Comprehensive Guide to Atomic Structure, Isotopes, Atomic Models, and Density Calculations

Atomic Symbol Notation and Subatomic Structure

• Comparison of Notation Systems:

  • Periodic Table Layout: Elements are ordered sequentially by atomic number, placing the smaller number (atomic number) at the top of the element box.
  • Atomic Symbol Notation Layout: The positions are inverted relative to the periodic table layout. The larger number (mass number) is written at the top left, and the smaller number (atomic number) is written at the bottom left.

• Components of the Atomic Symbol:

  • Mass Number ($A$): Placed at the top. Represents the total sum of protons and neutrons in the atomic nucleus ($A = \text{protons} + \text{neutrons}$). Virtually all of an atom's mass is concentrated in the nucleus. Electrons are omitted from mass calculations due to their extremely light weight.
  • Atomic Number ($Z$): Placed at the bottom. Represents the total number of protons in the nucleus. The number of protons is the unique identifier of an atom; changing the number of protons changes the identity of the element (which occurs exclusively in nuclear chemistry).
  • Ion Charge: Written at the top right. Indicates whether an atom has gained or lost electrons.
    • Positively Charged Ions (Cations): Formed when an atom loses electrons (resulting in more protons than electrons).
    • Negatively Charged Ions (Anions): Formed when an atom gains electrons (resulting in more electrons than protons).

• Detailed Example — Nitrogen Ion Notation (${}^{15}_7\text{N}^{3+}$):

  • Atomic Number ($Z$): $7$ (indicates all nitrogen atoms strictly possess $7$ protons).
  • Mass Number ($A$): $15$ (sum of protons and neutrons).
  • Neutron Count: Calculated by subtracting the atomic number from the mass number:         $\text{Neutrons} = 15 - 7 = 8\,\text{neutrons}$
  • Electron Count: A charge of $+3$ indicates that the atom has lost $3$ electrons relative to its neutral proton count:         $\text{Electrons} = 7 - 3 = 4\,\text{electrons}$

Subatomic Particle Determination Across Atoms and Ions

• Hydrogen ($\text{H}$):

  • Atomic Number ($Z$): $1$
  • Atomic Weight (Periodic Table): $1.008\,\text{amu}$
  • Mass Number ($A$): $1\,\text{amu}$
  • Protons: $1$
  • Neutrons: Calculated as Mass Number minus Atomic Number:         $\text{Neutrons} = 1 - 1 = 0\,\text{neutrons}$
    • Exception Note: Hydrogen-1 is the only atom in existence that contains zero neutrons. All other elements and isotopes possess at least one neutron.
  • Electrons: $1$ (neutral atom).

• Hydrogen Cation ($\text{H}^+$):

  • Atomic Number ($Z$): $1$
  • Atomic Weight: $1.008\,\text{amu}$
  • Mass Number ($A$): $1$
  • Protons: $1$
  • Neutrons: $0$
  • Electrons: Calculated as Protons minus Charge:         $\text{Electrons} = 1 - 1 = 0\,\text{electrons}$
    • Structure Note: A positive hydrogen ion ($\text{H}^+$) consists solely of an isolated proton.

• Carbon-12 (${}^{12}\text{C}$):

  • Atomic Number ($Z$): $6$
  • Atomic Weight: Rounded to $12.01\,\text{amu}$
  • Mass Number ($A$): $12$ (explicitly given)
  • Protons: $6$
  • Neutrons:         $\text{Neutrons} = 12 - 6 = 6\,\text{neutrons}$
  • Electrons: $6$ (neutral atom).

