7-1 Dr. Wolf’s CHM 101 Chapter 7 Quantum Theory and Atomic Structure

7-1 Dr. Wolf’s CHM 101

Chapter 7

Quantum Theory and Atomic Structure


Quantum Theory and Atomic Structure

7.1 The Nature of Light

7.2 Atomic Spectra

7.3 The Wave-Particle Duality of Matter and Energy

7.4 The Quantum-Mechanical Model of the Atom


Wavelength, the distance from one crest to the next in the wave. Measured in units of distance.

c =

Frequency, the number of complete cycles per sec., cps, Hz

Electromagnetic Radiation (light) - Wave like

Speed of Light, C, same for all EM radiation.3.00 x 1010 cm/sec in vacuum


Amplitude (Intensity) of a Wave


Regions of the Electromagnetic Spectrum c =

= c /

1000 kHz100 MHz

3 m 300 m


Sample Problem 7.1

SOLUTION:PLAN:

Interconverting Wavelength and Frequency

PROBLEM: A dental hygienist uses x-rays (= 1.00A) to take a series of dental radiographs while the patient listens to a radio station ( = 325cm) and looks out the window at the blue sky (= 473nm). What is the frequency (in s-1) of the electromagnetic radiation from each source? (Assume that the radiation travels at the speed of light, 3.00x108m/s.)

wavelength in units given

wavelength in m

frequency (s-1 or Hz)

1A = 10-10m1cm = 10-2m1nm = 10-9m

= c/

Use c = 1.00A

325cm

473nm

10-10m1A

10-2m1cm

10-9m1nm

= 1.00x10-10m

= 325x10-2m

= 473x10-9m

=3x108m/s

1.00x10-10m= 3x1018s-1

=

=

3x108m/s

325x10-2m= 9.23x107s-1

3x108m/s

473x10-9m= 6.34x1014s-1


Different behaviors of waves and particles.


The diffraction pattern caused by light passing through two adjacent slits.


Electromagnetic Radiation - Particle like

The view that EM was wavelike could not explain certain phenomena like :

1) Blackbody radiation - when objects are heated, they give off shorter and more intense radiation as the temperature increases, e.g. dull red hot, to hotter orange, to white hot.......But wavelike properties would predict hotter temps would continue to give more and more of shorter and shorter wavelengths. But instead a bell shape curve of intensities is obtained with the peak at the IR- Vis part of the spectrum.


Blackbody Radiation

E = h

Electromagnetic Radiation

E = hc/

Planck’s constanth = 6.626 x 10-34 J-s




2) Photoelectron Effect - light shinning on certain metal plates caused a flow of electrons.However the the light had a minimum frequency to cause the effect, i.e. not any color would work.

And although bright light caused more electron flow than weak light, electron flow started immediately with both strong or weak light.


Demonstration of the photoelectric effect



The better explanation for these experiments was that EM consisted of packets of energy called photons (particle-like) that had wave-like properties as well. And that atoms could have only certain quantities of energy, E = nhwhere n is a positive integer, 1, 2, 3, etc. This means energy is quantized.

Ephoton = h= Eatom


Sample Problem 7.2

SOLUTION:

PLAN:

Calculating the Energy of Radiation from Its Wavelength

PROBLEM: A cook uses a microwave oven to heat a meal. The wavelength of the radiation is 1.20cm. What is the energy of one photon of this microwave radiation?

After converting cm to m, we can use the energy equation, E = h combined with = c/ to find the energy.

E = hc/

E =6.626X10-34J-s 3x108m/s

1.20cm 10-2m

cm

x= 1.66x10-23J




3) Atomic Spectra - Electrical discharges in tube of gaseous elements produces light (EM).But not all wavelengths of light were produced but just a few certain wavelengths (or frequencies).

And different elements had different wavelengths associated with them. Not just in the Visible but also IR and UV regions.


The line spectra of several elements


= RRydberg equation -1

1

n22

1

n12

R is the Rydberg constant = 1.096776 x 107 m-1for the visible series, n1 = 2 and n2 = 3, 4, 5, ...

Three Series of Spectral Lines of Atomic Hydrogen

Looking for an equation that would predict the wavelength seen in H spectrum

But WHY does this equation work?


Bohr Model of the Hydrogen Atom

Assumed the H atom has only certain allowable energy levels for the electron orbits. (quantized because it made the equations work))

When the electron moves from one orbit to another, it has to absorb or emit a photon whose energy equals the difference in energy between the two orbits.

By assuming the electron traveled in circular orbits, the energy level for each orbits was

E = -2.18 x 10-18 J / n2 where n = 1, 2, 3,....Note: because of negative sign, lowest energy (most stable) when n = 1 and

highest energy is E = 0 when n = .

So energy released when the electron moves from one n level to another isE= h = hc/ = -2.18 x 10-18 J ( 1 / n2

final - 1 / n2initial )


Quantum staircase


The Bohr explanation of the three series of spectral lines.


If EM can have particle-like properties in addition to being wave-like, what if the electron particles have wave-like properties?

Quantization is a natural consequence of having wave-like properties.

But why must the electron’s energy be quantized?


