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Chemistry Lecture Notes: Atomic Structure, Isotopes, and Practice Problems

Schedule and Assessments

  • Quiz: scheduled for 09:11 on the next Thursday (online); must be completed before it closes.
  • Exam: scheduled for 09:18, one week after the quiz; will cover material up to the section 1a/1b; 2a and 2b are each worth 10 points.
  • 1a and 1b: already posted; you have enough information to complete both.
  • The instructor will pull some problems from the summertime test to illustrate typical question styles.
  • Homework and practice on iCollege:
    • There are four practice sets labeled as practice (not the homework). These are due at 05:00 on test day.
    • The homework (actual graded) is due at 05:00 on test day and is split into part a and part b, with separate point values (part 1 and part 2, around 10 points each).
  • It’s common for questions to use two-part problems where you can deduce one part from the other if you know one of the quantities; always ensure the two pieces sum to 100% when dealing with isotopic abundances.

Review of Key Concepts from Today

  • Dalton and Rutherford are two central figures in early atomic theory.
  • Dalton
    • Law of definite proportions: Elements in a given compound are always in the same proportion by mass.
    • Law of multiple proportions: When two elements form more than one compound, the ratios of the masses of the second element, which combine with a fixed mass of the first element, are ratios of small whole numbers.
  • Thomson
    • Cathode ray experiment led to the discovery of the electron.
  • Rutherford
    • Gold foil experiment led to the nuclear model: a dense positively charged nucleus in the center with electrons surrounding it.
  • Nuclear model components
    • Nucleus contains protons (positive charge) and neutrons (neutral).
    • Electrons (negative charge) are located outside the nucleus.
    • Protons, neutrons have masses ≈ 1 amu each; electrons have negligible mass in comparison.
  • Subatomic particles
    • Proton: positive charge, ~1 amu.
    • Neutron: neutral charge, ~1 amu.
    • Electron: negative charge, ~5.485 × 10^-4 amu (negligible for most mass calculations).
  • Atomic structure terminology
    • Atom consists of a nucleus (protons + neutrons) and surrounding electrons.
    • Atomic number Z: number of protons in the nucleus.
    • Mass number A: total number of protons and neutrons in the nucleus (A = Z + N).
    • Isotopes: atoms of the same element (same Z) with different numbers of neutrons (different A).
  • Isotopes and Abundances
    • Isotopes have the same Z but different N, hence different masses.
    • Natural isotopes: those found in nature; some elements have multiple natural isotopes; some isotopes are radioactive (e.g., carbon-14) while others are stable (e.g., carbon-12, carbon-13).
    • Atomic mass unit (amu or u): defined as 1/12 the mass of a carbon-12 atom, i.e., 1~ ext{amu} = rac{1}{12} m( ext{^{12}C}). Carbon-12 is defined to be exactly 12 amu, so 1 amu is exactly one twelfth of the carbon-12 mass. Other elements have atomic masses that are weighted averages of their naturally occurring isotopes.
    • The proton and neutron masses are approximately 1 amu; the electron mass is about 1/1836 of a proton mass (much smaller).
    • The periodic table arrangement places elements by increasing Z; symbols have 1 or 2 letters, with the first letter capitalized and the second (if present) lowercase.
    • The symbol W stands for tungsten (from the German word Wolfram).
    • Many very old element names/symbols are from historical origins; the standard two-letter symbols reflect those origins and capitalization rules.

