Chapter 2: The Components of Matter

Classification of Matter

• Matter Definition: Matter is defined as anything that occupies space and has mass.

• Primary Categorization: The first classification criterion for any sample of matter is whether its composition is constant or variable, dividing it into Pure Substances or Mixtures.

• Pure Substances:

  • Composed of a single type of atom or molecule throughout.

  • Composition is invariant (does not vary from sample to sample).

  • Divided based on chemical decomposability:

    • Element: A pure substance that cannot be broken down into simpler substances by chemical reactions. Composed of a single species of atom. Example: Helium in a Goodyear blimp.

    • Compound: A pure substance composed of two or more elements combined in fixed, definite proportions. Can be chemically decomposed into simpler substances. Example: Pure water ($\text{H}_2\text{O}$).

• Mixtures:

  • Composed of two or more different types of atoms or molecules in proportions that can vary from one sample to another.

  • Divided based on uniformity of distribution:

    • Homogeneous Mixture: Composition is completely uniform throughout the sample. The individual components are mixed at the molecular or atomic level and cannot be visually distinguished. Example: Sweetened tea (tea with sugar).

    • Heterogeneous Mixture: Composition varies from one region of the sample to another. Distinct phases or components are visually identifiable. Example: Wet sand at a beach.

Fundamental Chemical Laws

• Law of Conservation of Mass:

  • Proposed by Antoine Lavoisier in 1789 based on quantitative combustion experiments.

  • Statement: Matter is neither created nor destroyed in a chemical reaction.

  • Mathematical Expression:         $\text{Total Mass of Reactants} = \text{Total Mass of Products}$         $\text{Total Number of Reactant Atoms} = \text{Total Number of Product Atoms}$

• Law of Definite Proportions:

  • Proposed by Joseph Proust in 1797.

  • Statement: All samples of a given compound, regardless of their source or how they were prepared, have the exact same proportions of their constituent elements by mass.

  • Methane Example:

    • Mass analysis of a $20.00\,\text{g}$ sample of methane yields $15.00\,\text{g}$ of carbon and $5.00\,\text{g}$ of hydrogen.

    • Mass ratio: $\frac{5.00\,\text{g H}}{15.00\,\text{g C}} = \frac{1.00\,\text{g H}}{3.00\,\text{g C}}$.

    • This $1:3$ mass ratio of hydrogen to carbon holds true for any pure sample of methane.

• Law of Multiple Proportions:

  • Published by John Dalton in 1804.

  • Statement: When two elements (designated A and B) form two different compounds, the masses of element B that combine with $1.00\,\text{g}$ of element A can be expressed as a ratio of small whole numbers.

  • Carbon-Oxygen Compounds Example:

    • Carbon dioxide ($\text{CO}_2$): $2.67\,\text{g}$ of oxygen combines with $1.00\,\text{g}$ of carbon.

    • Carbon monoxide ($\text{CO}$): $1.33\,\text{g}$ of oxygen combines with $1.00\,\text{g}$ of carbon.

    • Ratio of the masses of oxygen:             $\frac{2.67\,\text{g O}}{1.33\,\text{g O}} = 2.00$

    • This whole-number ratio ($2:1$) reflects the atomic ratio of oxygen atoms per carbon atom in the respective compounds ($\text{CO}_2$ versus $\text{CO}$).

Development of Atomic Theory and Historical Timeline

• Historical Timeline of Atomic Models:

*   **Democritus (460 B.C.)**: Proposed that matter is composed of small, indivisible particles termed *atomos* (meaning uncuttable or indivisible). Lacked empirical support.
*   **John Dalton (1803–1808 A.D.)**: Formulated the first scientifically testable atomic model based on fundamental mass laws.
*   **J. J. Thomson (1897)**: Discovered the electron via cathode ray deflections; proposed the Plum-Pudding Model.
*   **Ernest Rutherford (1912)**: Discovered the atomic nucleus via the Gold Foil Experiment; proposed the Nuclear/Planetary Model.
*   **Niels Bohr (1913)**: Formulated quantized electron orbit models.
*   **Quantum Cloud Model (Post-1930)**: Developed by Erwin Schrödinger, Werner Heisenberg, Albert Einstein, Max Planck, and Richard Feynman; treats electrons as wave-particle probability clouds.

