CHEM 1030 Chapter 2: Light, Quantum Numbers, and Orbitals

Electromagnetic Radiation and Wave Properties

• Speed of Light ($c$): All electromagnetic radiation travels at the speed of light in a vacuum, $c \rightleftharpoons 3.00 \times 10^8\,m\,s^{-1}$. Yellow light and red light travel at the exact same speed.

• Wave Equation: Wavelength ($\lambda$) and frequency ($\nu$) are inversely related by the speed of light:   $c = \lambda \nu$

• Wavelength Unit Conversion: Wavelength must be converted to meters ($m$) to calculate frequency. The conversion factor is $1\,m = 10^9\,nm$ (or $1\,nm = 10^{-9}\,m$).

• Electromagnetic Spectrum Order:

  • Increasing Wavelength: Gamma waves < UV waves < Visible light waves < Microwaves < Radio waves

  • Increasing Energy / Frequency: Radio waves < Microwaves < Visible light waves < UV waves < Gamma waves

Energy and Frequency Relationships

• Planck's Energy Equation: Energy ($E$) is directly proportional to frequency ($\nu$) and inversely proportional to wavelength ($\lambda$):   $E = h\nu = \frac{hc}{\lambda}$

• Constants and Units:

  • Planck's constant ($h$): $6.626 \times 10^{-34}\,J\,s$

  • Energy ($E$) units: Joules ($J$

• Proportionality Effects (if frequency is doubled):

  • Energy ($E$) doubles.

  • Wavelength ($\lambda$) is halved.

  • Speed of light ($c$) remains constant.

  • Planck's constant ($h$) remains constant.

Atomic Energy Transitions

• Absorption vs. Emission:

  • Absorption: Electron moves from a lower principal energy level to a higher level ($n_{initial} < n_{final}$), absorbing energy.

  • Examples: $n = 4 \rightarrow n = 5$, $n = 3 \rightarrow n = 5$

  • Emission: Electron moves from a higher principal energy level to a lower level ($n_{initial} > n_{final}$), releasing energy.

  • Examples: $n = 2 \rightarrow n = 1$, $n = 4 \rightarrow n = 1$

• Energy Change Magnitude: Transitions involving lower principal shells (especially $n = 1$) over larger intervals ($\Delta n$) correspond to larger energy changes.

Quantum Numbers and Subshells

• Four Quantum Numbers:

  1. Principal quantum number ($n$): $n = 1, 2, 3, 4, \dots$

  2. Angular momentum quantum number ($l$): $l = 0, 1, \dots, (n - 1)$

  3. Magnetic quantum number ($m_l$): $m_l = -l, \dots, 0, \dots, +l$

  4. Spin quantum number ($m_s$): $m_s = +\frac{1}{2}, -\frac{1}{2}$

• Angular Momentum Codes ($l$):

  • $l = 0 \rightarrow s$

  • $l = 1 \rightarrow p$

  • $l = 2 \rightarrow d$

  • $l = 3 \rightarrow f$

• Maximum Electron Capacity per Subshell:

  • $s$ subshell: $2$

  • $p$ subshell: $6$

  • $d$ subshell: $10$

  • $f$ subshell: $14$

• Orbital Existence:

  • Do NOT Exist: $1p$, $2d$, $3f$ (because $l$ must be strictly less than $n$).

  • Exist: $2s$, $3p$, $3d$, $6s$.

• Invalid Quantum Sets:

  • $n = 3, l = -2, m_l = 0, m_s = +\frac{1}{2}$ (Invalid: $l$ cannot be negative).

  • $n = 2, l = 2, m_l = -1, m_s = -\frac{1}{2}$ (Invalid: $l$ must be less than $n$).

Orbital Shapes

• $s$ Orbital: Spherical shape ($l = 0$).

• $p$ Orbital: Dumbbell shape ($l = 1$).

• $d$ Orbital: Cloverleaf shape ($l = 2$).