Calculate results from titrations and gravimetry, compare methods statistically, read chromatograms and spectra, validate methods to ICH Q2, plan sampling, and work in GMP, GLP, EPA and ISO 17025 labs.

Analytical Chemistry Methods

## Why it matters

Titration is one of the oldest quantitative methods, and it is still everywhere: acid content of foods and wines, water hardness, the assay of a drug substance in a pharmacopoeia, the moisture in a solvent by Karl Fischer titration. It is cheap, accurate to a few parts per thousand in good hands, and traceable to a weighed primary standard. Every analytical chemist is expected to calculate a titration result correctly and to know where its errors come from.

> **Safety and scope.** Acids, bases, silver salts and other reagents in this lesson are hazardous. Bench work happens only in a teaching or employer lab, with your instructor or supervisor, after the lab's chemical-hygiene training ([OSHA laboratories](https://www.osha.gov/laboratories), checked on 2026-10-23). The activity is a data exercise: never attempt these procedures at home.

## The core idea

A **titrant** of known concentration is added from a burette to a measured volume (or mass) of **analyte** until the reaction between them is complete.

- The **equivalence point** is where the moles of titrant exactly match the moles of analyte, according to the balanced equation.
- The **endpoint** is what you actually observe: an indicator color change, or a jump in a pH or potential curve. A good method makes the endpoint fall as close to the equivalence point as possible.

The calculation is stoichiometry:

```text
moles titrant = M_titrant x V_titrant (L)
moles analyte = moles titrant x (mol analyte / mol titrant)
C_analyte     = moles analyte / V_analyte (L)
```

Because both volumes are in the same unit, you can work in mL directly: C = M x V_titrant x ratio / V_analyte.

## Worked example 1: a monoprotic acid

25.00 mL of hydrochloric acid is titrated with **0.1000 M NaOH**; the endpoint is at **21.40 mL**. HCl + NaOH -> NaCl + H2O, so the ratio is 1.

C(HCl) = 0.1000 x 21.40 x 1 / 25.00 = **0.08560 M**

## Worked example 2: a diprotic acid

25.00 mL of sulfuric acid needs **18.72 mL** of the same NaOH. H2SO4 + 2 NaOH -> Na2SO4 + 2 H2O, so there is 1 mol of acid per 2 mol of titrant: ratio = 0.5.

C(H2SO4) = 0.1000 x 18.72 x 0.5 / 25.00 = **0.03744 M**

Getting this ratio upside down (using 2 instead of 0.5) is the most common titration error on exams and in notebooks.

## Standardization and blanks

NaOH solutions absorb carbon dioxide and change concentration, so they are **standardized** against a primary standard such as **potassium hydrogen phthalate** (KHP, 204.22 g/mol, monoprotic). If 0.5105 g of KHP needs 24.85 mL of NaOH:

M(NaOH) = (0.5105 / 204.22) / (24.85 / 1000) = **0.1006 M**

A **blank titration** (the same procedure with no analyte) measures how much titrant the solvent and indicator consume. Subtract the blank volume from each sample volume before calculating.

## Endpoint detection and titration types

- **Acid-base:** indicators such as phenolphthalein, or a pH electrode (potentiometric titration), where the endpoint is the steepest point of the curve (the maximum of the first derivative).
- **Redox:** for example permanganate or iodine-thiosulfate titrations.
- **Complexometric:** EDTA titrations for calcium and magnesium (water hardness).
- **Precipitation:** silver nitrate for chloride.
- **Back titration:** add a known excess of reagent, then titrate what is left. Useful when the analyte reacts slowly or is insoluble.

Modern labs use **autotitrators** that dose titrant in small steps, record the curve and pick the endpoint mathematically, but the chemistry and the calculation are the same.

## Replicates and reporting

Titrate at least three replicates. Report the mean with an appropriate number of significant figures and the relative standard deviation (RSD). Replicates that agree within a few tenths of a percent are typical for a well-run manual titration.

## Common mistakes

- Using the wrong mole ratio for polyprotic acids or redox reactions.
- Reading the burette at the top of the meniscus instead of the bottom, or with parallax.
- Forgetting the blank correction.
- Using an indicator whose color change does not match the pH at equivalence (for example methyl orange for a weak acid with a strong base).
- Mixing units: milliliters in one place and liters in another.

## Quick recap

- C_analyte = M_titrant x V_titrant x (mol analyte / mol titrant) / V_analyte.
- Standardize titrants against a primary standard such as KHP; subtract blanks.
- The endpoint approximates the equivalence point; potentiometric detection uses the steepest part of the curve.
- Run replicates and report the mean and RSD.

