Atomic Structure, Isotopes, Electron Configurations, and Periodic Trends

Atomic Structure, Mass Number, and Isotopic Atomic Mass Calculations

• Definition of Isotopes:

  • Isotopes are defined as atoms of the same chemical element that possess the identical atomic number but differ in their number of neutrons.
  • Because atoms of the same element have the same atomic number, they contain an equal number of protons and an equal number of electrons, which gives the identity of that element.

• Naturally Occurring Carbon Isotopes:

  • Carbon has an atomic number of $6$ (meaning every carbon atom contains $6$ protons and $6$ electrons).
  • There are three naturally occurring isotopes of carbon, differentiated by their mass numbers:
    • Carbon-12 ($_6^{12}\text{C}$): Mass number = $12$. Neutrons = $12 - 6 = 6\,neutrons$.
    • Carbon-13 ($_6^{13}\text{C}$): Mass number = $13$. Neutrons = $13 - 6 = 7\,neutrons$.
    • Carbon-14 ($_6^{14}\text{C}$): Mass number = $14$. Neutrons = $14 - 6 = 8\,neutrons$.

• Mass Number vs. Atomic Mass (Atomic Weight):

  • Mass Number: Always a whole integer because it represents the discrete count of subatomic particles (protons plus neutrons).
  • Atomic Mass (Atomic Weight): A decimal number found on the periodic table, representing the weighted average mass of all naturally occurring isotopes of an element due to their natural abundance.

• Chlorine Isotope and Atomic Weight Calculation Case Study:

  • Chlorine has an atomic number of $17$.
  • Chlorine possesses two naturally occurring isotopes:
    • Chlorine-35: Mass number = $35$. Neutrons = $35 - 17 = 18\,neutrons$. Isotopic mass = $34.97\,amu$. Natural abundance = $75.78\,\%$.
    • Chlorine-37: Mass number = $37$. Neutrons = $37 - 17 = 20\,neutrons$. Isotopic mass = $36.97\,amu$. Natural abundance = $24.22\,\%$.
    • Total relative percent abundance sum: $75.78\,\% + 24.22\,\% = 100.00\,\%$.
  • Atomic Weight Calculation Equation:         $\text{Atomic Mass} = \sum \left( \text{Isotopic Mass} \times \frac{\text{Percent Abundance}}{100} \right)$
  • Step-by-Step Numerical Substitution:         $\text{Atomic Mass} = \left( 34.97 \times \frac{75.78}{100} \right) + \left( 36.97 \times \frac{24.22}{100} \right)$$\text{Chlorine-35 Contribution} = 26.5003\,amu$$\text{Chlorine-37 Contribution} = 8.9541\,amu$$\text{Sum} = 26.5003 + 8.9541 = 35.4544\,amu$
  • Rounding to four significant figures yields an atomic weight of $35.45\,amu$, matching the atomic mass reported on the periodic table for chlorine.

Principal Energy Levels, Shells, and Orbital Capacity

• Anatomy of the Atom:

  • Nucleus: Dense core situated at the center of the atom containing protons and neutrons; accounts for virtually all of the atom's mass.
  • Electron Cloud: Surroundings occupied by electrons; accounts for virtually all of the atom's volume.

• Bohr Model Example (Nitrogen Atom):

  • Nitrogen symbol and notation: $_7^{15}\text{N}$ (Atomic number = $7$, Mass number = $15$).
  • Subatomic particle composition: $7\,protons$, $7\,electrons$, and $15 - 7 = 8\,neutrons$ inside the nucleus.
  • Electrons reside in concentric circular orbits termed shells surrounding the central nucleus.

• Principal Energy Levels (Shells):

  • Shells are designated by integer principal quantum numbers $n = 1, 2, 3, 4\dots$
  • $n = 1$ is the shell closest to the nucleus, possessing the lowest energy.
  • As shell number $n$ increases:
    • Distance from the nucleus increases.
    • Energy level increases.
    • Total maximum electron capacity increases.

• Maximum Shell Capacity Formula:     $\text{Maximum Electrons per Shell} = 2n^2$

  • Shell 1 ($n = 1$): $2(1)^2 = 2\,electrons\,maximum$.
  • Shell 2 ($n = 2$): $2(2)^2 = 8\,electrons\,maximum$.
  • Shell 3 ($n = 3$): $2(3)^2 = 18\,electrons\,maximum$.
  • Shell 4 ($n = 4$): $2(4)^2 = 32\,electrons\,maximum$.

• Subshells and Orbitals:

  • Shells are subdivided into subshells, designated by the letters $s, p, d, f$.
  • Subshells are further subdivided into orbitals (three-dimensional spaces where the probability of finding an electron is high).
  • Universal Rule: Each specific type of orbital can hold a maximum of $2\,electrons$.
  • Subshell Configurations:
    • $s\text{ subshell}$: Contains $1\text{ orbital}$ (holds maximum $2\,electrons$).
    • $p\text{ subshell}$: Contains $3\text{ orbitals}$ (holds maximum $6\,electrons$).
    • $d\text{ subshell}$: Contains $5\text{ orbitals}$ (holds maximum $10\,electrons$).
    • $f\text{ subshell}$: Contains $7\text{ orbitals}$ (holds maximum $14\,electrons$).