• Dalton's Atomic Theory (1808):

  1. Each element is composed of extremely small, indestructible particles called atoms.

  2. All atoms of a given element are identical in mass, size, and chemical properties. Atoms of different elements possess different properties.

  3. Atoms cannot be subdivided, created, or destroyed in chemical processes.

  4. Atoms of different elements combine in simple, fixed whole-number ratios to form chemical compounds ($\text{AB}$, $\text{AB}_2$, $\text{AB}_3$, etc.).

  5. Atoms of one element cannot change into atoms of another element in a chemical reaction. In chemical reactions, atoms separate, combine, or rearrange, altering their bonds to form new substances.

Discovery of Subatomic Particles

• Cathode Rays and J. J. Thomson:

  • In the late 19th century, J. J. Thomson used a cathode ray tube—a partially evacuated glass tube outfitted with two metal electrodes connected to a high-voltage electrical power supply.

*   **Mechanism**: A stream of particles (cathode rays) travels from the negatively charged electrode (cathode) toward the positively charged electrode (anode).
*   **Detection**: Striking the far end of the tube, coated with a fluorescent phosphor material, produces a bright visual glow.
*   **Observed Properties of Cathode Ray Particles**:
    1.  They travel in straight lines.
    2.  Their properties are completely independent of the metallic composition of the cathode electrode.
    3.  They carry a negative electrical charge.

• Determination of Charge-to-Mass Ratio:

  • Thomson applied opposing electric and magnetic fields to deflect the path of the cathode ray beam.

*   Measured the mass-to-charge ratio of the cathode ray particle:

        $\frac{m}{q} = -1.76 \times 10^8\,\text{C/g}$ * Significance: This value implied that cathode ray particles were roughly 2000 times less massive than a hydrogen atom (the lightest known element), demonstrating that the atom is divisible and composed of smaller subatomic components. * The Electron: Named as the fundamental negatively charged particle.

• Millikan Oil Drop Experiment (1909):

  • Robert Millikan sprayed microscopic oil drops into a chamber between two charged electric plates and ionized them using ionizing radiation.

*   By adjusting the electric field strength, Millikan suspended charged oil droplets in mid-air against gravity.
*   Determined that the charge on any oil droplet was always an integral whole-number multiple of a fundamental charge unit:

        $e = -1.60 \times 10^{-19}\,\text{C}$ * Calculation of Electron Mass:         $\text{Mass of Electron} = -1.60 \times 10^{-19}\,\text{C} \times \frac{1.00\,\text{g}}{-1.76 \times 10^8\,\text{C}} = 9.10 \times 10^{-28}\,\text{g} = 9.10 \times 10^{-31}\,\text{kg}$

Rutherford's Nuclear Model of the Atom

• The Plum-Pudding Model (J. J. Thomson):

  • Proposed around 1900 to account for atomic neutrality.

  • Posited that negative electrons were embedded within a uniform sphere of positive charge, akin to raisins in a plum pudding.

• Rutherford's Gold Foil Experiment (1909):

  • Ernest Rutherford directed highly energetic, positively charged alpha ($\alpha$) particles (emitted by radium inside a lead container) at an ultrathin sheet of gold foil surrounded by a circular fluorescent detector screen.

*   **Expected Result**: If the mass and positive charge were uniformly distributed (as predicted by Thomson), all $\alpha$ particles would pass straight through with minimal or no deflection.
*   **Actual Observations**:
    *   The vast majority of $\alpha$ particles passed straight through undeflected.
    *   A small fraction was deflected through moderate angles.
    *   Approximately 1 in 20,000 $\alpha$ particles was deflected backward at acute, large angles ($>90^\circ$).
    *   Rutherford remarked: *"It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."*

• Nuclear Theory Formulation:

1.  Most of the atom's mass and all of its positive charge are concentrated in a tiny central volume called the **nucleus**.
2.  Most of the atom's volume is empty space, throughout which tiny, negatively charged electrons are dispersed.
3.  The number of negatively charged electrons surrounding the nucleus equals the number of positively charged particles (protons) within the nucleus, maintaining electrical neutrality.

