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### Honors Chemistry

Chapter 7: Quantum Mechanics

7.1 Wave Properties

- Wavelength (l) = distance between two in-phase points
- Measured in meters
- Frequency (n) = number of waves per second
- Measured in Hertz (Hz)
- Amplitude (y) = distance of maximum displacement from rest position
- Amplitude corresponds to wave energy

7.1 The Wave Equation

- v = ln
- Find the wavelength of a 256 HZ (middle C) sound wave traveling at 343 m/s.
- v = ln
- 343 m/s = l (256 Hz)
- l = 1.34 m
- Try this….
- Find the frequency of a 25.0 cm wave traveling at 0.75 m/s.

7.1 Electromagnetic Radiation

- James Clerk Maxwell (1873)
- Mathematical description of light waves
- Light is an electromagnetic wave
- Speed of light (c) is constant
- c = 2.99792458 x 108 m/s
- To 3 sig dig, 3.00 x 108 m/s is fine
- Try this….
- Find the frequency of a 250 nm light wave. (Don’t forget about the “nano” prefix!)

7.1 Electromagnetic Spectrum

- Radio, micro, IR, ROYGBIV, UV, X, g
- long l ------------------------------- short l
- low n -------------------------------- high n
- Radio wave end of the spectrum is low energy radiation
- Gamma ray end is high energy radiation
- Black body radiation
- Wave theory fails to account for this!

7.1 Quantum Theory

- Max Planck (1900)
- Energy is emitted and absorbed only in small, discrete packets called quanta
- Energy of a quantum of energy given byE = hn
- h = 6.626 x 10-34 Js (Planck’s constant)
- Correctly accounts for blackbody curves
- Planck has no idea why it works!

7.1 Quantum Theory

- Find the energy of a 2.50 x 1014 Hz light wave.
- E = hn
- E = (6.626 x 10-34 Js)(2.50 x 1014 Hz)
- E = 1.66 x 10-19 J
- A quantum holds a tiny amount of energy!
- Try this….
- Find the energy of a 475 nm light wave.
- Hint: Use the wave equation first!

7.2 The Photoelectric Effect Bright light ejects more electrons Quantum theory explains results Light is made of quanta called photons

- Albert Einstein (1905)
- Electrons ejected from surface of metal exposed to light
- Depends on frequency of light
- Electrons ejected at a certain cutoff frequency
- Above cutoff n, electrons leave with more energy

7.3 Spectroscopy

- Emission spectra – light given off by glowing objects
- Can be continuous or discontinuous
- Line spectra – series of bright lines emitted by gas phase atoms
- Pattern of bright lines is characteristic of the element that is glowing
- Absorption spectra – dark lines in spectrum as light passes through a gas

7.3 Bohr’s Model

- Niels Bohr (1913)
- Electron energies are quantized
- Only certain orbits are allowed
- - RHEn = ------ n2
- RH = 2.18 x 10-18 J (Rydberg constant)
- n = 1, 2, 3, 4, ….

7.3 Bohr’s Model

- DE = Ef – E0
- -RH -RHDE = ----- - ----- nf2 n02
- Factor out RH
- 1 1 DE = RH (----- - ----- ) n02 nf2

Link to Hydrogen energy states

7.3 Bohr’s Model

- Find the energy of a photon of light emitted by an electron jumping from level 5 down to level 2.
- DE = RH (1/n52 – 1/n22)
- DE = (2.18 x 10-18 J)(1/25 – 1/4)
- DE = -4.58 x 10-19 J
- Try this….
- Find the energy of the jump from level 1 to level 4.
- Find the frequency of the light produced.

7.4 Duality

- Louis de Broglie (1924)
- Electrons can be treated as waves
- Each orbit must contain a whole number of waves…explains orbit quantization!
- hl = ---- mv
- mv is momentum (p), so we can write l = h/p
- Verified by Davisson, Germer, and Thomson

Link to quantum atom model

7.4 Duality

- Find the wavelength of a 3.00 kg duck flying at 5.00 m/s.
- l = h/mv
- l = (6.626 x 10-34 Js) / (3.00 kg)(5.00 m/s)
- l = 4.42 x 10-35 m
- Try this….
- Find the wavelength of an electron traveling at 500,000 m/s. (me = 9.11 x 10-31 kg)

7.5 Uncertainty Principle

- Werner Heisenberg (1926)
- Complementary variables cannot be known to arbitrary precision
- dp dq ≥ħ/2
- Minimum limits to uncertainties in values are inversely proportional
- Position and momentum are an important complementary pair
- dx dpx≥ħ/2

