Stoichiometry of Chemical Reactions

Writing and Balancing Chemical Equations

• A balanced chemical equation uses symbols to represent both the identities and the relative quantities of substances involved in a chemical change.

• Fundamental aspects of a chemical equation (using the example $CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O$):

  • Reactants: The formulas of substances undergoing the reaction are placed on the left side.

  • Products: The formulas of substances generated by the reaction are placed on the right side.

  • Separators: Plus signs ($+$) separate individual reactant or product species. An arrow ($\rightarrow$) separates the reactant and product sides of the equation.

  • Coefficients: These numbers show the relative numbers of reactant and product species. If a coefficient is $1$, it is typically omitted.

• Verification of Balance: An equation is balanced when the number of atoms for each element is the same on both sides.

  • Example: $CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O$

    • Carbon (C): $1 \times 1 = 1$ in reactants; $1 \times 1 = 1$ in products. Balanced ($1 = 1$).

    • Hydrogen (H): $1 \times 4 = 4$ in reactants; $2 \times 2 = 4$ in products. Balanced ($4 = 4$).

    • Oxygen (O): $2 \times 2 = 4$ in reactants; $(1 \times 2) + (2 \times 1) = 4$ in products. Balanced ($4 = 4$).

• Rules for Balancing Equations:

  • Subscripts define the identity of the substance and cannot be changed. Changing a subscript alters the meaning of the equation entirely.

  • Fractions may sometimes be used as intermediate coefficients before clearing them to reach the simplest whole-number ratio.

    • Intermediate Step: $C_2H_6 + \frac{7}{2}O_2 \rightarrow 3H_2O + 2CO_2$

    • Final Step (Multiplying by 2): $2C_2H_6 + 7O_2 \rightarrow 6H_2O + 4CO_2$

• Indicating Physical States: Physical states are noted parenthetically following the formula:

  • $(g)$: Gas

  • $(s)$: Solid

  • $(l)$: Liquid

  • $(aq)$: Aqueous (dissolved in water)

  • Example: $2Na(s) + 2H_2O(l) \rightarrow 2NaOH(aq) + H_2(g)$

Equations for Ionic Reactions

• Molecular Equation: Represents the substances as molecules or formula units without explicitly showing the ionic species.

  • Example: $CaCl_2(aq) + 2AgNO_3(aq) \rightarrow Ca(NO_3)_2(aq) + 2AgCl(s)$

• Ionic Dissociation in Water:

  • Soluble ionic compounds dissociate into their constituent ions when dissolved in water.

    • $CaCl_2(aq) \rightarrow Ca^{2+}(aq) + 2Cl^{-}(aq)$

    • $2AgNO_3(aq) \rightarrow 2Ag^{+}(aq) + 2NO_3^{-}(aq)$

    • $Ca(NO_3)_2(aq) \rightarrow Ca^{2+}(aq) + 2NO_3^{-}(aq)$

  • Insoluble compounds, such as $AgCl(s)$, do not dissociate.

• Complete Ionic Equation: Explicitly shows all dissolved ions in the reaction.

  • Example: $Ca^{2+}(aq) + 2Cl^{-}(aq) + 2Ag^{+}(aq) + 2NO_3^{-}(aq) \rightarrow Ca^{2+}(aq) + 2NO_3^{-}(aq) + 2AgCl(s)$

• Spectator Ions: Ions that appear in identical forms on both the reactant and product sides of a complete ionic equation. They do not participate in the actual chemical change.

  • In the example above, $Ca^{2+}(aq)$ and $NO_3^{-}(aq)$ are spectator ions.

• Net Ionic Equation: Formed by eliminating spectator ions from the complete ionic equation to show only the species involved in the reaction.

  • Example: $Ag^{+}(aq) + Cl^{-}(aq) \rightarrow AgCl(s)$

Classifying Chemical Reactions: Precipitation

• Precipitation Reaction: A process where dissolved substances react to form one or more solid products (precipitates). These are also called double displacement, double replacement, or metathesis reactions.

• Solubility Concepts:

  • Solubility: The maximum concentration of a substance that can be achieved under specific conditions.

  • Precipitate: Forms when the concentration of a substance exceeds its solubility.

  • Insoluble: Term for substances with very low solubility that readily precipitate.

• Solubility Rules for Common Ionic Compounds:

  • Soluble Compounds Contain:

    • Group 1 metal cations (e.g., $Li^{+}, Na^{+}, K^{+}, Rb^{+}, Cs^{+}$) or ammonium ion ($NH_4^{+}$).

    • Halide ions ($Cl^{-}, Br^{-}, I^{-}$), EXCEPT those of capital $Ag^{+}, Hg_2^{2+},\text{ and } Pb^{2+}$.

    • Acetate ($C_2H_3O_2^{-}$), bicarbonate ($HCO_3^{-}$), nitrate ($NO_3^{-}$), and chlorate ($ClO_3^{-}$) ions.

    • Sulfate ion ($SO_4^{2-}$), EXCEPT sulfates of $Ag^{+}, Ba^{2+}, Ca^{2+}, Hg_2^{2+}, Pb^{2+},\text{ and } Sr^{2+}$.