• Discovery of the Neutron:

  • An anomaly in mass ratios existed: a hydrogen atom contains 1 proton, while a helium atom contains 2 protons. If mass were solely due to protons, the Helium:Hydrogen mass ratio should be $2:1$. However, the experimental ratio is $4:1$

  • Rutherford later demonstrated that the extra mass is accounted for by neutrons—subatomic particles located within the nucleus that possess mass equal to protons but carry zero electrical charge.

Subatomic Particle Properties and Atomic Structure

• Summary of Subatomic Particle Characteristics:

  • Proton:

    • Mass: $1.67262 \times 10^{-27}\,\text{kg} \approx 1.00727\,\text{amu}$

    • Charge: $+1.60 \times 10^{-19}\,\text{C}$

    • Relative Charge: $+1$

  • Neutron:

    • Mass: $1.67493 \times 10^{-27}\,\text{kg} \approx 1.00866\,\text{amu}$

    • Charge: $0\,\text{C}$

    • Relative Charge: $0$

  • Electron:

    • Mass: $9.10938 \times 10^{-31}\,\text{kg} \approx 0.00055\,\text{amu}$

    • Charge: $-1.60 \times 10^{-19}\,\text{C}$

    • Relative Charge: $-1$

• Atomic Mass Unit (amu) Definition:

  • $1\,\text{amu}$ is defined as exactly $\frac{1}{12}$ the mass of a single carbon-12 atom containing 6 protons and 6 neutrons.

Elements, Symbols, and Nomenclature Origins

• Atomic Number ($Z$): The number of protons in an atom's nucleus. Defines the chemical identity of an element.

• Mass Number ($A$): The total number of protons plus neutrons in an atom's nucleus:     $A = \text{Protons} + \text{Neutrons}$

• Chemical Notation Symbolism:

• Origins of Element Symbols:

  • English Names: Carbon (C), Chlorine (Cl).

  • Latin Roots: Sodium (Natrium = Na), Gold (Aurum = Au).

  • German Roots: Tungsten (Wolfram = W).

  • Descriptive Properties: Argon (argos = inactive, lazy).

  • Mythological References: Mercury (Mercury, Roman god of commerce).

  • Geographic Locations: Ytterby, Sweden (Erbium, Terbium, Ytterbium, Yttrium); Poland (Polonium, honor of Marie Curie).

  • Renowned Scientists: Einsteinium (Albert Einstein), Bohrium (Niels Bohr).

Isotopes and Atomic Mass Calculations

• Isotopes: Atoms of the same element containing identical numbers of protons ($Z$) but differing numbers of neutrons, resulting in different mass numbers ($A$).

• Natural Abundance: The relative percentage of each specific isotope found in a naturally occurring sample of an element. Example: Natural bromine consists of $50.7\%$ ${}^{79}_{35}\text{Br}$ (44 neutrons) and $49.3\%$ ${}^{81}_{35}\text{Br}$ (46 neutrons).

• Average Atomic Mass: The weighted average of the atomic masses of all naturally occurring isotopes of an element based on their fractional abundances:     $\text{Atomic Mass} = \sum_{i} \left( \text{Fractional Abundance}_i \times \text{Isotopic Mass}_i \right)$

Ions and Charge Predictability

• Ion Formation: Atoms lose or gain electrons during chemical reactions, acquiring an overall net charge.

  • Cation: A positively charged ion formed when a neutral atom loses one or more electrons.         $\text{Na} \rightarrow \text{Na}^+ + e^-$

  • Anion: A negatively charged ion formed when a neutral atom gains one or more electrons.         $\text{Br} + e^- \rightarrow \text{Br}^-$

• Predictable Charges in Main-Group Elements:

  • Elements gain or lose electrons to match the electron configuration of the nearest noble gas (Group 8A).

  • Group 1A (Alkali Metals): Form $+1$ cations ($\text{Li}^+$, $\text{Na}^+$, $\text{K}^+$, $\text{Rb}^+$, $\text{Cs}^+$).