## Try it

Open the **Do** tab. A Python titration calculator is written except for the concentration formula.


Calculate an analyte's concentration from titration data, including mole ratios, standardization and blanks.

Titrations

Practice Problem: Titration Calculations

Acid Base Titration Problems, Basic Introduction, Calculations, Examples, Solution Stoichiometry

## Why it matters

Before instruments, chemists measured composition with a balance. **Gravimetric analysis** is still used where accuracy and traceability matter most: sulfate in cement and water, loss on drying and residue on ignition in pharmaceuticals, ash and moisture in foods, and reference measurements that calibrate other methods. Because the result depends on masses and molar masses, a good gravimetric method needs no calibration curve. Classical methods also teach the habits (constant mass, blanks, stoichiometric factors) that every instrumental method relies on.

> **Safety and scope.** Acids, bases, silver salts and other reagents in this lesson are hazardous. Bench work happens only in a teaching or employer lab, with your instructor or supervisor, after the lab's chemical-hygiene training ([OSHA laboratories](https://www.osha.gov/laboratories), checked on 2026-10-23). The activity is a data exercise: never attempt these procedures at home.

## The core idea

In **precipitation gravimetry**, the analyte is converted into a sparingly soluble compound of known composition, which is filtered, washed, dried or ignited, and weighed:

1. Dissolve a weighed sample.
2. Add a precipitating reagent in slight excess (for example silver nitrate to precipitate chloride as AgCl).
3. Digest the precipitate (keep it warm) so particles grow larger and purer.
4. Filter through a weighed crucible, wash away soluble impurities.
5. Dry or ignite **to constant mass**: weigh, heat again, weigh again until two weighings agree within a set tolerance (often 0.3 mg).

The result uses a **gravimetric factor (GF)** that converts the mass of precipitate into the mass of analyte:

```text
GF = (molar mass of analyte x a) / (molar mass of precipitate x b)
% analyte = mass of precipitate x GF / mass of sample x 100
```

where a and b balance the atoms of interest (for chloride as AgCl, a = b = 1).

## Worked example: chloride as AgCl

A 0.5012 g sample gives **0.2866 g** of AgCl.

- GF = 35.45 / 143.32 = **0.2474**
- Mass of Cl = 0.2866 x 0.2474 = 0.07090 g
- % Cl = 0.07090 / 0.5012 x 100 = **14.14 %**

## Worked example: a factor with a ratio

Iron is often weighed as Fe2O3 after ignition. Each Fe2O3 contains two iron atoms:

- GF = (2 x 55.845) / 159.69 = **0.6994**
- If 0.3510 g of Fe2O3 comes from a 1.2000 g ore sample: % Fe = 0.3510 x 0.6994 / 1.2000 x 100 = **20.46 %**

Forgetting the 2 would halve the answer.

## Other classical methods

- **Volatilization gravimetry:** measure mass lost or gained on heating. **Loss on drying** gives moisture; **residue on ignition** (sulfated ash) gives inorganic content in drug substances.
- **Kjeldahl nitrogen:** digest a sample in sulfuric acid to convert organic nitrogen to ammonium, distill the ammonia, and titrate it. Protein is estimated as nitrogen x a conversion factor (6.25 for many foods). It is classical chemistry, but it is still a regulatory reference method.
- **Electrogravimetry:** deposit a metal onto a weighed electrode and weigh it.

## Sources of error

- **Solubility losses:** some precipitate dissolves, especially with too much wash water. Washing with a dilute electrolyte reduces this.
- **Co-precipitation:** impurities trapped in or on the precipitate. Digestion and reprecipitation help.
- **Incomplete drying** or **hygroscopic** precipitates gaining water from the air; cool in a desiccator before weighing.
- **Light sensitivity:** AgCl decomposes slowly in light, so it is kept in subdued light.
- **Balance technique:** weigh warm crucibles and convection currents give false readings. Always let them cool to room temperature.

A **reagent blank** carried through the whole procedure corrects for any precipitate formed from impurities in the reagents.

## Common mistakes

- Using the wrong atoms ratio in the gravimetric factor.
- Weighing once and assuming constant mass.
- Reporting more significant figures than the balance or sample mass supports.
- Skipping the blank.

## Quick recap

- Precipitate, digest, filter, wash, dry or ignite to constant mass, weigh.
- GF = (M analyte x a) / (M precipitate x b); % analyte = mass precipitate x GF / sample mass x 100.
- Loss on drying, residue on ignition and Kjeldahl nitrogen are classical methods still used in regulated labs.
- Control solubility, co-precipitation, moisture and blanks.