Atomic Mass Unit and Isotopes: Foundational Details

  • Isotopes of an element
    • Have the same number of protons (Z) but different numbers of neutrons (N).
    • This leads to different atomic masses for each isotope.
  • Carbon isotopes as a canonical example
    • Common natural isotopes: ${}^{12} ext{C}, {}^{13} ext{C}, {}^{14} ext{C}$.
    • Carbon-12 is exactly 1 amu by definition (one twelfth of the mass of a ${}^{12} ext{C}$ atom).
    • Carbon-13 and carbon-14 masses are close to 13 amu and 14 amu respectively but are defined relative to the carbon-12 standard.
  • Atomic mass of an element on the periodic table
    • It is a weighted average of the masses of the naturally occurring isotopes, using their fractional abundances.
    • Example concept: if an element has two naturally occurring isotopes with masses $ma$ and $mb$ and abundances $fa$ and $fb$ (with $fa + fb = 1$), then the atomic mass $M$ is M = fa ma + fb mb. When a third isotope is present, the same weighted-average approach applies with more terms.
  • Practical point about notation
    • Notation like ${}^{A}_{Z} ext{X}$ indicates an isotope of element X with mass number A and atomic number Z (i.e., A = Z + N).
    • If the charge is shown (e.g., ${}^{A}_{Z} ext{X}^{n+}$), the charge indicates the net number of electrons relative to protons.
  • Isotope abundance problem-solving approach
    • Given two isotopes with masses $ma$ and $mb$ and a measured average mass $M$, solve for the unknown fraction (abundance) of one isotope using the relation M = fa ma + fb mb, ext{ with } fa + fb = 1. A convenient rearrangement is $fb = 1 - fa$ and $fa = rac{M - mb}{ma - mb}$ (assuming $ma eq mb$).
    • If you’re given $M$, $ma$, and $mb$, you can compute the unknown fraction directly. Then the other fraction is the complement.

Worked Examples from Today’s Transcript

  • Boron isotopes: ${}^{10} ext{B}$ and ${}^{11} ext{B}$
    • Masses: $m{10} = 10$ amu, $m{11} = 11$ amu.
    • Atomic mass of boron in nature (weighted average): $M = 10.81$ amu.
    • Let $f{10}$ be the fraction of ${}^{10} ext{B}$ and $f{11} = 1 - f_{10}$ for ${}^{11} ext{B}$.
    • Solve: M = f{10} imes 10 + (1 - f{10}) imes 11 = 11 - f{10} \ herefore f{10} = 11 - M = 11 - 10.81 = 0.19.
    • Thus: f{10} = 0.19 ext{ (19%), } f{11} = 0.81 ext{ (81%)}.
  • Bromine isotopes: ${}^{79} ext{Br}$ and ${}^{81} ext{Br}$
    • Masses: $m{79} = 79$, $m{81} = 81$ amu.
    • Weighted average mass given: $M ext{ (approximately }79.9 ext{ amu)}$.
    • Solve: M = f{79} imes 79 + (1 - f{79}) imes 81 = 81 - 2f{79} \ herefore f{79} = rac{81 - M}{2}.
    • If $M = 79.9$, then $f{79} = rac{81 - 79.9}{2} = rac{1.1}{2} = 0.55$ (55%), and $f{81} = 0.45$ (45%).
    • The heavier isotope ${}^{81} ext{Br}$ mass is around 81 amu, as inferred from the calculation.
  • Lithium isotopes: ${}^{6} ext{Li}$ and ${}^{7} ext{Li}$
    • Given that the atomic mass of lithium is about $6.94$ amu and there are two naturally occurring isotopes with masses 6 and 7, determine which is more abundant.
    • Since the average is closer to 7, ${}^{7} ext{Li}$ is the more abundant isotope.
    • If you know the average and the masses, you can deduce the fractional abundances similarly to the boron example.
  • Carbon isotopes and dating context
    • Carbon has isotopes ${}^{12} ext{C}$, ${}^{13} ext{C}$, and ${}^{14} ext{C}$.
    • ${}^{12} ext{C}$ is exactly 1 amu by definition. ${}^{13} ext{C}$ and ${}^{14} ext{C}$ have masses near 13 and 14 amu, respectively, but the atomic mass of carbon on the periodic table is a weighted average of these naturally occurring isotopes.
    • ${}^{14} ext{C}$ is radioactive and used for dating (carbon dating).
    • The statement that there are “three isotopes” of carbon refers to the three naturally occurring isotopes, among which ${}^{12} ext{C}$ and ${}^{13} ext{C}$ are stable while ${}^{14} ext{C}$ is radioactive.
  • Isotope notation conventions and real-world relevance
    • In practice, you’ll encounter isotope notation in the form of ${}^{A}_{Z} ext{X}$ or sometimes ${}^{A} ext{X}$ for shorthand, where A is the mass number and Z is the atomic number.
    • In neutral atoms, the number of electrons equals the number of protons (Z). If the atom is ionized, the electron count differs by the net charge.
    • Understanding isotopes is crucial for fields like archaeology (carbon dating), geology, medicine, and environmental science.