  • Group 2A (Alkaline Earth Metals): Form $+2$ cations ($\text{Mg}^{2+}$, $\text{Ca}^{2+}$, $\text{Sr}^{2+}$, $\text{Ba}^{2+}$).

  • Group 3A: Aluminum forms $+3$ cations ($\text{Al}^{3+}$).

  • Group 5A (Pnictogens): Nonmetals form $-3$ anions ($\text{N}^{3-}$).

  • Group 6A (Chalcogens): Nonmetals form $-2$ anions ($\text{O}^{2-}$, $\text{S}^{2-}$, $\text{Se}^{2-}$, $\text{Te}^{2-}$).

  • Group 7A (Halogens): Form $-1$ anions ($\text{F}^-$, $\text{Cl}^-$, $\text{Br}^-$, $\text{I}^-$).

The Periodic Table: Structure and Chemical Families

• Dmitri Mendeleev (1869): Arranged the 65 then-known elements in order of increasing atomic mass. Noted recurring periodic property trends (Periodic Law). Left gaps for undiscovered elements (e.g., Gallium, Germanium) and accurately predicted their chemical properties.

• Modern Periodic Table: Arranges elements by increasing atomic number ($Z$) rather than atomic mass.

• Broad Classifications:

  • Metals: Located on the lower left and center. Properties: high thermal/electrical conductivity, malleability, ductility, lustrous appearance, tendency to lose electrons in reactions.

  • Nonmetals: Located on the upper right. Properties: poor conductors of heat and electricity, varied physical states, tendency to gain electrons in reactions.

  • Metalloids (Semimetals): Lie along the bold diagonal zigzag border separating metals and nonmetals. Exhibit mixed metallic and nonmetallic semiconductor properties.

• Major Periodic Families/Groups:

  • Alkali Metals (Group 1A / Group 1): Highly reactive metals.

  • Alkaline Earth Metals (Group 2A / Group 2): Fairly reactive metals.

  • Transition Metals (Groups 3B–2B / Groups 3–12): Less predictable charges.

  • Halogens (Group 7A / Group 17): Highly reactive nonmetals.

  • Noble Gases (Group 8A / Group 18): Unreactive, extremely stable monoatomic gases.

  • Inner Transition Series: Lanthanides and Actinides.

Chemical Bonding: Ionic vs. Covalent

• Ionic Bonding:

  • Occurs between metals (which lose electrons) and nonmetals (which gain electrons).

  • Electrons are completely transferred from metal to nonmetal.

  • Resulting oppositely charged cations and anions are held together by strong, non-directional electrostatic forces.

*   Form three-dimensional crystalline lattices consisting of alternating positive and negative ions.

• Covalent Bonding:

  • Occurs between two or more nonmetals.

  • Neither atom fully surrenders electrons; instead, valence electrons are shared between bonded nuclei.

  • Shared electrons interact with both nuclei, lowering potential energy.

Chemical Formulas and Molecular Representation

• Chemical Formula: Indicates the specific elements present in a compound and the relative number of atoms or ions of each.

• Formula Classifications:

  • Empirical Formula: Gives the simplest whole-number ratio of atoms of each element in a compound.

  • Molecular Formula: Gives the actual number of atoms of each element present in a molecule of a compound. Example: Hydrogen peroxide empirical formula is $\text{HO}$, while its molecular formula is $\text{H}_2\text{O}_2$

  • Structural Formula: Shows how atoms are bonded together in space, using lines to represent shared electron pairs.

*   **Ball-and-Stick Model**: Represents atoms as spheres and chemical bonds as rods.
*   **Space-Filling Model**: Displays atoms scaled to their relative Van der Waals sizes to show the space occupied by the electron cloud.

• Forms of Natural Existence:

  • Atomic Elements: Exist in nature as single unbonded atoms (e.g., $\text{He}$, $\text{Ne}$, $\text{Ar}$).

  • Molecular Elements: Exist in nature as multi-atom molecules.

