Solution stoichiometry: molarity, titration, and gravimetric analysis

Intuition Beginner

Many chemical reactions happen in solution -- dissolved in water or another solvent. To do stoichiometry on these reactions, you need a way to count moles of dissolved substances. That is what molarity does.

Molarity ($M$) is moles of solute per litre of solution: $M=n/V$. A 1.00 $M$ solution of NaCl contains 1.00 mol of NaCl dissolved in enough water to make exactly 1.00 L of solution. You can think of molarity as the "density of moles" -- how many moles are packed into each litre.

When you dilute a solution by adding more solvent, the moles of solute do not change. Only the volume increases. This gives the dilution equation: $M_1{V}_1=M_2{V}_2$, where the subscripts denote "before" and "after" dilution. The product $M\times V$ always equals moles, and moles are conserved during dilution.

Solution stoichiometry works just like the stoichiometry you already know. Convert volume and molarity to moles, use the balanced equation's mole ratios, then convert back. The only difference is the starting point: instead of weighing a solid, you measure the volume of a solution of known concentration.

A titration is a technique that uses solution stoichiometry to find an unknown concentration. You slowly add a solution of known concentration (the titrant) to a solution of unknown concentration (the analyte) until the reaction between them is exactly complete. That point is called the equivalence point. By measuring how much titrant was needed, you can calculate the unknown concentration.

Visual Beginner

The key relationships in solution stoichiometry form a triangle:

Quantity	Symbol	Unit
Moles of solute	$n$	mol
Molarity	$M$	mol/L
Volume of solution	$V$	L

The equation connecting them: $n=M\times V$. Given any two, you find the third.

In a titration curve (pH vs. volume of titrant added), the equivalence point appears as a sharp vertical step. For a strong acid--strong base titration, the equivalence-point pH is exactly 7.00. The indicator is a dye that changes colour near the equivalence point so you can see when to stop.

Worked example Beginner

You titrate 25.00 mL of an unknown HCl solution with 0.100 $M$ NaOH. The endpoint is reached after 22.40 mL of NaOH has been added. What is the molarity of the HCl?

Step 1. Write the balanced equation:

$$\text{HCl}+\text{NaOH}\to \text{NaCl}+{\text{H}}_2\text{O}$$

The mole ratio is 1:1.

Step 2. Calculate moles of NaOH used:

$$n(\text{NaOH})=M\times V=0.100\text{mol/L}\times 0.02240\text{L}=0.002240\text{mol}$$

Step 3. Use the 1:1 ratio to find moles of HCl:

$$n(\text{HCl})=0.002240\text{mol}$$

Step 4. Calculate the molarity of HCl:

$$M(\text{HCl})=n/V=0.002240/0.02500=0.0896\text{mol/L}$$

The HCl solution is 0.0896 $M$.

Check your understanding Beginner

Formal definition Intermediate+

Molarity. The molar concentration of a solute $A$ in a solution is

$$c_A=\frac{n_A}{V_{\text{solution}}},$$

where $n_A$ is the amount of solute in moles and $V_{\text{solution}}$ is the total volume of the solution (not the volume of solvent added). Molarity has SI units of mol/L, also written as $M$. Because volume depends on temperature, molarity is temperature-dependent.

Dilution. When a solution of initial concentration $c_1$ and volume $V_1$ is diluted to a new volume $V_2$, the amount of solute is conserved:

$$c_1{V}_1=c_2{V}_2.$$

This is not a new physical law; it is the statement that $n=c_1{V}_1=c_2{V}_2$ because dilution changes only the volume, not the moles.

Solution stoichiometry. For a reaction $aA+bB\to cC+dD$ occurring in solution, the stoichiometric relationship between two reactants is

$$\frac{n_A}{a}=\frac{n_B}{b},$$

where $n_A=c_A{V}_A$ and $n_B=c_B{V}_B$. The limiting-reagent logic from 14.03.01 applies identically: whichever reactant provides the smaller stoichiometrically-scaled amount of product is limiting.

Equivalence point. In a titration, the equivalence point is reached when the moles of titrant added exactly satisfy the stoichiometric ratio with the moles of analyte:

$$\frac{n_{\text{titrant}}}{{\nu }_{\text{titrant}}}=\frac{n_{\text{analyte}}}{{\nu }_{\text{analyte}}}.$$

The endpoint is the experimentally observed indicator colour change, which ideally coincides with the equivalence point but may differ slightly. The titration error is the difference between the endpoint and the true equivalence point.

Gravimetric analysis. The analyte is converted to an insoluble product of known composition, which is filtered, dried, and weighed. The mass of the precipitate and its known stoichiometric relationship to the analyte determine the amount of analyte:

$$n_{\text{analyte}}=\frac{{\nu }_{\text{analyte}}}{{\nu }_{\text{precipitate}}}\cdot \frac{m_{\text{precipitate}}}{M_{\text{precipitate}}}.$$

Other concentration units

Molality $b=n_{\text{solute}}/m_{\text{solvent}}$ (mol per kg of solvent, not solution) is temperature-independent because mass does not change with temperature. Mass fraction $w=m_{\text{solute}}/m_{\text{solution}}$ and mole fraction $x_A=n_A/n_{\text{total}}$ are also temperature-independent. These units are preferred in colligative-property calculations and thermodynamic contexts [see 14.09.01].

Parts per million (ppm) and parts per billion (ppb) are used for trace concentrations. For dilute aqueous solutions, 1 ppm $\approx$ 1 mg/L and 1 ppb $\approx$ 1 $\mu$g/L.

Counterexamples to common slips

• "Molarity equals moles of solute divided by volume of solvent." Wrong. Molarity divides by the total volume of the solution after mixing. Adding 1 mol of solute to 1 L of water produces a solution whose volume is not exactly 1 L due to volume changes on mixing.

• "The equivalence point always coincides with the endpoint." The equivalence point is a stoichiometric condition; the endpoint is an experimental observation based on an indicator. A good indicator is chosen so that these nearly coincide, but they are distinct concepts.

• "Diluting an acid is just adding water to the acid." For concentrated acids (especially ${\text{H}}_2{\text{SO}}_4$), always add the acid to the water, never the reverse. Adding water to concentrated acid releases large amounts of heat and can cause violent spattering. The dilution equation is the same either way, but the safe procedure matters.

• "All precipitation reactions go to completion." Precipitation removes product from solution, driving the reaction forward, but the precipitate has a finite solubility product $K_{sp}$. Some product always remains dissolved. In gravimetric analysis the $K_{sp}$ must be small enough that the dissolved fraction is negligible compared to the weighing precision.

Key theorem with proof Intermediate+

Theorem (Titration stoichiometry). Consider the titration of an analyte $A$ with a titrant $T$ according to the balanced equation

$${\nu }_{A}A+{\nu }_{T}T\to \text{products}.$$

If $V_A$ litres of analyte at concentration $c_A$ are titrated to the equivalence point with titrant at concentration $c_T$, the volume of titrant required is

$$V_T=\frac{{\nu }_T}{{\nu }_A}\cdot \frac{c_A{V}_A}{c_T}.$$

Proof. At the equivalence point, the stoichiometric ratio is satisfied:

$$\frac{n_A}{{\nu }_A}=\frac{n_T}{{\nu }_T}.$$

Substituting $n_A=c_A{V}_A$ and $n_T=c_T{V}_T$:

$$\frac{c_A{V}_A}{{\nu }_A}=\frac{c_T{V}_T}{{\nu }_T}.$$

Solving for $V_T$:

$$V_T=\frac{{\nu }_T}{{\nu }_A}\cdot \frac{c_A{V}_A}{c_T}.\square$$

Corollary (Limiting reagent in solution). If the titrant is added in excess (past the equivalence point), the analyte is the limiting reagent. If the titrant is insufficient, the titrant is limiting. At the equivalence point, neither is in excess.

Worked example at intermediate level

A 0.5143 g sample of an unknown acid is dissolved in water and titrated with 0.1000 $M$ NaOH. The endpoint is reached at 28.47 mL. Is the acid monoprotic or diprotic, and what is its molar mass?

Moles of NaOH: $n=0.1000\times 0.02847=0.002847\text{mol}$.

If monoprotic (HA + NaOH $\to$ NaA + ${\text{H}}_2\text{O}$, ratio 1:1):

$M_{\text{HA}}=0.5143/0.002847=180.6\text{g/mol}$.

If diprotic (${\text{H}}_2\text{A}$ + 2 NaOH $\to$ ${\text{Na}}_2\text{A}$ + 2 ${\text{H}}_2\text{O}$, ratio 1:2):

$n({\text{H}}_2\text{A})=0.002847/2=0.001424\text{mol}$.

$M_{{\text{H}}_2\text{A}}=0.5143/0.001424=361.2\text{g/mol}$.

A molar mass of 180.6 g/mol matches a known monoprotic acid (ascorbic acid, ${\text{C}}_6{\text{H}}_8{\text{O}}_6$, $M=176.12$; the discrepancy suggests a slightly impure sample or a different acid). A molar mass of 361.2 g/mol has no common match. The acid is most likely monoprotic.

Bridge. Solution stoichiometry builds directly toward 14.09.01 where molarity and molality define the composition of solutions for colligative-property calculations (boiling-point elevation, freezing-point depression, osmotic pressure). The titration framework also connects to 14.10.01 where the pH calculations at each stage of a titration curve are derived from acid-base equilibrium expressions. The gravimetric technique relies on precipitation equilibria governed by $K_{sp}$, a concept developed further in the solutions-and-phase unit. Redox titrations connect to 14.11.01 electrochemistry, where the Nernst equation provides the theoretical basis for potentiometric endpoints.

Exercises Intermediate+

Titration theory and indicator selection Master

A titration curve plots pH against the volume of titrant added. The shape of the curve depends on the strength of the acid and base involved. For each combination, the calculation at a given point in the titration uses different equilibrium expressions.

Strong acid--strong base

The reaction ${\text{H}}^{+}+{\text{OH}}^{-}\to {\text{H}}_2\text{O}$ has $K=1/(K_w)=10^{14}$ at 25$°$C, so the reaction is essentially complete. Before the equivalence point, excess ${\text{H}}^{+}$ determines the pH. After, excess ${\text{OH}}^{-}$ determines the pH. At the equivalence point, the solution is neutral (pH = 7.00 at 25$°$C) because neither the cation of the strong base nor the anion of the strong acid hydrolyses appreciably.

The pH change near the equivalence point is abrupt. For the titration of 50.00 mL of 0.100 $M$ HCl with 0.100 $M$ NaOH, the pH jumps from 3.00 (at 49.95 mL) to 11.00 (at 50.05 mL) -- a change of 8 pH units for 0.10 mL of titrant. This steep transition is what makes indicator-based detection possible.

Weak acid--strong base

Consider the titration of a weak acid HA ($K_a$) with NaOH. Three regions of the curve require different calculations:

1. Before titration. The pH is determined by the weak-acid dissociation: $[{\text{H}}^{+}]\approx \sqrt{K_a{c}_0}$ (assuming $c_0/K_a>100$).

2. Buffer region (before equivalence point). The solution contains both HA and its conjugate base ${\text{A}}^{-}$. The Henderson-Hasselbalch equation applies: $\text{pH}=\text{p}{K}_a+\log ([{\text{A}}^{-}]/[\text{HA}])$. At the half-equivalence point ($V_T=V_{\text{eq}}/2$), $\text{pH}=\text{p}{K}_a$.

3. At the equivalence point. All HA has been converted to ${\text{A}}^{-}$, a weak base that hydrolyses: ${\text{A}}^{-}+{\text{H}}_2\text{O}\rightleftharpoons \text{HA}+{\text{OH}}^{-}$. The pH is above 7, typically 8--10, depending on $K_a$ and the concentration of ${\text{A}}^{-}$ (which is diluted by the titrant volume).

4. After the equivalence point. Excess ${\text{OH}}^{-}$ dominates and the pH is determined by the concentration of excess strong base.

Indicator selection

An indicator is a weak acid (or base) whose protonated and deprotonated forms have different colours. The colour transition occurs over approximately 2 pH units centered on the indicator's $\text{p}{K}_{\text{In}}$:

$$\text{pH}=\text{p}{K}_{\text{In}}\pm 1.$$

The indicator must be chosen so that its transition range overlaps the steep part of the titration curve at the equivalence point.

Titration type	Equivalence pH	Suitable indicator
Strong acid--strong base	7.0	Bromothymol blue (6.0--7.6), phenolphthalein (8.2--10.0)
Weak acid--strong base	8--10	Phenolphthalein (8.2--10.0)
Strong acid--weak base	4--6	Methyl red (4.4--6.2)

Using phenolphthalein for a strong acid--strong base titration introduces a small systematic error because the indicator transition (8.2--10.0) does not exactly match the equivalence point (7.0). However, the titration curve is so steep near pH 7 that the volume difference between pH 7 and pH 9 is negligible (typically $<0.02$ mL for 0.1 $M$ reagents), and the error is within normal burette-reading precision ($\pm 0.05$ mL).

For weak acid--strong base titrations, phenolphthalein is the standard choice because the equivalence point (pH 8--10) falls within its transition range. Methyl orange (3.1--4.4) would be a poor choice here because it changes colour well before the equivalence point.

Redox titrations Master

Redox titrations use an oxidation-reduction reaction as the stoichiometric basis. The titrant is either an oxidising agent (e.g., ${\text{KMnO}}_4$, ${\text{K}}_2{\text{Cr}}_2{\text{O}}_7$) or a reducing agent (e.g., ${\text{FeSO}}_4$, sodium thiosulfate ${\text{Na}}_2{\text{S}}_2{\text{O}}_3$).

The stoichiometry of a redox reaction is determined by balancing electron transfer. For the permanganate--iron titration:

$${\text{MnO}}_4^{-}+5{\text{Fe}}^{2+}+8{\text{H}}^{+}\to {\text{Mn}}^{2+}+5{\text{Fe}}^{3+}+4{\text{H}}_2\text{O}$$

Each ${\text{MnO}}_4^{-}$ gains 5 electrons; each ${\text{Fe}}^{2+}$ loses 1 electron. The 5:1 ratio governs the titration stoichiometry. At the equivalence point:

$$n({\text{Fe}}^{2+})=5\times n({\text{MnO}}_4^{-}).$$

${\text{KMnO}}_4$ is its own indicator: the deep purple ${\text{MnO}}_4^{-}$ solution turns colourless as it is reduced to ${\text{Mn}}^{2+}$, and the first persistent pink colour signals the endpoint. This is an advantage over acid-base titrations that require a separate indicator.

Iodometric titrations are a two-step indirect method. In step 1, an oxidising analyte oxidises excess ${\text{I}}^{-}$ to ${\text{I}}_2$:

$$\text{Ox}+2{\text{I}}^{-}\to \text{Red}+{\text{I}}_2.$$

In step 2, the liberated ${\text{I}}_2$ is titrated with ${\text{Na}}_2{\text{S}}_2{\text{O}}_3$:

$${\text{I}}_2+2{\text{S}}_2{\text{O}}_3^{2-}\to 2{\text{I}}^{-}+{\text{S}}_4{\text{O}}_6^{2-}.$$

Starch is added near the endpoint as an indicator: it forms a deep blue complex with ${\text{I}}_2$ that disappears when all ${\text{I}}_2$ has been consumed. Iodometric methods are versatile because many oxidising agents liberate ${\text{I}}_2$ quantitatively from ${\text{I}}^{-}$, and the thiosulfate titration is precise and reliable.

Gravimetric analysis Master

Gravimetric analysis achieves high precision (often $\pm 0.1\%$ or better) because mass can be measured more accurately than volume. The analyte is converted to a sparingly soluble compound of known stoichiometry, which is filtered, washed to remove impurities, dried or ignited to a constant composition, and weighed.

Precipitation requirements. The precipitate must satisfy:

1. Low solubility. The $K_{sp}$ must be small enough that the dissolved fraction is below analytical significance. For a precipitate $M_m{X}_x$ with $K_{sp}=[M^{x+}{]}^m[X^{m-}{]}^x$, the solubility $s$ satisfies $K_{sp}=(ms{)}^m(xs{)}^x=m^m{x}^x{s}^{m+x}$. The mass of dissolved precipitate must be negligible compared to the precision of the balance (typically 0.1 mg).

2. Known composition. The precipitate must have a definite stoichiometry that does not vary with conditions. Hydrates that gain or lose water unpredictably are unsuitable unless they can be ignited to a fixed anhydrous form.

3. Large particle size. Large particles are easier to filter and wash without loss. Colloidal precipitates (small particles) can pass through filter paper or adsorb impurities. Digestion (heating the precipitate in the mother liquor) promotes Ostwald ripening, where small particles dissolve and re-deposit on larger ones.

4. Purity. Coprecipitation -- the inclusion of impurities within or on the surface of the precipitate -- is the main source of systematic error. Common-ion effect (adding excess precipitating reagent) reduces solubility but can increase coprecipitation of the excess reagent.

Classical gravimetric determinations include chloride (as AgCl), sulfate (as ${\text{BaSO}}_4$), nickel (as Ni(DMG)${}_2$), and iron (as ${\text{Fe}}_2{\text{O}}_3$ after ignition of ${\text{Fe(OH)}}_3$). Each exploits a reaction with near-quantitative yield and a precipitate of definite composition.

The gravimetric factor $F$ converts the mass of the weighed form to the mass of the analyte:

$$F=\frac{M_{\text{analyte}}}{M_{\text{weighed form}}}\times \frac{{\nu }_{\text{analyte}}}{{\nu }_{\text{weighed form}}}.$$

For example, determining Al as ${\text{Al}}_2{\text{O}}_3$: the gravimetric factor is $F=(2\times 26.98)/101.96=0.5293$. The mass of Al = $0.5293\times m({\text{Al}}_2{\text{O}}_3)$.

Back titration Master

Back titration is used when the direct titration is impractical: the analyte is insoluble, reacts slowly, or has no suitable indicator. The analyte is treated with a known excess of a reagent that reacts with it, and the unreacted excess is then titrated.

The mole balance is:

$$n_{\text{analyte}}=\frac{n_{\text{reagent}}^{\text{total}}-n_{\text{titrant}}\cdot ({\nu }_{\text{reagent}}/{\nu }_{\text{titrant}})}{({\nu }_{\text{reagent}}/{\nu }_{\text{analyte}})}.$$

Common applications include:

• Antacid tablets. An excess of HCl dissolves the antacid (${\text{CaCO}}_3$, ${\text{Mg(OH)}}_2$, NaHCO${}_3$). The remaining HCl is titrated with NaOH. The direct reaction between the solid antacid and HCl is heterogeneous and slow; back titration avoids the need to wait for complete dissolution during the titration itself.

• Nitrogen analysis (Kjeldahl method). Organic nitrogen is converted to ${\text{NH}}_4^{+}$ by acid digestion. The ${\text{NH}}_4^{+}$ is treated with excess NaOH, and the liberated ${\text{NH}}_3$ is trapped in a known excess of HCl. The unreacted HCl is back-titrated with NaOH. This is the standard method for protein determination in food (protein $\approx N\times 6.25$).

• Insoluble carbonates. ${\text{CaCO}}_3$ in limestone is determined by adding excess HCl, boiling to expel ${\text{CO}}_2$, and back-titrating as in Exercise 7.

Connections Master

• Stoichiometry and gas laws 14.03.01 introduces the mole concept, limiting-reagent logic, and balanced equations. Solution stoichiometry applies the same framework to reactions in solution, where moles are counted by measuring volumes of solutions of known concentration rather than weighing solids or measuring gas volumes.

• Solutions and phase equilibria 14.09.01 expands on concentration units (molality, mole fraction) and introduces colligative properties that depend on solute concentration. The molarity concept developed here generalises to activity, which corrects for non-ideal solution behaviour at higher concentrations.

• Acid-base chemistry 14.10.01 provides the equilibrium expressions needed to calculate pH at each point on a titration curve. The Henderson-Hasselbalch equation used in buffer-region calculations comes directly from the weak-acid dissociation equilibrium treated in that unit. Titration curves are the graphical bridge between stoichiometry (equivalence point) and equilibrium (buffer regions).

• Electrochemistry 14.11.01 provides the Nernst equation that governs potentiometric endpoint detection. In a potentiometric titration, the potential of an indicator electrode is monitored as titrant is added, and the equivalence point is located from the inflection in the potential-vs-volume curve. Redox titrations also rely on the standard electrode potentials tabulated in that unit to determine feasibility and stoichiometry.

• Quantitative analysis depends on the statistical treatment of replicate measurements, calibration curves, and uncertainty propagation. The precision of a titration (repeatability) and its accuracy (closeness to the true value) are evaluated using the statistical methods covered in the statistics section of the curriculum.

Historical and philosophical context Master

Volumetric analysis began with Francois Antoine Henri Descroizilles, who around 1795 developed the first burette (a graduated glass tube) for determining the chlorine content of bleaching solutions. Joseph Louis Gay-Lussac systematised the method in the 1820s, introducing the terms "pipette" and "burette" and establishing titration as a quantitative analytical technique ^{[Gay-Lussac 1824]}.

Friedrich Mohr's 1855 textbook Lehrbuch der chemisch-analytischen Titrimethode consolidated titration methods and introduced improved apparatus designs, including the pinchcock burette. Mohr also developed several classic titration methods, including the Mohr method for chloride determination using chromate indicator, and the Mohr method for determining calcium and magnesium in water (a precursor to complexometric titration).

Soren Sorensen introduced the pH scale in 1909 at the Carlsberg Laboratory, providing the quantitative framework for understanding acid-base titration curves. Before Sorensen, indicator selection was empirical; the pH scale allowed systematic prediction of which indicator would change colour at the equivalence point of any given titration.

Karl Fischer developed his eponymous titration method for water determination in 1935, using the reaction of water with iodine and sulfur dioxide in the presence of a base. The Karl Fischer titration remains the most precise method for water determination in many industrial and pharmaceutical contexts, capable of measuring water content down to parts per million.

Gravimetric analysis predates volumetric analysis. The earliest quantitative gravimetric work was done by Torbern Bergman in the late 18th century, who used precipitation and weighing to determine the composition of mineral waters. Martin Heinrich Klaproth, considered the father of analytical chemistry, used gravimetric methods to determine the composition of minerals and discover several elements (uranium, zirconium, cerium) between 1789 and 1804. The high precision of gravimetric methods made them the gold standard for atomic-weight determinations throughout the 19th century; Theodore Richards received the 1914 Nobel Prize for his meticulous gravimetric redeterminations of atomic weights, which contributed to the discovery of isotope effects.

The development of analytical chemistry reflects a recurring theme: increased precision reveals previously hidden phenomena. Richards' atomic-weight measurements were precise enough to show that lead from different mineral sources had different atomic weights, a discrepancy later explained by the natural variation in isotopic composition. This pattern -- improved measurement exposing new structure -- repeats throughout the physical sciences and connects to the epistemological point that precision is not merely a practical virtue but a driver of discovery.

The distinction between the equivalence point (a stoichiometric condition) and the endpoint (an experimental observation) carries philosophical weight. Analytical chemistry treats measurement as a sequence of operations, each with its own uncertainty, and the total uncertainty is propagated through the calculation. A titration result depends on the accuracy of the standard solution, the precision of the burette readings, the appropriateness of the indicator, and the completeness of the reaction. None of these is individually sufficient; the result is a product of the entire analytical system. This systems-thinking approach to measurement is one of analytical chemistry's deepest conceptual contributions.

Bibliography Master

Primary literature:

• Descroizilles, F. A. H., "Memoire sur l'alcalimetrie", Ann. Chim. 20 (1797). Early volumetric analysis.

• Gay-Lussac, J. L., "Essai de titre de la potasse et de la soude du commerce", Ann. Chim. 55 (1824). First systematic description of volumetric titration.

• Mohr, F., Lehrbuch der chemisch-analytischen Titrimethode (Braunschweig, 1855). Systematic titration methods.

• Sorensen, S. P. L., "Enzymstudien II", Biochem. Z. 21 (1909), 131--200. Introduction of the pH scale.

• Fischer, K., "Neues Verfahren zur massanalytischen Bestimmung des Wassergehaltes von Fluessigkeiten und festen Stoffen", Angew. Chem. 48 (1935), 394--396. The Karl Fischer titration.

• Kolthoff, I. M. & Stenger, V. A., Volumetric Analysis, 2e, 3 vols. (Interscience, 1942--1947). Comprehensive treatise.

• Richards, T. W., "Atomic Weights", Nobel Lecture (1920). Gravimetric determination of atomic weights.

Modern references:

• Harris, D. C., Quantitative Chemical Analysis, 9e (W. H. Freeman, 2015), Ch. 6--12. Titration theory, gravimetric analysis, and analytical methodology.

• Skoog, D. A., West, D. M., Holler, F. J. & Crouch, S. R., Analytical Chemistry: An Introduction, 8e (Brooks/Cole, 2004). Comprehensive analytical chemistry textbook.

• Zumdahl, S. S. & DeCoste, D. J., Chemical Principles, 8e (Cengage, 2017), Ch. 4. Introductory solution stoichiometry.

• Atkins, P. & Jones, L., Chemical Principles, 2e (W. H. Freeman, 2010), Ch. 3. Stoichiometry and concentration.