• Apartment Building Analogy:

  • The atomic nucleus functions like a parking garage at the ground level.
  • First Floor ($n = 1$): Contains $1\text{ large room}$ ($1s\text{ subshell}$), housing a maximum of $2\text{ tenants}$ (electrons).
  • Second Floor ($n = 2$): Divided into $2\text{ rooms}$ ($2s\text{ and } 2p\text{ subshells}$), housing a maximum of $8\text{ tenants}$.
  • Third Floor ($n = 3$): Divided into $3\text{ rooms}$ ($3s, 3p, \text{and } 3d\text{ subshells}$), housing a maximum of $18\text{ tenants}$.
  • Fourth Floor ($n = 4$): Divided into $4\text{ rooms}$ ($4s, 4p, 4d, \text{and } 4f\text{ subshells}$), housing a maximum of $32\text{ tenants}$.

• Orbital Geometries and Shapes:

  • $s\text{ orbitals}$: Spherical shape.
  • $p\text{ orbitals}$: Dumbbell-shaped along three orientation axes ($p_x, p_y, p_z$).
  • $d\text{ orbitals}$: More complex geometries.
  • $f\text{ orbitals}$: Even more complex geometries.

Rules for Electron Configuration and Orbital Filling

• Ground State Electron Configuration Principles:

  • Electron configuration describes how electrons are distributed across atomic orbitals.
  • Rule 1 (Aufbau Principle): Electrons are placed first in the lowest energy orbitals (closest to the nucleus) and fill progressively according to increasing energy.
  • Rule 2: Each orbital can hold a maximum of $2\,electrons$.
  • Rule 3 (Hund's Rule): When orbitals of equal energy (half-filled states) are available (such as $p_x, p_y, p_z$), electrons are placed singly into each orbital before pairing them up.

• Detailed Distribution Case Studies:

  • Nitrogen Atom ($Z = 7$):
    • Total electrons = $7$.
    • First shell ($n = 1$): Receives $2\,electrons$ ($1s^2$).
    • Second shell ($n = 2$): Receives remaining $5\,electrons$ ($2s^2 2p^3$).
    • $2p\text{ subshell filling}$: One electron is placed in $2p_x$, one in $2p_y$, and one in $2p_z$ (all pointing upwards as single electrons before pairing).
  • Neon Atom ($Z = 10$):
    • Total electrons = $10$.
    • First shell ($n = 1$): Receives $2\,electrons$ ($1s^2$).
    • Second shell ($n = 2$): Receives maximum capacity of $8\,electrons$ ($2s^2 2p^6$).
    • $2p\text{ subshell filling}$: Each orbital ($2p_x, 2p_y, 2p_z$) contains $2\,electrons$ (paired up-arrow and down-arrow).

• Orbital Filling Order Sequence:

  • Orbitals fill in order of increasing energy:         $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \rightarrow 5s \dots$
  • Note: The $3d$ subshell is filled after the $4s$ subshell because $3d$ is higher in energy than 4s$.\n\n* **Electron Pairing and Spin Orientation**:\n * Electrons possess negative charges and naturally exert repulsive forces against one another.\n * Two electrons occupy the exact same orbital by exhibiting paired, opposite spins (one spinning in one direction represented as an up arrow, the other spinning in the opposite direction represented as a down arrow).\n * Opposite spin orientations allow repulsive like charges to stably co-exist within the same orbital.\n\n# Periodic Table Structure, Subshell Blocks, and Notation Methods\n\n* **Periodic Table Subshell Blocks**:\n * **s\text{-Block}**: Groups 1A and 2A (leftmost columns).\n * **p\text{-Block}**: Groups 3A through 8A (rightmost columns).\n * **d\text{-Block}**: Transition elements situated in the central region.\n * **f\text{-Block}**: Inner transition elements at the bottom (Lanthanides and Actinides).\n * Period numbers (1$through$7$) directly correspond to the principal shell number$n$.

• Chlorine Configuration Analysis ($\text{Cl}$, $Z = 17$):

  • Electron fill path: $1s^2 2s^2 2p^6 3s^2 3p^5$.
  • Sum verification: $2 + 2 + 6 + 2 + 5 = 17\,electrons$.
  • Highest principal shell level $n = 3$ confirms Period 3 location.