    • Diatomic molecules: $\text{H}_2$, $\text{N}_2$, $\text{O}_2$, $\text{F}_2$, $\text{Cl}_2$, $\text{Br}_2$, $\text{I}_2$

    • Polyatomic molecules: $\text{P}_4$, $\text{S}_8$

Nomenclature of Inorganic Compounds

• Binary Ionic Compounds (Type I - Invariant Charge Metals):

  • Naming convention: [Name of Metal] + [Base Name of Nonmetal + -ide]

  • Example: $\text{CaF}_2$ is named Calcium fluoride.

• Binary Ionic Compounds (Type II - Variable Charge Metals):

  • Naming convention: [Name of Metal] + (Roman Numeral for Charge) + [Base Name of Nonmetal + -ide]

  • Example: $\text{FeCl}_2$ is Iron(II) chloride; $\text{FeCl}_3$ is Iron(III) chloride.

• Transition Metals with Variable Charges:

  • Chromium: $\text{Cr}^{2+}$ Chromium(II) [Chromous], $\text{Cr}^{3+}$ Chromium(III) [Chromic]

  • Iron: $\text{Fe}^{2+}$ Iron(II) [Ferrous], $\text{Fe}^{3+}$ Iron(III) [Ferric]

  • Cobalt: $\text{Co}^{2+}$ Cobalt(II) [Cobaltous], $\text{Co}^{3+}$ Cobalt(III) [Cobaltic]

  • Copper: $\text{Cu}^+$ Copper(I) [Cuprous], $\text{Cu}^{2+}$ Copper(II) [Cupric]

  • Tin: $\text{Sn}^{2+}$ Tin(II) [Stannous], $\text{Sn}^{4+}$ Tin(IV) [Stannic]

  • Mercury: $\text{Hg}_2^{2+}$ Mercury(I) [Mercurous], $\text{Hg}^{2+}$ Mercury(II) [Mercuric]

  • Lead: $Pb^{2+}$ Lead(II) [Plumbous], $Pb^{4+}$ Lead(IV) [Plumbic]

• Common Polyatomic Ions Reference:

  • Acetate: $\text{C}_2\text{H}_3\text{O}_2^-$

  • Carbonate: $\text{CO}_3^{2-}$

  • Hydrogen carbonate (Bicarbonate): $\text{HCO}_3^-$

  • Hydroxide: $\text{OH}^-$

  • Nitrite: $\text{NO}_2^-$

  • Nitrate: $\text{NO}_3^-$

  • Chromate: $\text{CrO}_4^{2-}$

  • Dichromate: $\text{Cr}_2\text{O}_7^{2-}$

  • Phosphate: $\text{PO}_4^{3-}$

  • Hydrogen phosphate: $\text{HPO}_4^{2-}$

  • Dihydrogen phosphate: $\text{H}_2\text{PO}_4^-$

  • Ammonium: $\text{NH}_4^+$

  • Hypochlorite: $\text{ClO}^-$

  • Chlorite: $\text{ClO}_2^-$

  • Chlorate: $\text{ClO}_3^-$

  • Perchlorate: $\text{ClO}_4^-$

  • Permanganate: $\text{MnO}_4^-$

  • Sulfite: $\text{SO}_3^{2-}$

  • Hydrogen sulfite (Bisulfite): $\text{HSO}_3^-$

  • Sulfate: $\text{SO}_4^{2-}$

  • Hydrogen sulfate (Bisulfate): $\text{HSO}_4^-$

  • Cyanide: $\text{CN}^-$

  • Peroxide: $\text{O}_2^{2-}$

• Oxyanion Naming Rules:

  • Two-member series: The ion with more oxygen ends in -ate; the ion with fewer oxygen ends in -ite.

  • Extended series (four members):

    • per- + base name + -ate (most oxygen, e.g., $\text{ClO}_4^-$ Perchlorate)

    • base name + -ate (e.g., $\text{ClO}_3^-$ Chlorate)

    • base name + -ite (e.g., $\text{ClO}_2^-$ Chlorite)

    • hypo- + base name + -ite (least oxygen, e.g., $\text{ClO}^-$ Hypochlorite)

Nomenclature of Hydrates, Acids, and Molecular Compounds

• Hydrated Ionic Compounds:

  • Contain a specific number of bound water molecules in their crystal lattice (waters of hydration).

