Chemistry - Quantum-Mechanical Model of the Atom
Beginnings of Quantum Mechanics
- Classical physics believed physical phenomena were deterministic. Quantum mechanics discovered that for subatomic particles, the present condition does not determine the future condition.
- Quantum mechanics is the foundation of chemistry, explaining the periodic table, chemical bonding, and applications like lasers and computers.
Electron Behavior
- Electrons are incredibly small; a dust speck has more electrons than the number of people who have lived on Earth.
- Observing electrons changes their behavior, making direct observation impossible.
- The quantum-mechanical model explains how electrons exist and behave in atoms, helping predict atomic properties.
Nature of Light
- Light is electromagnetic radiation composed of perpendicular oscillating electric and magnetic waves.
- All electromagnetic waves travel at the speed of light: 3.00 \times 10^8 m/s.
Characterizing Waves
- Amplitude is the height of the wave, determining light intensity.
- Wavelength (\lambda) is the distance covered by the wave.
- Frequency (\nu) is the number of waves passing a point per time unit (Hz or s\text{^{-1}}).
- c = \nu \lambda: Wavelength and frequency are inversely proportional for electromagnetic waves.
Electromagnetic Spectrum
- Visible light is a small part of the electromagnetic spectrum.
- Shorter wavelength (high-frequency) light has higher energy.
- High-energy electromagnetic radiation can damage biological molecules (ionizing radiation).
Interference
- Constructive interference: Waves add to make a larger wave (in phase).
- Destructive interference: Waves cancel each other (out of phase).
Diffraction
- Diffraction occurs when waves bend around obstacles or openings of comparable size to their wavelength.
- Diffraction of light through two slits creates an interference pattern.
Photoelectric Effect
- Metals emit electrons when light shines on them (photoelectric effect).
- Einstein proposed light energy comes in packets called quanta or photons.
- E = \frac{hc}{\lambda}: Energy of a photon is proportional to its frequency and inversely proportional to its wavelength.
Atomic Spectroscopy and Bohr Model
- Atoms release energy as light when excited.
- Emission spectrum: A unique pattern of wavelengths emitted by an element.
- Rydberg's equation: \frac{1}{\lambda} = R \left( \frac{1}{m^2} - \frac{1}{n^2} \right), where R is the Rydberg constant (1.097 \times 10^7 m^{\text{-1}}), and m and n are integers.
- Bohr's model: Electrons exist in quantized orbits at fixed distances from the nucleus.
Wave Behavior of Electrons
- De Broglie proposed particles have wavelike character.
- \lambda = \frac{h}{mv}: Wavelength of a particle is inversely proportional to its momentum.
Complementary Properties
- Wave-particle duality: You cannot simultaneously observe the wave and particle nature of an electron.
Uncertainty Principle
- Heisenberg's uncertainty principle: \Delta x \cdot \Delta (mv) \geq \frac{h}{4\pi}. The more accurately you know the position of a small particle, the less you know about its velocity, and vice versa.
Determinacy Versus Indeterminacy
- Classical physics: Particles move in a path determined by velocity, position, and forces.
- Quantum mechanics: We can only predict the probability of finding an electron in a region.
Schrödinger’s Equation
- Schrödinger’s equation calculates the probability of finding an electron with a particular amount of energy at a location.
- Solutions produce wave functions (\psi) describing the wavelike nature of the electron.
- H \psi = E \psi
Quantum Numbers
- Three integer terms in the wave function determine the size, shape, and orientation of an orbital:
- Principal quantum number (n)
- Angular momentum quantum number (l)
- Magnetic quantum number (m_l)
- Spin quantum number (m_s)
Principal Quantum Number (n)
- Characterizes the size and energy of the electron; can be any integer \geq 1.
- E_n = -2.18 \times 10^{\text{-18}} J \left( \frac{1}{n^2} \right)
Angular Momentum Quantum Number (l)
- Determines the shape of the orbital; values from 0 to (n-1).
- l = 0: s orbitals (spherical)
- l = 1: p orbitals (dumbbell-shaped)
- l = 2: d orbitals
- l = 3: f orbitals
Magnetic Quantum Number (m_l)
- Specifies the orientation of the orbital in space; integers from -l to +l, including zero.
Spin Quantum Number (m_s)
- Specifies the orientation of the electron spin; either +1/2 (spin up) or -1/2 (spin down).
Orbitals and Energy Levels
- Orbitals with the same n are in the same principal energy level (shell).
- Orbitals with the same n and l are in the same sublevel (subshell).
Energy Levels and Sublevels
- Number of sublevels in a level = n.
- Number of orbitals in a sublevel = (2l + 1).
- Number of orbitals in a level = n^2.
Atomic Spectroscopy
- Each wavelength corresponds to an electron transition between quantum-mechanical orbitals.
- Excitation: Electron transitions from lower to higher energy level.
- Relaxation: Electron transitions from higher to lower energy level, releasing a photon.
- E{\text{photon}} = \Delta E{\text{electron}}
Predicting the Spectrum of Hydrogen
- Wavelengths can be predicted by calculating the energy difference between states.
Probability Density
- Represents the probability of finding an electron at a point in space.
Radial Distribution Function
- The total probability of finding an electron within a thin spherical shell at a distance r from the nucleus.
S Orbitals
- Spherical shape; number of nodes = (n - 1).
P Orbitals
- Two-lobed; one node at the nucleus, total of n nodes.
D Orbitals
F Orbitals
Phase of an Orbital
- Wave functions can have positive or negative values (phase).
- Interacting orbitals can be in phase (same sign) or out of phase (opposite signs).