• Lithium Ion ($\text{Li}^+$):

  • Atomic Number ($Z$): $3$
  • Atomic Weight: $6.94\,\text{amu}$
  • Mass Number ($A$): $7$
  • Protons: $3$
  • Neutrons:         $\text{Neutrons} = 7 - 3 = 4\,\text{neutrons}$
  • Electrons:         $\text{Electrons} = 3 - 1 = 2\,\text{electrons}$

• Potassium ($\text{K}$):

  • Atomic Number ($Z$): $19$
  • Mass Number ($A$): $39$
  • Protons: $19$
  • Neutrons:         $\text{Neutrons} = 39 - 19 = 20\,\text{neutrons}$
  • Electrons: $19$ (neutral atom; electron symbol is $\text{e}^-$).

• Magnesium Ion ($\text{Mg}^{2+}$):

  • Atomic Number ($Z$): $12$
  • Atomic Weight: $24.31\,\text{amu}$
  • Charge Impact: A $+2$ charge indicates the loss of $2$ electrons.

• Arsenide Ion ($\text{As}^{3-}$):

  • Atomic Number ($Z$): $33$
  • Atomic Weight: $74.92\,\text{amu}$
  • Mass Number ($A$): $75$ (rounded from $74.92$ to the nearest whole number when not explicitly provided)
  • Protons: $33$
  • Neutrons:         $\text{Neutrons} = 75 - 33 = 42\,\text{neutrons}$
  • Electrons: A $-3$ charge indicates gaining $3$ electrons:         $\text{Electrons} = 33 + 3 = 36\,\text{electrons}$

• Silver ($\text{Ag}$):

  • Atomic Number ($Z$): $47$
  • Atomic Weight: $107.868\,\text{amu}$
  • Mass Number ($A$): $108$
  • Protons: $47$
  • Neutrons:         $\text{Neutrons} = 108 - 47 = 61\,\text{neutrons}$

Isotopes and Weighted Average Atomic Mass

• Definition of Isotopes:

  • Isotopes are atoms of the same chemical element that possess the exact same number of protons but different numbers of neutrons.
  • Proton numbers never change for a given element. Electron numbers change to form ions; neutron numbers change to form isotopes.

• Explanation of Decimal Atomic Mass Values:

  • An individual atom cannot contain a fraction of a neutron or proton.
  • The atomic weights printed on the periodic table are decimal numbers because they represent a weighted average of all naturally occurring isotopes of that element based on their natural percent abundances on Earth.
  • The atomic weight value reflects the most abundant isotope. For example, terrestrial carbon exists as Carbon-12, Carbon-13, and Carbon-14. Carbon-12 constitutes almost the entire global percentage, while Carbon-13 and Carbon-14 make up roughly $1\%$ combined. Consequently, the weighted average atomic weight of carbon is $12.011\,\text{amu}$.

• Calculation Procedure for Weighted Average Atomic Mass:

  1. Convert each percentage abundance into a decimal multiplier by dividing by $100$:         $\text{Decimal Abundance} = \frac{\text{Percent}}{100}$
  2. Multiply each isotope's mass number by its corresponding decimal abundance.
  3. Sum the resulting values to calculate the total weighted atomic mass in atomic mass units ($\text{amu}$).
  4. Significant Figures Rule for Practice: Abundance percentages obtained from mass spectrometers are highly precise (often to 4 or more digits). In these introductory calculations, numbers are treated as exact values with no uncertainty.

• Worked Isotope Practice Problems:

  • Cesium Sample:

    • Cesium-133: $75\%$ abundance $\rightarrow 0.75 \times 133 = 99.75\,\text{amu}$
    • Cesium-132: $20\%$ abundance $\rightarrow 0.20 \times 132 = 26.4\,\text{amu}$
    • Cesium-134: $5\%$ abundance $\rightarrow 0.05 \times 134 = 6.7\,\text{amu}$
    • Total Weighted Average:             $\text{Atomic Weight} = 99.75 + 26.4 + 6.7 = 132.85\,\text{amu}$(Note: $132.85\,\text{amu}$ is closest to $133\,\text{amu}$ because Cesium-133 makes up $75\%$ of the sample).