  • Heating drives off water to convert a hydrate into an anhydrous compound.

  • Greek Numerical Prefixes: Mono- (1), Di- (2), Tri- (3), Tetra- (4), Penta- (5), Hexa- (6), Hepta- (7), Octa- (8), Nona- (9), Deca- (10).

  • Example: $\text{MgSO}_4 \cdot 7\text{H}_2\text{O}$ is Magnesium sulfate heptahydrate.

• Acid Nomenclature:

  • Acids release hydrogen ions ($\text{H}^+$) when dissolved in water.

*   **Binary Acids** (Hydrogen + Nonmetal):
    *   Naming pattern: `hydro-` + `[base name of nonmetal]` + `-ic acid`
    *   Example: $\text{HBr}_{(aq)}$ is Hydrobromic acid.
*   **Oxyacids** (Hydrogen + Oxyanion):
    *   If oxyanion ends in **-ate**: `[base name of oxyanion]` + `-ic acid` (e.g., $\text{H}_2\text{SO}_4$ is Sulfuric acid).
    *   If oxyanion ends in **-ite**: `[base name of oxyanion]` + `-ous acid` (e.g., $\text{H}_2\text{SO}_3$ is Sulfurous acid).

• Molecular Compounds (Nonmetal + Nonmetal):

  • Naming pattern: [Prefix] + [1st Element] + [Prefix] + [2nd Element Base Name + -ide]

  • List the element with the lower group number (more left on periodic table) first.

  • Omit mono- for the first element, but retain it for the second element.

  • Examples: $\text{P}_4\text{S}_{10}$ is Tetraphosphorus decasulfide; $\text{CO}_2$ is Carbon dioxide; $\text{CO}$ is Carbon monoxide.

Introduction to Organic Chemistry and Hydrocarbons

• Organic Chemistry: The study of carbon-containing compounds.

• Properties of Carbon: Carbon always forms four covalent bonds. Over $99\%$ of the 30 million known chemical compounds contain carbon.

• Hydrocarbons: Organic compounds composed exclusively of carbon and hydrogen.

• Common Hydrocarbons:

  • Methane ($\text{CH}_4$): Primary constituent of natural gas.

  • Propane ($\text{C}_3\text{H}_8$): Liquefied petroleum (LP) gas for grills.

  • n-Butane ($\text{C}_4\text{H}_{10}$): Fuel for pocket lighters (n = straight-chain).

  • n-Pentane ($\text{C}_5\text{H}_{12}$): Component of gasoline.

  • Ethene ($\text{C}_2\text{H}_4$): Fruit ripening agent.

  • Ethyne ($\text{C}_2\text{H}_2$): Acetylene fuel for welding torches.

Comprehensive Worked Quantitative Examples

• Example 1: Mass Conservation in Magnesium Oxide Formation

  • Problem: Burning a $1.25\,\text{g}$ strip of magnesium yields $2.07\,\text{g}$ of $\text{MgO}$. How many grams of oxygen were consumed?

  • Solution:         $\text{Mass of Reactants} = \text{Mass of Products}$         $\text{Mass of Mg} + \text{Mass of O}_2 = \text{Mass of MgO}$         $1.25\,\text{g} + \text{Mass of O}_2 = 2.07\,\text{g}$         $\text{Mass of O}_2 = 2.07\,\text{g} - 1.25\,\text{g} = 0.82\,\text{g O}$

• Example 2: Mass Percent and Element Recovery (Law of Definite Proportions)

  • Problem: Mass analysis of a $78.80\,\text{g}$ sample of hematite reveals $55.08\,\text{g}$ of iron ($\text{Fe}$), with oxygen as the only other element. How many grams of iron and oxygen are in a $260.82\,\text{g}$ rock of hematite?