Periodic Table Essentials (Context from Today)

  • The periodic table arrangement is described as elements being placed along rows and columns with charges shown on the table for some students.
  • Tungsten’s symbol is W, derived from the German word Wolfram.
  • Element symbols are often one or two letters; the first letter is capitalized and the second (if present) is lowercase.
  • When naming and classifying elements, historical origins influence symbols and naming conventions.

Isotopes: Core Relationships and Notation

  • A (mass number) and Z (atomic number)
    • ${}^{A}_{Z} ext{X}$ denotes an isotope of element X with mass number A and atomic number Z.
  • Determining neutrons from A and Z
    • Neutrons N = A - Z.
  • Charge and electron count
    • For a neutral isotope, the number of electrons equals Z; for an ion, electrons = Z ± |charge| depending on whether it’s an anion or cation.

Quick Reference: Key Concepts to Remember

  • Atomic mass unit (amu)
    • Defined via carbon-12: 1~ ext{amu} = rac{1}{12} m( ext{^{12}C}).
    • Protons and neutrons ~ 1 amu; electrons ~ 5.485 × 10^-4 amu (negligible for most mass calculations).
  • Isotopes have the same Z but different N; masses differ accordingly.
  • The atomic mass shown on the periodic table is a weighted average of natural isotopes.
  • The nucleus contains protons and neutrons; electrons orbit the nucleus.
  • Isotope abundance problems require solving simple linear equations with the constraint that abundances sum to 1 (or 100%).

Practical Tips and Exam Strategies

  • When solving isotope-abundance problems, always check that abundances sum to 1 (or 100%). If you’re given one abundance, you can find the other using the weighted-average mass.
  • For problems with two isotopes, use the relation M = fa ma + (1 - fa) mb to solve for the fractional abundance $f_a$.
  • Express abundances as decimals (fractions) or percentages consistently; remember to convert to the opposite form if needed.
  • Recognize that many exam questions will present the problem in the form of A/Z notation and ask you to extract Z, or to determine N from A and Z.
  • Be comfortable with unit conversions (mm → cm → m → km) and with dimensional analysis to avoid errors in the number of significant figures.

Worked Practice Concept Recap

  • Example: Density calculation from a cube
    • Given edge length in millimeters, convert to centimeters, compute volume in cm^3, and then density from mass/volume.
    • For 11.4 mm edge length:
    • Edge in cm: 11.4~ ext{mm} o 1.14~ ext{cm}.
    • Cube volume: V = (1.14~ ext{cm})^3 \ = 1.48~ ext{cm}^3 ext{ (approximately)}.
    • Mass: 25 g.
    • Density:
      ho = rac{m}{V} = rac{25~ ext{g}}{1.48~ ext{cm}^3} \
      ho
      oughly 16.9~ ext{g/cm}^3.
  • Example: Millimeters to kilometers
    • Given 1.23 mm: convert to kilometers via meters and then to kilometers:
    • 1.23~ ext{mm} = 1.23 imes 10^{-3}~ ext{m} = 1.23 imes 10^{-6}~ ext{km}.$$

Final Note on Relevance and Ethics (Practical Implications)

  • Understanding isotopes and atomic structure informs real-world applications like medical imaging, radiometric dating, environmental tracing, and materials science.
  • Accurate unit handling, notation, and problem-solving skills are foundational for exams and for responsible laboratory work.
  • Ethical and philosophical reflections may arise in contexts such as nuclear energy, radiometric dating’s role in archaeology, and the societal implications of scientific knowledge; these were not covered explicitly in today’s lecture but are important to consider in broader coursework.