  • Iodine Sample ($\text{I}$, Atomic Number $53$):

    • Iodine-127: $80\%$ abundance $\rightarrow 0.80 \times 127 = 101.6\,\text{amu}$
    • Iodine-126: $17\%$ abundance $\rightarrow 0.17 \times 126 = 21.42\,\text{amu}$
    • Iodine-128: $3\%$ abundance $\rightarrow 0.03 \times 128 = 3.84\,\text{amu}$
    • Total Weighted Average:             $\text{Atomic Weight} = 101.6 + 21.42 + 3.84 = 126.86\,\text{amu}$

  • Gold Sample ($\text{Au}$):

    • Gold-197: $50\%$ abundance $\rightarrow 0.50 \times 197 = 98.5\,\text{amu}$
    • Gold-198: $50\%$ abundance $\rightarrow 0.50 \times 198 = 99.0\,\text{amu}$
    • Total Weighted Average:             $\text{Atomic Weight} = 98.5 + 99.0 = 197.5\,\text{amu}$

  • Hydrogen Isotope Sample:

    • Hydrogen-1: $99\%$ abundance $\rightarrow 0.99 \times 1 = 0.99\,\text{amu}$
    • Hydrogen-2: $0.8\%$ abundance $\rightarrow \frac{0.8}{100} \times 2 = 0.008 \times 2 = 0.016\,\text{amu}$
    • Hydrogen-3: $0.2\%$ abundance $\rightarrow \frac{0.2}{100} \times 3 = 0.002 \times 3 = 0.006\,\text{amu}$
    • Total Weighted Average:             $\text{Atomic Weight} = 0.99 + 0.016 + 0.006 = 1.012\,\text{amu}$

  • Nitrogen Sample:

    • Nitrogen-14: $95\%$ abundance $\rightarrow 0.95 \times 14 = 13.3\,\text{amu}$
    • Nitrogen-15: $3\%$ abundance $\rightarrow 0.03 \times 15 = 0.45\,\text{amu}$
    • Nitrogen-13: $2\%$ abundance $\rightarrow 0.02 \times 13 = 0.26\,\text{amu}$
    • Total Weighted Average:             $\text{Atomic Weight} = 13.3 + 0.45 + 0.26 = 14.01\,\text{amu}$

  • Carbon Sample:

    • Carbon-12: $98\%$ abundance $\rightarrow 0.98 \times 12 = 11.76\,\text{amu}$
    • Carbon-14: $2\%$ abundance $\rightarrow 0.02 \times 14 = 0.28\,\text{amu}$
    • Total Weighted Average:             $\text{Atomic Weight} = 11.76 + 0.28 = 12.04\,\text{amu}$

Historical Evolution of Atomic Theory

• Democritus ($400\,\text{BC}$):

  • Coined the term atomos.
  • Proposed that matter consists of small, indivisible particles that cannot be broken down further.
  • Possessed no empirical evidence or experimental proof (approached atomic structure purely as a philosopher), but his concept was correct.

• John Dalton ($1803$):

  • Proposed the Billiard Ball Model.
  • Provided experimental evidence showing that chemical compounds combine in fixed ratios.
  • Inaccurately claimed that atoms were the smallest indivisible pieces of matter (unaware of subatomic particles).

• J.J. Thomson ($1897$):

  • Proposed the Plum Pudding Model.
  • Discovered the electron using the Cathode Ray Tube experiment.
  • Observed that the cathode ray beam deflected toward a positively charged plate, proving that electrons are negatively charged particles.

• Ernest Rutherford:

  • Conducted the Gold Foil Experiment by shooting positively charged alpha particles at thin gold foil.
  • Discovered the nucleus: a extremely small, dense, positively charged central core containing protons and holding virtually all the atom's mass.
  • Demonstrated that the atom is comprised mostly of empty space.