  • Solution:         $\frac{55.08\,\text{g Fe}}{78.80\,\text{g sample}} = \frac{x\,\text{g Fe}}{260.82\,\text{g sample}}$         $78.80x = 55.08 \times 260.82$         $78.80x = 14365.9656$         $x = 182.3\,\text{g Fe}$         $\text{Mass of O} = 260.82\,\text{g total} - 182.3\,\text{g Fe} = 78.5\,\text{g O}$

• Example 3: Identification of Unknown Metal Oxide

  • Problem: A $1.563\,\text{g}$ sample of an unknown metal $\text{M}$ burns consuming $127.4\,\text{mL}$ of $\text{O}_2$ gas (density $1.429\,\text{g/L}$) to produce a metal oxide $\text{MO}$. What is the atomic mass and identity of metal $\text{M}$?

  • Step 1: Calculate mass of consumed oxygen gas:         $\text{Mass of O}_2 = 127.4\,\text{mL} \times \frac{1\,\text{L}}{1000\,\text{mL}} \times 1.429\,\text{g/L} = 0.1820546\,\text{g O}_2$

  • Step 2: Apply Law of Definite Proportions ($\text{MO}$ stoichiometry implies 1 atom $\text{M}$ per 1 atom $\text{O}$):         $\frac{\text{Mass of M}}{\text{Mass of O}} = \frac{\text{Atomic Mass of M}}{\text{Atomic Mass of O}}$         $\frac{1.563\,\text{g}}{0.1820546\,\text{g}} = \frac{x\,\text{amu}}{16.00\,\text{amu}}$         $0.1820546x = 1.563 \times 16.00 = 25.008$         $x = 137.4\,\text{amu}$

  • Conclusion: Metal $\text{M}$ is Barium ($\text{Ba}$, atomic mass $137.33\,\text{amu}$).

• Example 4: Calculating Average Atomic Mass of Iron

  • Data: ${}^{54}\text{Fe}$ ($5.85\%$, $53.94\,\text{amu}$), ${}^{56}\text{Fe}$ ($91.75\%$, $55.93\,\text{amu}$), ${}^{57}\text{Fe}$ ($2.12\%$, $56.94\,\text{amu}$), ${}^{58}\text{Fe}$ ($0.28\%$, $57.93\,\text{amu}$).

  • Calculation:         $\text{Average Mass} = (0.0585 \times 53.94) + (0.9175 \times 55.93) + (0.0212 \times 56.94) + (0.0028 \times 57.93)$         $\text{Average Mass} = 3.15549 + 51.315775 + 1.207128 + 0.162204 = 55.84\,\text{amu}$

• Example 5: Determining Isotopic Natural Abundances of Boron

  • Data: ${}^{10}\text{B}$ ($10.01\,\text{amu}$), ${}^{11}\text{B}$ ($11.01\,\text{amu}$), Average atomic mass = $10.81\,\text{amu}$.

  • Calculation:         $x(10.01) + (1 - x)(11.01) = 10.81$         $10.01x + 11.01 - 11.01x = 10.81$         $-1.00x = -0.20 \implies x = 0.20$

  • Abundances: ${}^{10}\text{B} = 20\%$, ${}^{11}\text{B} = 80\%$.

• Example 6: Molecular Mass Determination for $\text{POF}_3$

  • $1 \times \text{P} = 1 \times 30.97\,\text{amu} = 30.97\,\text{amu}$

  • $1 \times \text{O} = 1 \times 16.00\,\text{amu} = 16.00\,\text{amu}$

  • $3 \times \text{F} = 3 \times 19.00\,\text{amu} = 57.00\,\text{amu}$

  • $\text{Total Molecular Mass} = 103.97\,\text{amu}$

• Example 7: Formula Mass Determination for Copper(II) Phosphate, $\text{Cu}_3(\text{PO}_4)_2$

  • $3 \times \text{Cu} = 3 \times 63.55\,\text{amu} = 190.65\,\text{amu}$

  • $2 \times \text{P} = 2 \times 30.97\,\text{amu} = 61.94\,\text{amu}$

  • $8 \times \text{O} = 8 \times 16.00\,\text{amu} = 128.00\,\text{amu}$

  • $\text{Total Formula Mass} = 380.59\,\text{amu}$