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Quantum Mechanics

- Or why mad scientists have all the fun
- Quantum Mechanics describes the behavior of electrons in an atom.
- The arrangement of electrons in atoms is termed the electronic structure of atoms.
- We will examine how quantum theory is used to explain the trends in the periodic table and the formation of bonds in molecules

The wave nature of light

- The light we see with our eyes makes a small portion of the electromagnetic spectrum called visible light.
- All electromagnetic radiation travels in a vacuum at 299 792 458 m / s. That’s fast.
- Wavelength is the distance between two successive peaks. Frequency is the number of wavelengths that pass a given point per second.
- Light carries energy that is inversely proportional to its wavelength and directly proportional to frequency.

Wavelength and Frequency

- The electromagnetic spectrum is a chart of increasing wavelength.
- The spectrum spans an enormous range, from the size of atoms to more than a mile (km)
- Frequency is expressed in cycles per second in a unit called the hertz (Hz)
- WBAP radio station at 820 on your AM radio dial is 820 kHz or 820,000 Hz or 820,000 wavelengths per second.

Quantized Energy & Photons

- The wave/particle duality of light
- In 1900 a German physicist named Max Plank (1858-1947) discovered that energy can be absorbed or emitted in discrete chunks or quanta.
- E = hv
- The constant h is called Plank’s constant and has a value of 6.626 x 10-34 J-s

Quantized Energy & Photons

- According to Plank’s theory matter can emit or absorb light in only whole-number multiples of hv.
- Each energy packet of electromagnetic radiation is called a photon – the particle aspect of the wave/particle nature of light.
- In the Photoelectric effect there is a minimum energy requirement to eject an electron, called the work function.

Bohr’s Model

- Rutherford’s discovery of the nuclear nature of the atom suggested that the atom can be thought of as a microscopic solar system.
- Bohr based his model on three postulates
- Only orbits of certain radii or energy level are permitted
- An electron in a permitted orbit has a “allowed” energy state. An electron in an “allowed” energy state will not radiate energy.
- Energy is emitted or absorbed by the electron only as the electron changes from one allowed energy state to another.

Energy States of the Hydrogen Atom

- Bohr calculated the energies corresponding to each allowed orbit for the electron in the hydrogen atom.
- E = (-2.18 x 10-18 J)(1/n2)
- The number -2.18 x 10-18J is a product of three constants.
- The number n is called the principal quantum number and ranges from 1 to ∞

Energy States of the Hydrogen Atom

- The lower (more negative) the energy the more stable the atom will be.
- N = 1 is the lowest energy state and is called the ground state of the atom.
- When n > 1 the atom is said to be in an excited state.
- When n = ∞ the energy is zero and the electron is completed separated from the atom.

Energy states of the hydrogen atom

- Energy must be absorbed for an electron to be moved into a higher orbit. (higher value of n)
- Energy is emitted when an electron falls from a higher orbit to a lower orbit.
- From Bohr’s postulates only specific frequencies of light can be absorbed or emitted by the atom.
- ΔE = Ef – Ei = Ephoton = hv

Limitations of Bohr Model

- The Bohr Model only explains the hydrogen atom
- Subsequent atoms get further away from Bohr’s model.
- But Bohr’s model introduces two very important aspects
- Electrons exist only in certain discrete energy levels
- Energy is involved in moving at electron from one level to the next

The wave behavior of matter

- Louis de Broglie suggested that all matter has both wavelike and particle behavior.
- λ = h/mv
- where h is Plank’s constant, m is mass of the object and v is the velocity,

The Heisenberg Uncertainty Principle

- German physicist Wener Heisenberg proposed that the dual nature of matter places a fundamental limitation on how precisely we can know both the location and the momentum of any object.
- When applied to electrons we determine that it is impossible to know simultaneously both the exact momentum and exact position.
- Δx Δ(mv) ≥ h/4π(The uncertainty of an electron is 10-9 m)

Quantum mechanics and atomic orbitals

- Erwin Schrödinger proposed his wave equation that incorporates both the wavelike behavior and the particle-like behavior of the electron.
- If Schrödinger’s equations leads to a series of mathematical functions called wave functions.
- Wave functions yield a probability of electron density distribution

Orbitals and Quantum Numbers

- The solution to Schrodinger’s equation for the hydrogen atom yields a set of wave functions and energies called orbitals.
- There are three quantum numbers, n, l, m to describe an orbital.

Orbitals and Quantum Numbers

- Principal quantum number n can have a value of 1,2,3 … ∞
- The second quantum number l is the angular quantum number and can have values from 0 to n – 1
- The magnetic quantum number m can have values ranging from –l to l

Electron Shells and Subshells

- The collection of orbitals with the same value of n is called an electron shell.
- The set of orbitals that have the same n and l values is called a subshell
- Each subshell is designated by a number (the value of n) and a letter (s, p, d, f corresponding to the value of l.

Observations about quantum numbers

- The shell with principal quantum number n will consist of exactly n subshells. Each subshell corresponds to a different allowed value of l from 0 to n – 1.
- Each subshell consists of a specific number of orbitals. Each orbital corresponds to a different allowed value of ml
- The total number of orbitals is a shell is n2 where n is the principal quantum number.

The electron shell

- The collection of orbitals with the same value of n is called an electronic shell.
- The set of orbitals that have the same n and l is called a subshell.

Representations of orbitals

- The s orbital is spherical symmetric
- Plotting a radial probability function yields the probability of finding an electron versus the distance from the nucleus.

Many-Electron Atoms

- In a many electron atom, for a given value of n, the energy of an orbital increases with increasing value of l.

Electron Spin

- Each electron has an intrinsic property called electron spin that causes each electron to behave as if it were a tiny sphere spinning on its own axis.
- Electron spin is quantized and is denoted ms
- The only two possible values for ms are +1/2 and -1/2

Pauli exclusion principle

- No two electrons in an atom can have the same set of four quantum numbers n, l, ml, ms

Electron Configurations

- The way the electrons are distributed among the various orbitals of an atom is termed electronic configuration of the atom.
- Using the Pauli exclusion principle we can state that orbitals are filled in order of increasing energy with no more than two electrons per orbital.
- Hund’s rule states that for degenerate orbitals, the lowest energy is attained when the number of electrons with the same spin is maximized.

Condensed Electron Configuration

- He -2s2
- C - 1s2 2s2 2p2 or [He] 2s2 2p2
- Ne - 1s2 2s2 2p6
- Na – [Ne] 3s1
- Mn – [Ar] 4s2 3d5
- Zn – [Ar] 4s2 3d10
- After the d orbitals are filled the p orbitals are filled.

Electronic Configurations and the Periodic Table

- The periodic table is the best choice for selecting the order in which orbitals are filled.
- Exceptions to the rule – chromium, copper, molybdenum and silver
- The exceptions occur when there enough electrons to lead to precisely half-filled sets of degenerate orbitals or to completely fill a d subshell as in copper.

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