## Try it

Open the **Do** tab. A Python calculator is written except for the percent-analyte formula.


Compute a result from mass data with a gravimetric factor and know the classical methods still used in regulated labs.

Gravimetric and classical methods

Practice Problem: Gravimetric Analysis

## Why it matters

Every analytical result is a number with uncertainty. When a lab replaces an old method with a faster one, changes an instrument, or compares its results with another lab, someone must answer: **do the two methods give the same results, or is there a real difference?** Answering that with an eyeball is how labs drift. Answering it with the right statistical test, and the right sample design, is a core skill for analytical chemists. This lesson builds on `statistics-fundamentals` and `lab-data-quality`.

## Describing replicates

For n replicate results x1 ... xn:

- **Mean** (x-bar): the best estimate of the value.
- **Standard deviation** (s): the spread, with n - 1 in the denominator.
- **Relative standard deviation** (RSD, %): 100 x s / mean. Useful for comparing precision across concentrations.
- **Confidence interval** of the mean: mean +/- t x s / sqrt(n), where t comes from the t table for n - 1 degrees of freedom at the chosen confidence (95 % is typical).

## Accuracy vs precision

- **Precision** is how close replicates are to each other (random error).
- **Accuracy** is how close the result is to the true value. **Bias** (systematic error) is the difference between the mean and a reference value, often measured with a certified reference material (CRM).

A method can be precise and biased at the same time. Statistical tests help you separate a real bias from random scatter.

## Choosing the test

| Question | Test |
|---|---|
| Does my method's mean agree with a certified value? | One-sample t-test |
| Are two methods' precisions different? | F-test (ratio of variances) |
| Do two methods give the same mean on **independent** samples? | Two-sample t-test (pooled if variances are similar) |
| Do two methods give the same results on the **same** set of samples? | **Paired t-test** |

The **paired design** is the workhorse of method comparison. If you measure six different real samples with both methods, the samples have very different concentrations, so the between-sample spread would swamp any method difference in a two-sample test. Pairing removes that: you analyze the **differences** for each sample.

## Worked example: a paired comparison

Six fictional samples (mg/L) by the reference method A and a new method B:

| Sample | A | B | d = B - A |
|---|---|---|---|
| 1 | 10.2 | 10.4 | 0.2 |
| 2 | 12.5 | 12.9 | 0.4 |
| 3 | 8.9 | 9.0 | 0.1 |
| 4 | 15.1 | 15.6 | 0.5 |
| 5 | 11.3 | 11.5 | 0.2 |
| 6 | 9.8 | 10.1 | 0.3 |

- Mean difference d-bar = **0.283** mg/L; s_d = **0.147** mg/L; n = 6.
- t = d-bar / (s_d / sqrt(n)) = 0.283 / 0.0601 = **4.71**
- Critical t (two-tailed, 95 %, 5 degrees of freedom) = **2.571**.

Because |t| = 4.71 > 2.571, the difference is statistically significant: method B reads about 0.28 mg/L high. The next question is practical: is 0.28 mg/L acceptable for the method's purpose? Statistics tells you a difference is real; the **acceptance criteria** (set before the study) tell you whether it matters.

## Beyond the t-test

- When comparing methods across a wide concentration range, labs also use **regression** of B against A (slope near 1 and intercept near 0 means agreement) and **Bland-Altman** plots of difference vs mean. Ordinary least squares assumes A has no error; Deming or Passing-Bablok regression handle error in both methods.
- **Outlier tests** (Grubbs, Dixon's Q) exist, but an outlier is only rejected with a documented reason, never just because it is inconvenient (see `lab-data-quality`).
- In Python, `scipy.stats.ttest_rel` performs a paired t-test. The activity uses only the standard library so you see the formula.

## Common mistakes

- Using a two-sample test on paired data, which hides a real bias under sample-to-sample variation.
- Confusing statistical significance with practical importance.
- Running many tests and reporting only the one with p < 0.05.
- Comparing methods on too few samples, or only at one concentration.

## Quick recap

- Mean, s, RSD and confidence intervals describe replicates.
- Bias is systematic; precision is random scatter.
- Same samples measured by two methods: use a **paired t-test** on the differences.
- Decide acceptance criteria before the study.

## Try it

Open the **Do** tab. A Python method-comparison script is written except for the paired t statistic.


Describe replicate data, separate bias from scatter, and compare two methods with a paired t-test.

Statistics for analysts

Student's t-test

Statistics in Chemical Measurements - t-Test, F-test - Part 1 - The Analytical Chemistry Process

Calculate results from titrations and gravimetric data, and compare methods with the statistics analysts use.