithmetic (DF = V₂ ÷ V₁). Last checked: October 2026.

12.2 Serial dilution set-ups

FDA BAM Chapter 3 (Aerobic Plate Count, January 2026 edition) uses 90 ± 1 mL Butterfield's phosphate-buffered water blanks and 10 mL transfers. Last checked: October 2026.

12.3 Class A volumetric (transfer) pipettes

Source: Class A volumetric pipets to ASTM E969 and E542, as listed by Fisher Scientific (reusable Class A volumetric pipets). Last checked: October 2026.

12.4 Class A volumetric flasks

Source: ASTM E288 Class A, as listed by DWK Life Sciences (50 to 1000 mL) and Eisco Labs ASTM listings (10 and 25 mL). Calibrated to contain at 20 °C. Last checked: October 2026.

12.5 Plate counts: countable range and counting error

Source: FDA Bacteriological Analytical Manual, Chapter 3, Aerobic Plate Count (January 2026), which counts 15 to 300 colonies per plate and cites APHA's 25 to 250; counting statistics from Breed and Dotterrer (1916). Last checked: October 2026.

12.6 Concentrated acids and bases for dilution

Source: concentrated reagent table, George Mason University, Department of Chemistry and Biochemistry (S. Slayden). Read your bottle; lots vary. Last checked: October 2026.

12.7 X buffers and percent solutions

C₁V₁ = C₂V₂; 1 US gallon = 3.785 411 784 L exactly (NIST Handbook 44, Appendix C). Volumes of alcohol and water do not add: make up to the mark. Last checked: October 2026.

Eight worked problems that cover what a dilution calculator meets: textbook, AP lab, clinical, EPA water lab, US units, a limit case, an impossible request and a classic trap. Each number was checked in Python.

The written method, calculator keys and a short standard operating procedure for the headline problem: prepare 250.0 mL of 0.1000 M sodium chloride from a 1.000 M stock. School students do this under a teacher's supervision and the school's own risk assessment.

14.1 The written method

1. Write the equation. C₁V₁ = C₂V₂.
2. Check units. Both concentrations in M; both volumes in mL.
3. Rearrange. V₁ = C₂V₂ ÷ C₁.
4. Substitute. V₁ = 0.1000 M × 250.0 mL ÷ 1.000 M = 25.00 mL.
5. Dilution factor. DF = 1.000 ÷ 0.1000 = 10 (1 in 10).
6. Check. 1.000 × 25.00 = 25.00 = 0.1000 × 250.0 ✓

14.2 On a scientific calculator

14.3 The lab SOP

14.4 Practice problems

14.5 Mistakes that lose marks or fail an audit

• Different units for C₁ and C₂ (mM against µM).
• Adding factors across steps instead of multiplying.
• Reading 1:10 as 1 + 10 parts when the protocol means 1 in 10.
• Adding a fixed volume of water instead of making up to the mark.
• Pipetting below 2 µL instead of using an intermediate dilution.
• Reporting the diluted reading instead of the original (forgetting × DF).
• Water into acid instead of acid into water.

14.6 Pre-Lab and Post-Lab Questions

Research on chemistry students finds that most dilution errors come from misusing C₁V₁ = C₂V₂ and from final-concentration problems with mixtures, not from arithmetic (Chong, 2016; Vaccaro et al., 2022). The dilution calculator flags each one below.

Better methods for the highest accuracy: prepare dilutions by mass (gravimetric dilution), calibrate pipettes gravimetrically under ISO 8655-6, and count cells with a validated automated counter (Cadena-Herrera et al., 2015).

Your result

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What the dilution calculator does for the bench

Every dilution is the same equation, but the units, ratio conventions, multi-step factors and tiny volumes are where results go wrong. This dilution calculator keeps the units consistent, reads ratios both ways, multiplies steps, back-calculates readings and plate counts, and puts an uncertainty on the solution you actually make.

Watch out for

• Mixed units on the two sides of C₁V₁ = C₂V₂.
• The 1:N convention on the protocol you follow.
• Transfers below about 2 µL.
• Volumes that do not add: make up to the mark.
• Acid into water, never the reverse.

Next steps

Check a working solution by measurement and multiply by the dilution factor. For replicate readings, the mean calculator and standard deviation calculator give the average and spread; for a standard curve from a serial dilution, use the simple linear regression calculator. Whatever you dilute, start and finish with the dilution calculator.

Use these to report the statistics behind your dilutions: replicate spread, standard curves and counting error.

Peer-reviewed sources behind this dilution calculator: serial dilution and plate count statistics, cell counting, qPCR dilution series, nucleic acid and protein quantitation, uncertainty, and research on how students learn dilution. Standards (JCGM 100:2008, ASTM E969 and E288, FDA BAM Chapter 3, OSHA HazCom) are named where they are used. Every DOI checked October 2026.

