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Electron Transitions in Hydrogen

• Electrons in hydrogen atoms can gain energy and transition between energy levels.
    - Example: An electron in the ground state (n = 1) can be promoted to a higher energy state (n = 3).
    - When the electron falls back from n = 3 to n = 2, it emits light at a specific wavelength.

• Emission Spectrum of Elements
    - The light emitted during these transitions results in lines in the emission spectrum of hydrogen.
    - The emitted light encompasses various regions of the electromagnetic spectrum, including ultraviolet and infrared.

Focus on the Hydrogen Atom

• Hydrogen is particularly studied due to its simpler mathematical treatment compared to multi-electron atoms.
    - The wavelengths of the lines in hydrogen's emission spectrum are predictable by calculating energy differences between states.

• For an electron in energy state n, the number of possible transitions (lines) it can produce is n - 1:
    - E.g., if n = 3, possible transitions = 3 - 1 = 2 lines.

• The Bohr model and quantum mechanical models can accurately predict the emission spectrum for hydrogen, a single-electron system.

Electromagnetic Spectrum Regions

• Emission lines viewed through a spectroscope show only three visible lines due to hydrogen's transitions:
    - Ultraviolet Part: Emission when an electron transitions from n = 1 to a higher state and back to n = 1.
    - Visible Part: When the electron transitions from n = 2 to higher energy states and back to n = 2.
    - Infrared Part: Emission occurring when transitions from n = 3 to higher energy and back to n = 3.

Example Calculation of Wavelength

• Problem Statement: Determine the wavelength of light emitted when an electron transitions from n = 7 to n = 2.
    - Formulas Used:
      - Energy of emitted photon:
        E_{ ext{photon}} = 2.18 imes 10^{-18} ext{ joules} imes igg(\frac{1}{n_{final}^2} - \frac{1}{n_{initial}^2}\bigg)
    - Wavelength relation:
  $E = \frac{hc}{\lambda}$
        - Rearranged to:
  $\lambda = \frac{hc}{E_{ ext{photon}}}$

• Calculations:
    - Initial and final states identified:
      - n_final = 2
      - n_initial = 7
    - Squared values:
      - $n_{final}^2 = 2^2 = 4$
      - $n_{initial}^2 = 7^2 = 49$
    - Energy equation becomes:
  $E_{ ext{photon}} = 2.18 imes 10^{-18} imes \bigg(\frac{1}{4} - \frac{1}{49}\bigg)$
      - Calculation:
        - Compute the energy difference:
        - Result: $E_{ ext{photon}} = 5.01 imes 10^{-19} ext{ joules}$

• Finding the wavelength:
    - Constants:
      - Planck’s constant: $h = 6.626 imes 10^{-34} ext{ J s}$
      - Speed of light: $c = 2.998 imes 10^{17} ext{ nm/s}$
    - Plugging values into wavelength formula:
  $\lambda = \frac{(6.626 imes 10^{-34} ext{ J s})(2.998 imes 10^{17} ext{ nm/s})}{5.01 imes 10^{-19} ext{ J}}$
    - Result for wavelength:
      - $\lambda = 397 ext{ nm}$
        - This corresponds to light emitted during the transition from n = 7 to n = 2.

Orbital Shapes and Quantum Numbers

• s Orbitals (l = 0):
    - Spherical shape.
    - Each principal energy level (n) has one s orbital, which is the lowest energy level in its state.

• p Orbitals (l = 1):
    - Exist in each principal energy level above n = 1, with three p orbitals.
    - Shape resembles an hourglass, oriented along x, y, and z axes.

• d Orbitals (l = 2):
    - Present in principal energy levels above n = 2, comprising five d orbitals.
    - Fourth-lowest energy orbitals, complex shapes including toroidal regions.

• f Orbitals (l = 3):
    - Exist in principal energy levels above n = 3 with seven f orbitals.
    - Structures are considerably complex, some having eight lobes.

Why Atoms Are Spherical

• Atoms are typically depicted as spherical due to the distribution of orbitals which, when combined, create a roughly spherical shape, akin to multiple balloons tied together.

• Summary of Orbital Types:
    - A one s orbital is spherical.
    - p orbitals have hourglass shapes.
    - d orbitals involve more complex geometries with lobes and rings.
    - f orbitals present even more complicated structures with multiple lobes.

Transition to Electron Configuration

• Now moving on to Chapter Nine: Electron Configuration.
    - Refer to page seven of the fourth class handout for upcoming discussion details.