Wave motion in restricted systems


The de Broglie Wavelengths of Several Objects

Substance Mass (g) Speed, u, (m/s) (m)

slow electron

fast electron

alpha particle

one-gram mass

baseball

Earth

9x10-28

9x10-28

6.6x10-24

1.0

142

6.0x1027

1.0

5.9x106

1.5x107

0.01

25.0

3.0x104

7x10-4

1x10-10

7x10-15

7x10-29

2x10-34

4x10-63

E = hc/

h /mu

so = hc/ E but E = muc

de Broglie Wavelength - giving particles wave-like properties


Sample Problem 7.3

SOLUTION:

PLAN:

Calculating the de Broglie Wavelength of an Electron

PROBLEM: Find the deBroglie wavelength of an electron with a speed of 1.00x106m/s (electron mass = 9.11x10-31kg; h = 6.626x10-34 kg-m2/s).

Knowing the mass and the speed of the electron allows to use the equation = h/mu to find the wavelength.

= 6.626x10-34kg-m2/s

9.11x10-31kg x 1.00x106m/s= 7.27x10-10m


CLASSICAL THEORYCLASSICAL THEORY

Matter particulate,

massive

Energy continuous,

wavelike

Since matter is discontinuous and particulate perhaps energy is discontinuous and particulate.

Observation Theory

Planck: Energy is quantized; only certain values allowed

blackbody radiation

Einstein: Light has particulate behavior (photons)photoelectric effect

Bohr: Energy of atoms is quantized; photon emitted when electron changes orbit.

atomic line spectra

Summary of the major observations and theories leading from classical theory to quantum theory.


Since energy is wavelike, perhaps matter is wavelike

Observation Theory

deBroglie: All matter travels in waves; energy of atom is quantized due to wave motion of electrons

Davisson/Germer: electron diffraction by metal crystal

Since matter has mass, perhaps energy has mass

Observation Theory

Einstein/deBroglie: Mass and energy are equivalent; particles have wavelength and photons have momentum.

Compton: photon wavelength increases (momentum decreases) after colliding with electron

QUANTUM THEORY

Energy same as Matterparticulate, massive, wavelike


The Heisenberg Uncertainty Principle

x m u

h

4

Heisenberg Uncertainty Principle expresses a limitation on accuracy of simultaneous measurement of observables such as the position and the momentum of a particle.

.


Sample Problem 7.4

SOLUTION:

PLAN:

Applying the Uncertainty Principle

PROBLEM: An electron moving near an atomic nucleus has a speed 6 x 106 ± 1% m/s. What is the uncertainty in its position (x)?

The uncertainty (u) is given as ±1% (0.01) of 6 x 106 m/s. Once we calculate this, plug it into the uncertainty equation.

u = (0.01) (6 x 106 m/s) = 6 x 104 m/s

x m u h

4

x 4 (9.11 x 10-31 kg) (6 x104 m/s)

6.626 x 10-34 kg-m2/s 10-9m

.


The Schrödinger Equation - Quantum Numbers and Atomic Orbitals

i.e., an atomic orbital is specified by three quantum numbers.

n the principal quantum number - a positive integer

l the angular momentum quantum number - an integer from 0 to n-1

ml the magnetic moment quantum number - an integer from -l to +l

ms the spin quantum number, + 1/2 or - 1/2

A complicated equation with multiple solutions which describes the probability of locating an electron at the various allowed energy levels. Solutions involve three interdependent variables to describe an electron orbital.


Electron probability in the ground-state H atom

“Orbital” showing 90% of electron probability

n = 1 l = 0 m = 0


Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals

Name, Symbol(Property) Allowed Values Quantum Numbers

Principal, n(size, energy)

Angular momentum, l

(shape)

Magnetic, ml

(orientation)

Positive integer(1, 2, 3, ...)

0 to n-1

-l,…,0,…,+l

1

0

0

2

0 1

0

3

0 1 2

0

0-1 +1 -1 0 +1

0 +1 +2-1-2

sublevel namesl = 0, called “s” l = 1, “p” l = 2, “d” l = 3, “f”


Sample Problem 7.5

SOLUTION:

PLAN:

Determining Quantum Numbers for an Energy Level

PROBLEM: What values of the angular momentum (l) and magnetic (ml) quantum numbers are allowed for a principal quantum number (n) of 3? How many orbitals are allowed for n = 3?

Follow the rules for allowable quantum numbers found in the text.

l values can be integers from 0 to n-1; ml can be integers from -l through 0 to + l.

For n = 3, l = 0, 1, 2

For l = 0 ml = 0

For l = 1 ml = -1, 0, or +1

For l = 2 ml = -2, -1, 0, +1, or +2

There are 9 ml values and therefore 9 orbitals with n = 3.


Sample Problem 7.6

SOLUTION:

PLAN:

Determining Sublevel Names and Orbital Quantum

NumbersPROBLEM: Give the name, magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers:

(a) n = 3, l = 2 (b) n = 2, l = 0 (c) n = 5, l = 1 (d) n = 4, l = 3

Combine the n value and l designation to name the sublevel. Knowing l, we can find ml and the number of orbitals.

l sublevel name possible ml values # of orbitals

(a)

(b)

(c)

(d)

3d

2s

5p

4f

-2, -1, 0, 1, 2

0

-1, 0, 1

-3, -2, -1, 0, 1, 2, 3

n

3 2

2 0

5 1

4 3

5

1

3

7


s orbitals

1s 2s 3s


The 2p orbitals n = 2, l = 1

p orbitals - three of them

Combination


The 3d orbitals n = 3 l = 2

d orbitals - five of them


d orbitals - five of them

Combination


One of the seven possible 4f orbitals

f orbitals - seven of them


End of Chapter 7

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7-1 Dr. Wolf’s CHM 101 Chapter 7 Quantum Theory and Atomic Structure