1. Altman, D. G., & Bland, J. M. (2005). Standard deviations and standard errors. BMJ, 331(7521), 903. https://doi.org/10.1136/bmj.331.7521.903
2. Ben-David, A., & Davidson, C. E. (2014). Estimation method for serial dilution experiments. Journal of Microbiological Methods, 107, 214–221. https://doi.org/10.1016/j.mimet.2014.08.023
3. Breed, R. S., & Dotterrer, W. D. (1916). The number of colonies allowable on satisfactory agar plates. Journal of Bacteriology, 1(3), 321–331. https://doi.org/10.1128/jb.1.3.321-331.1916
4. Bustin, S. A., Benes, V., Garson, J. A., Hellemans, J., Huggett, J., Kubista, M., Mueller, R., Nolan, T., Pfaffl, M. W., Shipley, G. L., Vandesompele, J., & Wittwer, C. T. (2009). The MIQE guidelines: Minimum information for publication of quantitative real-time PCR experiments. Clinical Chemistry, 55(4), 611–622. https://doi.org/10.1373/clinchem.2008.112797
5. Cadena-Herrera, D., Esparza-De Lara, J. E., Ramírez-Ibañez, N. D., López-Morales, C. A., Pérez, N. O., Flores-Ortiz, L. F., & Medina-Rivero, E. (2015). Validation of three viable-cell counting methods: Manual, semi-automated, and automated. Biotechnology Reports, 7, 9–16. https://doi.org/10.1016/j.btre.2015.04.004
6. Chong, S. H. (2016). Wither the concepts of mole and concentration: Conceptual confusion in applying M₁V₁ = M₂V₂. Universal Journal of Educational Research, 4(5), 1158–1162. https://doi.org/10.13189/ujer.2016.040527
7. Cumming, G., Fidler, F., & Vaux, D. L. (2007). Error bars in experimental biology. Journal of Cell Biology, 177(1), 7–11. https://doi.org/10.1083/jcb.200611141
8. Desjardins, P., & Conklin, D. (2010). NanoDrop microvolume quantitation of nucleic acids. Journal of Visualized Experiments, (45), e2565. https://doi.org/10.3791/2565
9. Gültepe, N., & Erdem, A. R. (2026). Diagnosing teacher candidates’ reasoning on solution concentration: A theoretical analysis of definitional and procedural difficulties. Chemistry Education Research and Practice, 27(3), 1073–1088. https://doi.org/10.1039/d5rp00422e
10. Herigstad, B., Hamilton, M., & Heersink, J. (2001). How to optimize the drop plate method for enumerating bacteria. Journal of Microbiological Methods, 44(2), 121–129. https://doi.org/10.1016/S0167-7012(00)00241-4
11. Jett, B. D., Hatter, K. L., Huycke, M. M., & Gilmore, M. S. (1997). Simplified agar plate method for quantifying viable bacteria. BioTechniques, 23(4), 648–650. https://doi.org/10.2144/97234bm22
12. Kragten, J. (1994). Calculating standard deviations and confidence intervals with a universally applicable spreadsheet technique. Analyst, 119(10), 2161–2165. https://doi.org/10.1039/AN9941902161
13. Pace, C. N., Vajdos, F., Fee, L., Grimsley, G., & Gray, T. (1995). How to measure and predict the molar absorption coefficient of a protein. Protein Science, 4(11), 2411–2423. https://doi.org/10.1002/pro.5560041120
14. Prohaska, T., Irrgeher, J., Benefield, J., Böhlke, J. K., Chesson, L. A., Coplen, T. B., Ding, T., Dunn, P. J. H., Gröning, M., Holden, N. E., Meijer, H. A. J., Moossen, H., Possolo, A., Takahashi, Y., Vogl, J., Walczyk, T., Wang, J., Wieser, M. E., Yoneda, S., Zhu, X.-K., & Meija, J. (2022). Standard atomic weights of the elements 2021 (IUPAC Technical Report). Pure and Applied Chemistry, 94(5), 573–600. https://doi.org/10.1515/pac-2019-0603
15. Raviolo, A., Farré, A. S., & Traiman Schroh, N. (2021). Students’ understanding of molar concentration. Chemistry Education Research and Practice, 22(2), 486–497. https://doi.org/10.1039/d0rp00344a
16. Rusek, M., Koperová, D., Bártová, I., Machková, V., Chytrý, V., Sirotek, V., Štrofová, J., & Šmejkal, P. (2026). Challenges in chemistry calculations at university entry: Evidence from a multi-institutional study. Journal of Chemical Education, 103(4), 1826–1834. https://doi.org/10.1021/acs.jchemed.6c00022
17. Strober, W. (2015). Trypan blue exclusion test of cell viability. Current Protocols in Immunology, 111(1). https://doi.org/10.1002/0471142735.ima03bs111
18. Svec, D., Tichopad, A., Novosadova, V., Pfaffl, M. W., & Kubista, M. (2015). How good is a PCR efficiency estimate: Recommendations for precise and robust qPCR efficiency assessments. Biomolecular Detection and Quantification, 3, 9–16. https://doi.org/10.1016/j.bdq.2015.01.005
19. Tanaka, M., Girard, G., Davis, R., Peuto, A., & Bignell, N. (2001). Recommended table for the density of water between 0 °C and 40 °C based on recent experimental reports. Metrologia, 38(4), 301–309. https://doi.org/10.1088/0026-1394/38/4/3
20. Vaccaro, E., Stella, C., & Alonso, M. (2022). Difficulties of novice students in solving the final concentration value of a mixture of solutions. Chemistry Teacher International, 4(4), 297–305. https://doi.org/10.1515/cti-2021-0026