• Strontium Configuration Analysis ($\text{Sr}$, $Z = 38$):

  • Element Properties: Located in Period 5, Group 2A, $s\text{-block}$, atomic number 38$.\n * **Long-hand Electron Configuration**:\n        1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2\n * Electrons Sum Verification: 2 + 2 + 6 + 2 + 6 + 2 + 10 + 6 + 2 = 38\,electrons.\n * Highest principal shell level n = 5 confirms Period 5 location.\n * Termination in 5s^2 confirms Group 2A location.\n * **Short-hand (Noble Gas) Notation**:\n * Identify the nearest preceding noble gas: Krypton (\text{Kr}$,$Z = 36).\n * \text{Kr}$accounts for the core inner-shell electron configuration of$36\,electrons$($1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6).\n * Shorthand expression: [\text{Kr}] 5s^2.\n\n* **Special Group Classifications**:\n * Group 1A: Alkali metals.\n * Group 2A: Alkaline earth metals.\n * Group 7A: Halogens.\n * Group 8A: Noble gases.\n * Central block: Transition metals.\n * Bottom rows: Lanthanides and Actinides.\n\n# Valence Shells, Valence Electrons, and Lewis Dot Symbols\n\n* **Valence Shell and Valence Electron Definitions**:\n * **Valence Shell**: The outermost occupied principal energy level shell (the shell with the highest value of n).\n * **Valence Electrons**: Electrons located inside the outermost/valence shell. These electrons determine the chemical behavior and properties of the element.\n\n* **Valence Electron Determinations**:\n * **Strontium** (\text{Sr}$,$Z = 38$): Outer shell is shell$5$($5s^2$). Contains$2\text{ valence electrons}.\n * **Chlorine** (\text{Cl}$,$Z = 17$): Configuration$1s^2 2s^2 2p^6 3s^2 3p^5$. Highest shell$n = 3$. Valence electrons =$2 + 5 = 7\text{ valence electrons}.\n * **Beryllium** (\text{Be}$,$Z = 4$): Configuration$1s^2 2s^2$. Highest shell$n = 2$. Valence electrons =$2\text{ valence electrons}.\n * **Nitrogen** (\text{N}$,$Z = 7$): Configuration$1s^2 2s^2 2p^3$. Highest shell$n = 2$. Valence electrons =$2 + 3 = 5\text{ valence electrons}. Belongs to Group 5A.\n * **Neon** (\text{Ne}$,$Z = 10$): Configuration$1s^2 2s^2 2p^6$. Highest shell$n = 2$. Valence electrons =$2 + 6 = 8\text{ valence electrons}. Belongs to Group 8A.\n\n* **Main Group Valence Shortcut Rule**:\n * For main group elements (A-group elements, Groups 1A through 8A):\n        \text{Group Number} = \text{Number of Valence Electrons}\n * *Exception*: Helium is in Group 8A but contains only 2\text{ valence electrons}.\n\n* **Lewis Dot (Electron Dot) Structures**:\n * Represent valence electrons as dots drawn around the atomic chemical symbol.\n * **1 to 4 Valence Electrons**: Drawn as single unpaired dots around the symbol.\n * **5 to 8 Valence Electrons**: Dots begin pairing up on the sides of the symbol.\n * **Group Summary for Main Group Elements**:\n * Group 1A: 1\text{ dot}$(e.g.,$\text{H}).\n * Group 2A: 2\text{ unpaired dots}.\n * Group 3A: 3\text{ unpaired dots} (e.g., Boron to Gallium).\n * Group 4A: 4\text{ unpaired dots}$(e.g.,$\text{C}).\n * Group 5A: 5\text{ dots}$($1\text{ pair}$,$3\text{ unpaired}).\n * Group 6A: 6\text{ dots}$($2\text{ pairs}$,$2\text{ unpaired}$; e.g.,$\text{O}).\n * Group 7A: 7\text{ dots}$($3\text{ pairs}$,$1\text{ unpaired}$; e.g.,$\text{Cl}).\n * Group 8A: 8\text{ dots}$($4\text{ pairs}$; except Helium which has$2\text{ dots}).\n\n# Periodic Trends in Atomic Size\n\n* **Atomic Size Model**:\n * Atoms are modeled as spheres; size is quantified by atomic radius or atomic diameter.\n\n* **Trend Across a Period (Left to Right)**:\n * **Behavior**: Atomic size **decreases** moving from left to right across a period.\n * **Mechanism**: As protons are added to the nucleus across a period, the increasing positive nuclear charge pulls the outer valence electrons closer to the nucleus.\n * **Period 2 Example** (\text{B}, \text{C}, \text{N}, \text{O}):\n * Largest atomic size: Boron (\text{B}).\n * Smallest atomic size: Oxygen (\text{O}).\n\n* **Trend Down a Group (Top to Bottom)**:\n * **Behavior**: Atomic size **increases** moving from top to bottom down a group/column.\n * **Mechanism**: Principal energy shells (n) are added with each lower period, placing the outermost valence electrons at greater distances from the nucleus.\n * **Group 1A Example** (\text{H}, \text{Li}, \text{Na}, \text{K}):\n * Smallest atomic size: Hydrogen (\text{H}).\n * Largest atomic size: Potassium (\text{K}$$).