  • Insoluble Compounds Contain:

    • Carbonate ($CO_3^{2-}$), chromate ($CrO_4^{2-}$), phosphate ($PO_4^{3-}$), or sulfide ($S^{2-}$) ions, EXCEPT when combined with Group 1 metal cations or ammonium.

    • Hydroxide ion ($OH^{-}$), EXCEPT when combined with Group 1 metal cations or $Ba^{2+}$.

• Precipitation Examples:

  • Molecular: $2KI(aq) + Pb(NO_3)_2(aq) \rightarrow PbI_2(s) + 2KNO_3(aq)$

  • Net Ionic: $Pb^{2+}(aq) + 2I^{-}(aq) \rightarrow PbI_2(s)$

  • Molecular: $NaCl(aq) + AgNO_3(aq) \rightarrow AgCl(s) + NaNO_3(aq)$

  • Net Ionic: $Ag^{+}(aq) + Cl^{-}(aq) \rightarrow AgCl(s)$

Acid-Base Reactions

• Definition: A reaction involving the transfer of a hydrogen ion ($H^{+}$) from one species to another.

• Acids:

  • Acid dissociation yields hydronium ions ($H_3O^{+}$).

  • Strong Acids: Completely react/dissociate in water.

    • Example: $HCl(aq) + H_2O(l) \rightarrow Cl^{-}(aq) + H_3O^{+}(aq)$

    • The Seven Known Strong Acids:

      1. $HBr(aq)$ (Hydrobromic acid)

      2. $HCl(aq)$ (Hydrochloric acid)

      3. $HI(aq)$ (Hydroiodic acid)

      4. $HNO_3(aq)$ (Nitric acid)

      5. $HClO_3(aq)$ (Chloric acid)

      6. $HClO_4(aq)$ (Perchloric acid)

      7. $H_2SO_4(aq)$ (Sulfuric acid)

  • Weak Acids: Only partially react with water.

    • Example: Acetic acid ($CH_3CO_2H(aq) + H_2O(l) \rightleftharpoons CH_3CO_2^{-}(aq) + H_3O^{+}(aq)$). At equilibrium, roughly $99\%$ is the unreacted acid and $1\%$ is the dissociated product.

• Bases:

  • A base dissolves in water to yield hydroxide ions ($OH^{-}$).

  • Strong Bases: Hydroxides of alkali or alkaline-earth cations that completely dissociate.

    • Examples: $NaOH$, $KOH$, $Ca(OH)_2$, $Ba(OH)_2$.

    • Dissociation: $NaOH(s) \rightarrow Na^{+}(aq) + OH^{-}(aq)$.

  • Weak Bases: Produce hydroxide ions by reacting partially with water molecules.

    • Example: Ammonia ($NH_3(aq) + H_2O(l) \rightleftharpoons NH_4^{+}(aq) + OH^{-}(aq)$).

• Neutralization Reactions:

  • General Form: $\text{Acid} + \text{Base} \rightarrow \text{Salt} + \text{Water}$.

  • Definition of Salt: Any chemical compound formed from the reaction of an acid with a base, where all or part of the hydrogen of the acid is replaced by a metal or other cation.

  • Example: $Mg(OH)_2(s) + 2HCl(aq) \rightarrow MgCl_2(aq) + 2H_2O(l)$. In this case, $MgCl_2$ is the salt.

Oxidation–Reduction (Redox) Reactions

• Basic Definitions:

  • Redox reactions involve changes in oxidation numbers due to the transfer of electrons or changes in chemical bonding.

  • Oxidation: Loss of electrons; increase in oxidation number.

  • Reduction: Gain of electrons; decrease in oxidation number.

  • Reducing Agent: The species that provides electrons (is oxidized) to another substance.

  • Oxidizing Agent: The species that removes electrons (is reduced) from another substance.

• Half-Reactions Example: $2Na(s) + Cl_2(g) \rightarrow 2NaCl(s)$

  • Oxidation half-reaction: $2Na(s) \rightarrow 2Na^{+}(s) + 2e^{-}$

  • Reduction half-reaction: $Cl_2(g) + 2e^{-} \rightarrow 2Cl^{-}(s)$

  • Note: Some redox processes forming covalent compounds, such as $H_2(g) + Cl_2(g) \rightarrow 2HCl(g)$, share electrons rather than transferring them entirely, but are still classified using oxidation numbers.

• Rules for Assigning Oxidation Numbers:

  1. Elemental substance: The oxidation number of an atom in its elemental form is zero (e.g., $Cu^0$, $H_2^0$).

  2. Monatomic ion: Equal to the ion's charge (e.g., $Ca^{2+} = +2$).

  3. Nonmetals:

     • Hydrogen: $+1$ with nonmetals, $-1$ with metals.

     • Oxygen: Usually $-2$. Exceptions: $-1$ in peroxides ($O_2^{2-}$), $-1/2$ in superoxides ($O_2^{-}$), or positive when combined with Fluorine.

     • Halogens: Fluorine is always $-1$. Others are $-1$ unless combined with Oxygen or other halogens (where they can be positive).

