5Study Notes on Atomic Structure and the Orbital Model

Rutherford’s Scattering Experiment and Bohr’s Atomic Model

• Rutherford’s Scattering Experiment (Streuversuch):

  • Execution: Ernest Rutherford bombarded a thin gold foil with positively charged alpha ($\alpha$) particles.
  • Core Observations:
    • The vast majority of alpha particles passed through the gold foil without any deflection.
    • A small portion of the particles were deflected from their straight-line path.
    • A very small fraction (approximately 1 in 8,000) were reflected back toward the source.
  • Conclusions/Statements:
    • The atom is not a solid sphere but consists mostly of empty space.
    • The atom contains a central, extremely dense, and positively charged region called the nucleus.
    • The nucleus contains almost the entire mass of the atom.
    • Electrons move in the space around the nucleus.

• Bohr’s Atomic Model:

  • Core Postulates:
    • Electrons move in specific, circular orbits (energy levels or shells) around the nucleus without radiating energy.
    • Angular momentum of the electron is quantized, meaning only certain orbits are stable.
    • Energy is absorbed or emitted only when an electron jumps from one allowed orbit to another ($\text{photon emission/absorption}$).
  • Definition of Electron Energy ($E$):
    • The energy of an electron in a specific orbit is defined by the principal quantum number $n$.
    • The total energy is the sum of kinetic energy and potential energy (Coulombic attraction).
    • For a hydrogen-like atom, the energy is defined as:           $E_n = -R_H \times \frac{Z^2}{n^2}$           where $R_H$ is the Rydberg constant, $Z$ is the atomic number, and $n$ is the principal quantum number.
    • The negative sign indicates that the electron is bound to the nucleus; energy is zero at an infinite distance.

Ionization Energy Calculations for Sodium ($Na$)

• Problem Context: To ionize $1\,mol$ of Sodium ($Na$), $459.8\,kJ$ of energy is required. We must determine the minimum frequency ($\nu$) and the corresponding wavelength ($\lambda$) of light capable of ionizing a single Sodium atom.

• Given Constants:

  • Planck’s Constant ($h$): $6.62618 \times 10^{-34}\,Js$
  • Speed of Light ($c$): $2.9979 \times 10^8\,m/s$
  • Avogadro's Constant ($N_A$): $\approx 6.022 \times 10^{23}\,mol^{-1}$
  • Molar Ionization Energy ($E_{molar}$): $459.8\,kJ\,mol^{-1} = 459,800\,J\,mol^{-1}$

• Step 1: Calculate Energy per Atom ($E_{atom}$):

  • $E_{atom} = \frac{E_{molar}}{N_A}$
  • $E_{atom} = \frac{459800\,J\,mol^{-1}}{6.02214 \times 10^{23}\,mol^{-1}} \times 10^3 \, \text{is not applicable here as kJ was already converted to J.}$
  • $E_{atom} \times N_A = 459.8 \times 10^3 \, J \implies E_{atom} \times 6.022 \times 10^{23} = 459,800 \, J$

• Step 2: Calculate Minimum Frequency ($\nu$):

  • Using the Einstein-Planck relation: $E = h \times \nu$
  • $\nu = \frac{E_{atom}}{h}$
  • Substituting values: $\nu = \frac{459800 / 6.022 \times 10^{23}}{6.62618 \times 10^{-34}}$
  • Calculation: $\nu \times 6.62618 \times 10^{-34} = 7.635 \times 10^{-19} \, J$
  • $\nu \times 10^{-14} \, Hz \approx 1.152 \times 10^{15}\,Hz$ ($s^{-1}$)

• Step 3: Calculate Wavelength ($\lambda$):

  • Using the wave equation: c = \nu \times \text{\lambda}
  • \text{\lambda} = \frac{c}{\nu}
  • \text{\lambda} = \frac{2.9979 \times 10^8\,m/s}{1.152 \times 10^{15}\,s^{-1}}
  • \text{\lambda} \times 10^9 \, nm \times \text{is common conversion unit.}
  • \text{\lambda} \times 10^{-7} \, m \times 2.602 \approx 260.2\,nm

The Orbital Model and Quantum Numbers

• Significance of Orbitals:

  • In the quantum mechanical model, electrons do not follow defined paths.
  • An orbital represents a mathematical wave function (Schrödinger equation) describing the region in 3D space where there is a high probability (usually $90\%$ or higher) of finding an electron.
  • Defined by the square of the wave function: |\text{\psi}|^2, known as the probability density.