7.5 Uncertainty Principle

- Find the uncertainty in velocity of an electron confined to a hydrogen atom (dx = 0.037 nm).
- dx dpx≥ħ/2
- (3.7 x 10-11 m) dp ≥ 5.27 x 10-35 Js
- dp ≥ 1.4 x 10-24 kg m/s
- p = mv
- 1.4 x 10-24 kg m/s = (9.11 x 10-31 kg) dv
- dv = 1.5 x 106 m/s

7.5 Uncertainty Principle Uncertainty limits are not significant for macroscopic objects, but they are significant to subatomic particles Cannot know the position and momentum of an electron at the same time! Concept of orbits will not work

- Try this…
- Find the uncertainty in position of a 20.0 mg fly whose position is known to within ±0.5 mm.

7.5 Quantum Mechanics

- Erwin Schrödinger (1926)
- Schrödinger equation – treat electron as a standing wave surrounding the nucleus
- Schrödinger equation is ugly!
- Solve for amplitude function (y)
- Remember – amplitude is energy
- Produces an energy diagram like Bohr’s, but this one actually works
- Wave function has no physical meaning

7.5 Copenhagen Interpretation

- Max Born (1926)
- y2 denotes probability
- Electron is delocalized
- Wave function collapses on observation
- electron density refers to magnitude of the probability wave for the electron
- Orbital = spatial probability distribution
- Electron clouds
- Objections: Schrödinger’s Cat
- Other interpretations

7.6 Quantum Numbers Describes energy level of the shell Defines the size of the electron cloud

- Set of numbers that describe the distribution of electrons in the atom
- Principal Quantum Number (n)
- n = 1, 2, 3, 4, …
- Corresponds to the n value used by Bohr

7.6 Quantum Numbers Sublevels of the energy level Angular distribution of electron cloud For hydrogen, sublevels are degenerate Correspond to fine structure spectral lines

- Angular Momentum Quantum Number (l)
- l = 0, 1, 2, … , n – 1
- There are a total of n values

- l = 0 is s orbital l = 2 is d orbital
- l = 1 is p orbital l = 3 is f orbital

7.6 Quantum Numbers Number of degenerate orbitals in sublevel Spatial orientation of the orbital Zeeman Effect Electron Spin Quantum Number (ms)

- Magnetic Quantum Number (ml)
- ml = -l, … , 0, …, +l
- There are a total of 2l + 1 values

- ms = +½, -½
- Two possible electron spin states
- Spin up, spin down

7.7 Atomic Orbitals

- s, p, d, f orbitals
- Radial probability distributions
- distance from nucleus of high e- probability

7.7 Atomic Orbitals

- Angular probability distributions
- Show regions of high e- probability
- Cool 3d pictures

7.7 Orbital Energies

- n determines energy
- For H, all subshells are degenerate
- Multielectron atomseach subshell liesat a different energy
- Shielding effect
- Fill lowest energyorbitals first

7.7 The Diagonal Rule Diamagnetic

- Rule of thumb
- Shows the orderin which orbitalsare filled
- Paramagnetic
- Unpaired e-
- Attracted to mag

- Paired e-
- Not attracted

7.8 Electron Configurations

- H has 1 electron
- Put it in 1s
- Write it 1s1
- Read “one-s-one”
- What about He?
- You got it…1s2
- Keep filling 1s until it is full
- But when is it full?

7.8 Pauli Exclusion Principle

- No two electrons may share the exact set of quantum numbers
- Consider Helium’s 1s2 configuration
- First electron: n = 1, l = 0, ml = 0, ms = +½
- Second electron: same n, l, ml
- ms must be -½
- No room for more electrons in 1s orbital
- Each orbital can hold only two electrons!

7.8 Electron Configurations

- What is the electron configuration of Li?
- 1s2 2s1
- What about N?
- 1s2 2s2 2p3
- But how are p electrons organized?
- Hund’s Rule – arrange electrons in such a way as to maximize total spin state
- Put e-’s in separate orbitals, same spin

7.9 Aufbau Principle

- Build up on previous e- configurations
- Each atom adds one more e-
- Express configuration with noble gas core
- Al = 1s2 2s2 2p6 3s2 3p1
- First 3 terms are the same as Ne config.
- Write it as [Ne] 3s2 3p1

7.9 Exceptions Some transition metals rearrange Cr = [Ar] 4s2 3d4 What other family would do this? Cu = [Ar] 4s2 3d9 Ar 4s1 3d10

- Particularly stable configurations
- Full sublevel
- Half-full sublevel

- Just missed the stable half-full 3d5
- Kick one e- up to d to get [Ar] 4s1 3d5

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