• Niels Bohr:

  • Proposed the Planetary Model (also referred to as the Solar System Model).
  • Pictured the nucleus as the sun at the center, with electrons orbiting around it like planets.
  • Introduced organization to electron pathways by defining distinct energy levels (replacing previous unstructured views that modeled electrons like bees buzzing around a beehive).

• Erwin Schrödinger:

  • Formulated quantum wave mechanics and famous for the Schrödinger's cat thought experiment ("is it dead or alive?").
  • Further detailed quantum electron behavior and light interactions.

Density, Mass, and Volume Calculations

• Formulas and Mathematical Relationships:

  • Density Formula:         $D = \frac{m}{V}$
  • Volume Formula:         $V = \frac{m}{D}$
  • Mass Formula:         $m = D \times V$

• Standard Units of Measurement:

  • Mass ($m$): Measured in grams ($\text{g}$).
  • Volume ($V$): Measured in milliliters ($\text{mL}$).
  • Density ($D$): Measured in grams per milliliter ($\text{g\,mL}^{-1}$).

• Mnemonic Density Triangle Device:

  • Place Mass ($m$) in the top peak of the triangle.
  • Place Density ($D$) and Volume ($V$) side-by-side in the bottom two corners.
  • Variables side-by-side are multiplied ($D \times V$); variables stacked vertically represent division (Mass over Volume or Mass over Density).

• Density Practice Problems:

  • Problem 1 — Finding Density:

    • Given: Mass $m = 32\,\text{g}$, Volume $V = 5.4\,\text{mL}$.
    • Calculation:             $D = \frac{32\,\text{g}}{5.4\,\text{mL}} = 5.9259...\,\text{g\,mL}^{-1}$
    • Rounding ($2$ significant figures): The digit $9$ is followed by a $2$, so it remains unchanged at $5.9\,\text{g\,mL}^{-1}$.

  • Problem 2 — Finding Mass:

    • Given: Density $D = 3.2\,\text{g\,mL}^{-1}$, Volume $V = 12.0\,\text{mL}$ (or $12.9\,\text{mL}$).
    • Calculation:             $m = 3.2\,\text{g\,mL}^{-1} \times 12.0\,\text{mL} = 38.4\,\text{g}$
    • Rounding ($2$ significant figures): The digit $8$ is followed by $4$ (less than 5), so it rounds to $38\,\text{g}$.

  • Problem 3 — Finding Volume:

    • Given: Density $D = 7.6\,\text{g\,mL}^{-1}$, Mass $m = 122\,\text{g}$.
    • Calculation:             $V = \frac{122\,\text{g}}{7.6\,\text{g\,mL}^{-1}} = 16.0526...\,\text{mL}$
    • Rounding ($2$ significant figures): Rounds to $16\,\text{mL}$.

Questions and Practical Clarifications

• Question: Why does the atomic symbol write the mass number on top when the periodic table places the atomic number on top?

  • Answer: This is a standard scientific convention. On the periodic table, elements are organized sequentially by atomic number, so the smaller atomic number is placed at the top. In formal nuclear symbol notation, the layout is deliberately switched so the larger mass number is on top and the atomic number is on the bottom.

• Question: What is a reliable trick to remember how electron gains and losses alter ion charges?

  • Answer: Remember that gaining electrons makes an ion negative because electrons carry a negative charge. A helpful conceptual trick: "It is negative to gain weight (gain electrons) and positive to lose weight (lose electrons)."

• Question: Is it required to round the atomic mass on the periodic table when recording atomic data?

  • Answer: Writing either the exact decimal value seen on the periodic table or rounding it demonstrates the ability to locate atomic weight. However, the mass number field specifically must be recorded as a rounded whole number, because an atom cannot contain a fraction of a proton or neutron.

• Question: What is the scope and structure of the upcoming exam?

  • Answer: The exam takes place at the end of class. It consists of fill-in-the-blank questions and calculation problems formatted like expanded versions of Quiz 1 and Quiz 2, covering Chapter 3 material in depth.