Chapter 7 Page 1

CHAPTER SEVEN: QUANTUM THEORY AND THE ATOM

Part One: Light Waves, Photons, and Bohr Theory

A. The Wave Nature of Light (Section 7.1) 1. Structure of atom had been established as cloud of electrons around a heavy nuclear

core = the nuclear atom picture. 2. Beyond that, nothing was known of arrangement of the electrons. 3. Final clues came from spectroscopy = interaction of matter with light. 4. Atomic emission spectra = light emitted from excited atoms, analyzed into

wavelength components. 5. Therefore some background facts about light are needed. 6. Light is energy propagating as an oscillating electromagnetic field. 7. The electromagnetic wave:

8. Wavelength λ = distance between successive peaks or troughs.

9. Frequency ν = oscillations per second or cycles per second. 10. Speed of light = c = 3.00 x 108 m/s.

λν = c So: λ = c/ν ν = c/λ

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11. Isaac Newton first separated light into its component wavelengths (or colors) using a

prism. Process is called dispersion.

12. The Electromagnetic Spectrum.

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B. Quantum Effects and Photons 1. Max Planck (1900) showed that light also has particle-like characteristics. Comes in

particles he called photons. Energy of a photon of light:

E = hν Planck’s constant: h = 6.626 x 10-34 J s, where J = joules and s = seconds. Could also write:

E = hc/λ 2. Problem: The yellow light emitted from excited Na atoms has a wavelength of 590

nm. Calculate its frequency and the energy of a photon of this light. λ = 590 nm λν = c ν = c/ λ = 3.00 x 108 meter/second 590 nm = 3.00 x 108 meter/second 590 x 10-9 meter = 5.08 x 1014 s-1

[106 = million; 109 = billion; 1012 = trillion] This ν = 508 x 1012 s-1 or 508 trillion cycles per sec!! Now: E = hν = 6.626 x 10-34 J s x 5.08 x 1014 s-1 = 3.37 x 10-19 J A whole mole of Na atoms emitting one photon would emit how much energy? 6.022 x 1023 particles/mol x 3.37 x 10-19 J = 2.03 x 105 J/mol

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Convert to calories: (4.184 J = 1 cal) 2.03 x 105 J/mol x 1 cal/4.184 J = 4.85 x 104 calories/mol

Enough energy to raise 4.85 x 104 grams or 48 kg of water by 1° C. C. Photoelectric Effect.

1. Led to confirmation of particle-like nature of light. 2. Observations:

a. e- ejected only if light has ν above some threshold frequency νο: ν > νο

b. If ν is below νο, no e- ejected no matter how bright (or intense) the light source. 3. Implication (Einstein - 1905): Electrons in the metal are interacting with individual particles of light (photons). This means energy of a single photon must be above the threshold energy φ:

E > φ φ = work function of the metal = min. energy required to eject an e-.

D. Atomic Spectra and the Bohr Atom. (Section 7.3)

1. Electric current through a gas in vacuum tube excites the atoms of gas. Can also be

done in a flame. 2. As they re-emit the light, the light is dispersed through a prism or diffraction grating

into its component colors (or wavelengths). This is the emission spectrum.

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3. Ordinary sunlight or incandescent light gives a continuous spectrum of light. 4. Excited atoms in a vacuum only emit certain characteristic wavelengths of light,

called a line spectrum. (see Figure 7.2). 5. Pattern of emission is different for every element. These spectra serve as

“fingerprints” for identification of elements.

6. Balmer (1885) showed that the wavelengths in the visible spectrum of Hydrogen can

be fit by a simple formula:

€

1λ

= 1.097 × 107m−1 1

22−

1

n2

- Balmer Equation

7. Rydberg found an equation that reproduces the λ of the all lines in Hydrogen emission

spectrum, including ultraviolet and infrared:

€

1λ

= R1

n12−

1

n22

- Rydberg Equation

R = 1.097 x 107 m-1. (“Rydberg constant”) n1, n2 are positive integers and n1 < n2

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8. Bohr (1913) explained Rydberg equation by his theory of H atom. a. Electron orbits the nucleus in circular orbits. b. Orbital energy is quantized; i.e., it can have only certain distinct values. electron’s energy: En = - a/n2 (n = 1, 2, 3...)

n = quantum number (tells what “quantum state” the electron inhabits)

Bohr’s energy level diagram: c. Bohr Frequency Rule: The atom can only absorb or emit light having just the

right energy (and thus frequency or wavelength) to move the e- between these energy levels.

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Therefore:

€

Ephoton = En2− En1

n2 > n1

=-a

n22−

-a

n12

€

Ephoton = a1

n12−

1

n22

Ephoton = hν = hc/λ

€

hc / λ = a1

n12−

1

n22

€

1 / λ =ahc

1

n12−

1

n22

where

€

ahc

= R

9. Whole H emission spectrum explained this way:

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10. Bohr theory failed to explain atoms bigger than Hydrogen. 11. “Orbit” idea also had important defects on physical grounds:

According to physics an orbiting charge should be continuously emitting radiation and losing energy. Orbit should spiral inward.

12. Orbiting particle idea had to be scrapped! Replaced by a wave picture of the electron.

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Part Two: Quantum Mechanics and Quantum Numbers

A. The Wave Nature of Matter. (Section 7.4)

1. 1925 - Louis de Broglie articulated the phenomenon called the wave/particle duality =

microscopic particles possess wave-like character, and vice versa. i.e. - just as light usually seems wave-like and yet has particle-like behavior (photons),

electrons also have both particle-like and wave-like behavior. λ = h/mv

λ = characteristic wavelength of particle of mass m traveling at speed v h = Planck’s constant 2. Predict λ of an electron having speed v typical of electrons in stable atoms: Given: m = 9.11 x 10-28 g v = 1.0 x 107 m/s

€

λ =6.626 × 10−34 J s

9.11× 10-28 g( ) × 1.0 × 107m/s( )

1 J = 1 kg m2/s2, so use mass in kg

€

λ =6.626 × 10−34 kg m2

s2 s

9.11× 10-31 kg( ) × 1.0 × 107 m/s( )

λ = 0.73 x 10-10 m = 0.73 x 10-2 nm = 0.73 Å This is about the size of an atom!! Therefore, on atomic length scale, electrons behave more like waves than particles!!!

