Stoichiometry of Chemical Reactions – Comprehensive Lecture Notes

4.1 Writing and Balancing Chemical Equations

A chemical equation is a symbolic sentence that conveys the identities of the reacting substances (reactants) and the substances formed (products) together with their relative quantities. Coefficients placed in front of formulas express the molar (and therefore molecular) ratios in which species react or are produced. In a balanced equation the number of atoms of each element is identical on both sides, in accordance with the Law of Conservation of Mass, which states that matter is neither created nor destroyed during a chemical change.

A balanced molecular example is
\mathrm{CH4(g)+2\,O2(g)\;\rightarrow\;CO2(g)+2\,H2O(g)}
Interpreting the coefficients gives the ratio 1:2:1:2 for \mathrm{CH4}, \mathrm{O2}, \mathrm{CO2} and \mathrm{H2O}, applicable at the molecular scale and, by Avogadro’s principle, at the molar scale. If fractions aid balancing they may be introduced temporarily and then cleared by multiplying through by a whole-number factor.

Guidelines for successful balancing are: write correct formulas first and never alter subscripts afterward; treat the BrINClHOF elements as diatomic in their elemental form \bigl(\mathrm{Br2, I2, N2, Cl2, H2, O2, F_2}\bigr); balance all atoms other than H and O first, then O, then H; and finally reduce coefficients to the smallest whole-number set.

Coefficients, States, and Special Conditions

Coefficients portray relative quantity; a coefficient of 1 is usually omitted. Physical states enhance the informational content: (s) solid, (l) liquid, (g) gas, (aq) aqueous. Special reaction conditions appear above or below the arrow—for example \Delta to indicate heating as in \mathrm{CaCO3(s)\xrightarrow{\;\Delta\;}CaO(s)+CO2(g)}.

Ionic Equations – Molecular, Complete, and Net

When ionic compounds dissolve, they dissociate into ions. A molecular equation shows formulas of all participants, for example
\mathrm{CaCl2(aq)+2\,AgNO3(aq)\;\rightarrow\;Ca(NO3)2(aq)+2\,AgCl(s)}
Dissociation leads to the complete ionic equation
\mathrm{Ca^{2+}(aq)+2\,Cl^{-}(aq)+2\,Ag^{+}(aq)+2\,NO3^{-}(aq)\;\rightarrow\;Ca^{2+}(aq)+2\,NO3^{-}(aq)+2\,AgCl(s)}
Species appearing unchanged on both sides are spectator ions (here \mathrm{Ca^{2+}} and \mathrm{NO_3^{-}}). Eliminating them yields the net ionic equation
\mathrm{Ag^{+}(aq)+Cl^{-}(aq)\;\rightarrow\;AgCl(s)}

4.2 Classifying Chemical Reactions

Seven broad classes recur throughout general chemistry.
Combination (or synthesis) joins reactants into a single product, e.g. \mathrm{2Na(s)+Cl2(g)\;\rightarrow\;2NaCl(s)}. Decomposition splits one reactant into two or more products, e.g. \mathrm{CaCO3(s)\;\rightarrow\;CaO(s)+CO2(g)}. Single-displacement involves an element replacing another in a compound: \mathrm{Zn(s)+2\,HCl(aq)\;\rightarrow\;H2(g)+ZnCl2(aq)}. Predicting single-displacement relies on the metal activity series (Li > K > … > Cu > Ag > Au); a higher element displaces a lower one. Double-displacement (metathesis) exchanges ions between two ionic reactants, often generating a precipitate, gas, or weak electrolyte. Predicting precipitation relies on solubility rules (Group 1 cations, ammonium, nitrates, acetates, and most perchlorates are soluble; carbonates, phosphates, chromates, sulfides are generally insoluble except with Group 1 or \mathrm{NH4^+}; \mathrm{OH^-} is insoluble except with Group 1 and \mathrm{Ba^{2+}}, etc.).
Acid–base reactions transfer \mathrm{H^+}: strong acids (HCl, HBr, HI, \mathrm{HNO3}, \mathrm{HClO4}, \mathrm{H2SO4}) ionize completely, whereas weak acids (e.g., acetic acid) do so partially. Strong bases are soluble hydroxides of Group 1 and some Group 2 metals; ammonia exemplifies a weak base.
Oxidation–reduction (redox) reactions exhibit changes in oxidation numbers; electron loss is oxidation, gain is reduction (OIL RIG). Agents are named oppositely: the reducing agent is oxidized, the oxidizing agent is reduced.
Combustion reactions, a redox subset, involve rapid combination with \mathrm{O2} producing heat and often light, e.g. \mathrm{CH4+2\,O2\;\rightarrow\;CO2+2\,H_2O}.

Oxidation States – Rules and Applications

1 The oxidation number (ON) of any free element is 0.
2 For a monatomic ion, the ON equals its charge.
3 Common non-metals have characteristic ONs: H is +1 with non-metals and -1 with metals; O is -2 normally, -1 in peroxides \bigl(\mathrm{O2^{2-}}\bigr), rarely -\tfrac12 in superoxides \bigl(\mathrm{O2^-}\bigr), and positive when bonded to F. Halogens are -1 except in compounds with O or other halogens where they assume positive values or are less electronegative.
4 The algebraic sum of ONs in a neutral molecule is 0; in a polyatomic ion it equals the ion charge. These rules enable identification of redox processes and the elements oxidized or reduced.
Example: In \mathrm{SO_3^{2-}} the O atoms contribute 3(-2)=-6, so S must be +4 to yield the ion charge -2.

4.3 Reaction Stoichiometry

A balanced equation functions like a recipe whose coefficients generate stoichiometric factors (mole ratios). The generic pathway for calculations is
\text{mass} \;\rightarrow\; \text{moles} \;\rightarrow\; \text{moles (other species via ratio)} \;\rightarrow\; \text{mass, molecules, volume, etc.}
For instance, to find moles of \mathrm{I2} needed for 4.00\,\text{mol} Al in \mathrm{2Al+3I2\;\rightarrow\;2AlI3}, the ratio \tfrac{3\,\text{mol }I2}{2\,\text{mol Al}} gives 6.00\,\text{mol }I_2.

Conversions can proceed to the molecular scale using Avogadro’s number 6.022\times10^{23}\,\text{mol}^{-1}, or to masses using molar masses, or to volumes for gases at specified conditions, or to solution concentrations via M=\dfrac{\text{mol}}{\text{L}}.

Limiting Reactant and Reaction Yields (4.4)

When reactants are present in non-stoichiometric amounts, the one that is consumed first is the limiting reactant (LR); the others are in excess. Determination involves four steps: (1) balance the equation, (2) convert each reactant mass to moles, (3) use mole ratios to compute the amount of product each could yield, (4) the smallest yield identifies the LR and dictates the theoretical yield. Any difference between starting moles of excess reactants and moles that actually react gives mass of excess left over.

Example: With 12.096\,\text{g} \mathrm{H2} and 283.6\,\text{g} \mathrm{Cl2}, step 2 gives 6.0000\,\text{mol H2} and 4.000\,\text{mol Cl2}. Step 3 (from \mathrm{H2+Cl2\;\rightarrow\;2HCl}) shows \mathrm{Cl_2} limits the reaction by permitting only 8.000\,\text{mol HCl}. Multiplying by the molar mass of \mathrm{HCl} (36.458\,\text{g mol}^{-1}) gives a theoretical yield of 291.7\,\text{g}.

Theoretical yield is the calculated maximal product; actual yield is what is isolated experimentally. Their ratio times 100 % is the percent yield: \text{percent yield}=\dfrac{\text{actual}}{\text{theoretical}}\times100\%. A copper recovery experiment yielding 0.392\,\text{g} against a theoretical 0.507\,\text{g} gives 77.3\% yield.

4.5 Quantitative Chemical Analysis

Quantitative chemical analysis measures concentration or composition.

Titration employs a titrant of known concentration and an analyte of unknown concentration. Volumetric data at the equivalence point, together with reaction stoichiometry, reveal the analyte molarity. For \mathrm{HCl+NaOH\;\rightarrow\;NaCl+H_2O}, addition of 35.23\,\text{mL} of 0.250\,M \mathrm{NaOH} to 50.00\,\text{mL} \mathrm{HCl} implies 0.0088075\,\text{mol HCl} and hence 0.176\,M.

Gravimetric analysis isolates the analyte as a precipitate whose mass can be linked stoichiometrically to the amount of target species. Precipitating \mathrm{BaSO4} (molar mass 233.37\,\text{g mol}^{-1}) from an unknown mixture allowed determination that the sample was 69.93\% \mathrm{MgSO4} by mass.

Combustion analysis determines empirical formulas of hydrocarbons or related compounds by collecting \mathrm{CO2} and \mathrm{H2O}. Converting their masses to moles of C and H, then scaling to the smallest mole ratio, produced the formula \mathrm{CH_2} for polyethylene and similar procedures for polystyrene.

Ethical, Practical, and Real-World Relevance

Accurate stoichiometry underpins industrial chemical production, environmental monitoring, and pharmaceutical synthesis, ensuring efficient use of resources and minimizing waste. Titrations verify water quality and food acidity; gravimetric determinations guide ore processing and pollution assessment. Redox concepts apply in energy storage, corrosion prevention, and metabolic biochemistry, while reaction-classification skills assist forensic scientists and materials engineers. Ethical practice demands precise reporting and acknowledgment of uncertainty to avoid detrimental consequences in environmental release, medical dosing, or consumer safety.

Connections and Recap

Chapter 4 integrates conservation laws, mole concepts (from Chapter 3), and chemical nomenclature (Chapter 2) to provide a toolkit for interpreting, predicting, and quantitatively analyzing chemical reactions. Mastery involves fluency with symbolic language (balanced equations, ionic notation, oxidation states), categorical thinking (reaction types), proportional reasoning (stoichiometry and molar ratios), and laboratory skills (titration, gravimetry). Together these competences enable students to classify matter and reactions, write and balance equations, solve stoichiometric problems, determine limiting and excess reagents, calculate theoretical, actual, and percent yields, and perform quantitative analyses—outcomes articulated in SLO 2, 6, 8, and 10.