• Visualizing Orbitals (Coordinate Systems):

  • $1s$-Orbital: A sphere centered at the origin ($x, y, z$). It reflects the lowest energy state with no radial nodes.
  • $2s$-Orbital: A larger sphere than $1s$. It contains one radial node (a region of zero electron probability) between the inner core and outer shell.
  • $2p_z$-Orbital: A dumbbell shape aligned exactly along the $z$-axis. It has a nodal plane in the $xy$-plane where the probability of finding an electron is zero.
  • $3p_z$-Orbital: Similar dumbbell shape along the $z$-axis but larger than the $2p_z$. It includes an additional radial node within the lobes.
  • $3d$-Orbitals: There are five distinct orientations:
    • $3d_{xy}, 3d_{xz}, 3d_{yz}$: Cloverleaf shapes positioned in the respective planes between the axes.
    • $3d_{x^2-y^2}$: Cloverleaf shape with lobes pointing directly along the $x$ and $y$ axes.
    • $3d_{z^2}$: A dumbbell shape along the $z$-axis with a torus (donut) in the $xy$-plane.

• Quantum Numbers for Principal Quantum Number $n=3$:

  • Principal Quantum Number ($n$): Specifies the main shell ($n=3$).
  • Azimuthal (Angular Momentum) Quantum Number ($l$): Range is $0$ to $n-1$.
    • $l=0$ ($s$-orbital)
    • $l=1$ ($p$-orbital)
    • $l=2$ ($d$-orbital)
  • Magnetic Quantum Number ($m_l$): Range is $-l$ to $+l$.
    • If $l=0$: $m_l = 0$
    • If $l=1$: $m_l = -1, 0, +1$
    • If $l=2$: $m_l = -2, -1, 0, +1, +2$
  • Spin Quantum Number ($m_s$): For every combination of $n, l, m_l$, there are two possible spins:
    • $m_s = +\frac{1}{2}$ and $m_s = -\frac{1}{2}$

The Aufbau Principle and Electron Configuration

• Fundamental Principles:

  • Pauli Exclusion Principle: No two electrons in an atom can have the same four quantum numbers. This implies an orbital can hold a maximum of two electrons with opposite spins.
  • Hund’s Rule: For degenerate orbitals (orbitals with the same energy, like the three $p$-orbitals), the lowest energy state is achieved when electrons occupy them singly with parallel spins before pairing begins.

• Examples in Carbon ($C$) and Titanium ($Ti$):

  • Carbon ($Z=6$): $1s^2 2s^2 2p^2$. Hund’s rule is visible in the $2p$ subshell: one electron goes into $2p_x$ and one into $2p_y$ with parallel spins, rather than pairing in $2p_x$.
  • Titanium ($Z=22$): $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^2$. Hund’s rule applies to the $3d$ subshell: the two electrons occupy different $d$-orbitals singly with parallel spins.

• Anomalous Configurations of Chromium ($Cr$) and Copper ($Cu$):

  • Chromium ($Z=24$): Expected: $[Ar] 4s^2 3d^4$; Actual: $[Ar] 4s^1 3d^5$.
  • Copper ($Z=29$): Expected: $[Ar] 4s^2 3d^9$; Actual: $[Ar] 4s^1 3d^{10}$.
  • Derived Rule: Half-filled ($d^5$) and fully-filled ($d^{10}$) subshells provide extra stability due to symmetry and reduced electron-electron repulsion.

• Electronic Configuration of the $Fe^{2+}$ Ion:

  • Neutral Iron ($Fe$, $Z=26$): $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^6$ or $[Ar] 4s^2 3d^6$.
  • $Fe^{2+}$ Ion: $1s^2 2s^2 2p^6 3s^2 3p^6 3d^6$ or $[Ar] 3d^6$.
  • Justification: When transition metals ionize, electrons are removed from the outer $s$-orbital ($4s$) before the $d$-orbital ($3d$), even though the $4s$ was filled first. This is because, in ionized states, the energy of the $3d$ orbitals drops below that of the $4s$ orbital.

• The Term "Formal Electron Configuration":

  • It is called "formal" because it is a simplified model used to track electrons.
  • In reality, electronic states in ions and complexes involve complex interactions, orbital mixing, and contraction of the electron cloud that the simple orbital filling sequence does not fully capture.
  • It serves as a bookkeeping method for oxidation states and magnetic properties.