  4. Sum of oxidation numbers: Must equal the total charge of the molecule (zero) or the polyatomic ion.

     • Example for Sulfite ($SO_4^{2-}$): $S = +6$, 4 Oxygen atoms ($4 \times -2 = -8$). Sum: $(+6) + (-8) = -2$.

• Specific Redox Types:

  • Combustion: A fuel (reductant) and oxidant react vigorously, producing heat and light.

    • Example (Rocket fuel): $10Al(s) + 6NH_4ClO_4(s) \rightarrow 4Al_2O_3(s) + 2AlCl_3(s) + 12H_2O(g) + 3N_2(g)$.

  • Single-Displacement: An ion in solution is replaced via the oxidation of a metallic element.

    • Example: $Cu(s) + 2AgNO_3(aq) \rightarrow Cu(NO_3)_2(aq) + 2Ag(s)$.

Reaction Stoichiometry

• Definition: The quantitative relationships between the amounts of reactants and products in a balanced chemical equation.

• Stoichiometric Factors: Ratios derived from the coefficients of the balanced equation used to compute desired quantities.

  • Example: $N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$.

  • The ratio of $NH_3$ to $H_2$ is $2:3$, expressed as $\frac{2\,mol\,NH_3}{3\,mol\,H_2}$.

• Sample Problems:

  • Example 4.8: Determining how many moles of $I_2$ react with $0.429\,mol$ of $Al$ using $2Al(s) + 3I_2(s) \rightarrow 2AlI_3(s)$.

  • Example 4.10: Calculating the mass of $NaOH$ needed to produce $16\,g$ of milk of magnesia ($Mg(OH)_2$) via $MgCl_2(aq) + 2NaOH(aq) \rightarrow Mg(OH)_2(s) + 2NaCl(aq)$.

Reaction Yields

• Limiting Reactants:

  • Limiting Reactant: The reactant that is entirely consumed first, limiting the amount of product formed.

  • Excess Reactant: The reactant that remains after the limiting reactant is used up.

  • Sandwich Analogy: If $1\text{ cheese slice} + 2\text{ bread slices} \rightarrow 1\text{ sandwich}$, and you have $11\text{ cheese slices}$ and $28\text{ bread slices}$, you can only make $11\text{ sandwiches}$. Cheese is the limiting reactant; bread is excess.

  • Chemical Approach: Compare the amount of product expected from the full consumption of each reactant. The one yielding the lesser amount is the limiting reactant.

  • Example: $H_2(g) + Cl_2(g) \rightarrow 2HCl(g)$. Combined $3\,mol\,H_2$ and $2\,mol\,Cl_2$.

    • $3\,mol\,H_2 \times \frac{2\,mol\,HCl}{1\,mol\,H_2} = 6\,mol\,HCl$

    • $2\,mol\,Cl_2 \times \frac{2\,mol\,HCl}{1\,mol\,Cl_2} = 4\,mol\,HCl$

    • $Cl_2$ is the limiting reactant.

• Yield Calculations:

  • Theoretical Yield: Maximum product amount calculated from stoichiometry.

  • Actual Yield: The amount of product truly obtained from the experiment. Often lower than theoretical due to side reactions, incomplete reaction, or recovery loss.

  • Percent Yield Formula: $\text{Percent Yield} = \frac{\text{Actual Yield}}{\text{Theoretical Yield}} \times 100$

  • Example 4.13: $1.274\,g$ copper sulfate yields $0.392\,g$ copper metal. Calculating percent yield.

  • Uranium Isolation Problem (4.73): Adding $0.4031\,g$ sodium oxalate to $1.481\,g$ uranyl nitrate resulting in $1.073\,g$ of $UO_2(C_2O_4)\cdot 3H_2O$. Requires identifying limiting reactant and percent yield.

Quantitative Chemical Analysis

• Definition: Determining the amount or concentration of a substance in a sample.

• Titration:

  • Titrant: Solution with a known concentration.

  • Analyte: Solution with an unknown concentration.

  • Equivalence Point: Point where titrant and analyte have reacted completely in stoichiometric proportions.

  • End Point: The volume of titrant measured when an indicator changes color. Ideally, the difference between the equivalence and end point is negligible.

  • Buret: Instrument used to deliver titrant; typically accurate to $\pm 0.02\,mL$.

  • Example 4.14: $50.00\,mL$ sample of $HCl$ reached end point with $35.23\,mL$ of $0.250\,M\,NaOH$. Calculating molarity of $HCl$.

• Gravimetric Analysis:

  • Involves separating a substance from a sample and measuring its mass.

  • Example 4.15: A $0.4550\,g$ mixture containing $MgSO_4$ is treated with excess $Ba(NO_3)_2$ to yield $0.6168\,g$ of $BaSO_4$ precipitate. Calculation used to find $MgSO_4$ content.

• Combustion Analysis:

  • Used to determine the empirical formula of hydrocarbons ($C_xH_y$).

  • Example 4.16: A $0.00126\,g$ sample of polyethylene burned in excess $O_2$ yields $0.00394\,g$ of $CO_2$ and $0.00161\,g$ of $H_2O$. Procedure determines the empirical formula.