3. Predict λ of a Nolan Ryan fast ball. Given: m = 5.25 oz v = 92.5 mi/h

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λ = 1.1 x 10-34 m Much too small to be measurable. Therefore, everyday objects like baseballs manifest apparently strictly particle-like

behavior. 4. De Broglie equation proved by Davidson-Germer experiment two years later. Electrons shown to diffract just like waves.

B. Quantum Mechanical Picture of the Atom. (Section 7.4) 1. 1920’s - Quantum mechanics (wave mechanics) replaced Newtonian mechanics. 2. Treats microscopic particles according to their wave-like character. 3. 1927 - Heisenberg Uncertainty Principle:

It is impossible to measure both the velocity and position of an e- simultaneously to an arbitrarily high degree of precision.

Therefore, cannot view the e- as following a precise trajectory around the nucleus. 4. The more precisely you measure the position, the less precisely you can

simultaneously measure its momentum (or velocity), and vice versa.

€

Δx ⋅ Δpx ≥h2π

Δx ⋅ Δvx ≥h

2πm

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5. Basic Postulates of Quantum Mechanics: a. Atoms and molecules can exist only in certain energy states. b. When they change energy, they must absorb or emit precisely the required

energy to place them in the new energy state.

c. The allowed energy states are indexed by sets of numbers called quantum

numbers. d. Electron in an atom is treated as a standing wave (not a traveling wave, like

light). e. It’s position is prescribed probabilistically according to a wave function ψ(x,y,z). f. ψ(x,y,z) is a mathematical function of spatial coordinates (x,y,z) which is found

by solving the Schrodinger equation (1926):

€

−h2

8π2m∂2ψ

∂x2+∂2ψ

∂y2+∂2ψ

∂z 2

+Vψ = Eψ

g. This equation has as many solutions as the atom has quantum states. h. The solutions are indexed by four quantum numbers: n, l, ml , ms. i. The wave function

€

ψn , l ,m l ,ms tells the size and shape of the region of space where

the probability of finding the electron is high. ψ2 ≈ atomic orbitals

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F. Quantum Numbers. (Section 7.5)

1. The principal quantum number, n, specifies energy level or shell an e- occupies.

Allowed values:

n = 1, 2, 3, 4, ...

K, L, M, N… shell 2. The angular momentum (azimuthal) quantum number, l, specifies the sublevel or

subshell an electron occupies, determines the shape of the region in space an electron occupies. Allowed values:

l = 0, 1, 2, ..., (n - 1)

We give a letter notation to each value of l. Each letter corresponds to a different kind

of atomic orbital.

l = 0, 1, 2, 3, ..., (n - 1) s p d f sublevel 3. The magnetic quantum number, ml, designates the spatial orientation of an atomic

orbital. Allowed values:

ml = (-l ), ... 0, ..., (+l ) 4. The spin quantum number, ms, refers to the spin of an electron and the orientation of

the magnetic field produced by this spin. Allowed values:

ms = ±1

Example from everyday life: At Tech football games I sit in Section J, Row 21, Seat

8. It takes 3 numbers to fully specify where I’m sitting!

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5. Figure out the various allowed electron states in hydrogen (Table 7.1):

6. Orbital energy level diagram.

G. Atomic Orbitals Shapes (AO).

1. An AO is specified by n, l, ml . 2. AO is a probability distribution function for finding the electron in space. 3. Look at ground state of H atom: n = 1 (so l = 0, ml = 0) l = 0 is an s-type AO; shape = spherical

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Called a 1s atomic orbital, where 1 is the n value (level) and s is an l value of zero (sublevel).

4. Appearance of 1s AO:

5. Excite e- to level n = 2. There it can inhabit either of two sublevels: l = 0 or l = 1 2s 2p 2s AO looks like 1s (still spherical, but larger);

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If it inhabits l = 1 (2p sublevel), it is in an AO shaped like this (no longer spherical, but bi-lobed):

Since l = 1, ml can be any of three values: ml = -1, 0, +1 These refer to orientation of the lobe-shaped p orbital:

2px 2py 2pz These are the three AO of the 2p sublevel.

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6. Now excite the H atom e- to a higher energy level n = 3. Here, l = 0 or l = 1 or l = 2 3s 3p 3d 3s and 3p look similar to 2s and 2p except larger. 3d sublevel (n = 3, l = 2) atomic orbitals have quadri-lobed shape:

Since l = 2, ml can be: ml = -2, -1, 0, +1, + 2 These again refer to orientation of d orbitals. There are then 5 d-type orbitals in the 3d sublevel:

€

3dxy 3dyz 3dzx 3dz2

3dx2y2

7. When n = 4, l can be as large as l = 3, which brings in f-type orbitals: 4f, where n = 4 and l = 3 -if l = 3, ml = -3, -2, -1, 1, 2, 3 -there are 7 f orbitals in any f sublevel -complicated shapes

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8. Summarize with energy level diagram of H-atom states, through n = 3 level:

notation = (n, l, ml) 9. To place the electron (↑) in one of these orbitals with spin ms = +1/2 (↑) or ms = - 1/2

(↓) is the last step in fully specifying which quantum state the electron occupies.

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Notes: