CHAPTER 7
QUANTUM THEORY AND ATOMIC STRUCTURE
The Nature of Light
Electromagnetic radiation (also called electromagnetic energy or radiant energy) – consists of energy propagated by electric and magnetic fields that increase and decrease in intensity as they move through space
· This classical wave model explains why rainbows form, how magnifying glasses work, and many other familiar observations.
The Wave Nature of Light
The wave properties of electromagnetic radiation are described by three variables and one constant…
Frequency – the number of cycles the wave undergoes per second [s-1, or hertz (Hz)]
Wavelength – the distance between any point on a wave and the corresponding point on the next crest or trough of a wave (the distance the wave travels in one cycle) [m, nm, pm, angstroms (A)]
Speed – the distance it moves per unit time (meters/second). Electromagnetic radiation moves at 2.99792458x108 m/s (3.00x108 m/s) in a vacuum. This is a physical constant and is referred to as the speed of light.
Amplitude – the height of the crest (or depth of the trough) For an electromagnetic wave, the amplitude is related to the intensity of the radiation. Light of a particular color has a specific frequency (and thus, wavelength) but it can be dimmer (lower amplitude, less intense) or brighter (higher amplitude, more intense)
The Electromagnetic Spectrum
Visible light is a small region of the electromagnetic spectrum
· All waves in the spectrum travel at the same speed through a vacuum but differ in frequency and wavelength
Long-wavelength, low frequency radiation…
· Used in microwave ovens, radios, and cell phones
Electromagnetic emissions are everywhere…
· Human artifacts such as light bulbs, x-ray equipment, and car motors
· Natural sources such as the Sun, lightning, radioactivity, and even the glow of fireflies!
Sample Problem 7.1 p. 263
Interconverting Wavelength and Frequency
1.00 A / 1.00 m / = 1.00x10-10 m1x1010 A
(x-ray)
Speed(c) = frequency(v) x wavelength(λ)
3.00x108 m/s = frequency(v) x 1.00x10-10 m
frequency(v) = 3.00x1018 s-1
(radio signal)
325 cm / 1 m / = 3.25 m100 cm
Speed(c) = frequency(v) x wavelength(λ)
3.00x108 m/s = frequency(v) x 3.25 m
frequency(v) = 9.23x1018 s-1
(Blue sky)
473 nm / 1.00 m / = 4.73x10-7 m1x109 nm
Speed(c) = frequency(v) x wavelength(λ)
3.00x108 m/s = frequency(v) x 4.73x10-7 m
frequency(v) = 6.34x1014 s-1
Follow-Up Problem 7.1 p. 263
Speed(c) = frequency(v) x wavelength(λ)
3.00x108 m/s = 7.23x1014 Hz x wavelength(λ)
wavelength(λ) = 4.15x10-7 m
4.15x10-7 m / 1x1010 A / = 4150 A1 m
4.15x10-7 m / 1x109 nm / = 415 nm
1 m
The Classical Distinction Between Energy and Matter
Some distinctions between the behavior of waves and matter…
· Refraction and dispersion
o Light of a given wavelength travels at different speeds through various transparent media – vacuum, air (even different temperatures of air), water, quartz, etc.
§ When a light wave passes from one medium into another, the speed of the wave changes
· Refraction – change in speed causes a change in direction (figure below)
· Dispersion – white light separates (disperses) into its component colors when it passes through a prism (or other refracting object) – rainbow is an example
In contrast to a wave of light, a particle of matter, like a pebble (above figure) does not undergo refraction…
· Diffraction and interference
o Diffraction – the bending of a wave when it strikes the edge of an object
§ If a wave passes through a slit about as wide as its wavelength, it bends around both edges of the slit and forms a semicircle on the other side of the opening
In contrast… if you throw a collection of particles (a hand full of sand for example) at a small opening, some hit the edge of the opening, while others go through the opening and continue in a narrower group.
o Interference – when waves of light pass through two adjacent slits (see above diagram), the nearby emerging semicircular waves interact through the process of interference
§ If the waves’ crests coincide (in phase), they interfere constructively (amplitudes add together to form brighter region)
§ If the waves’ troughs coincide (out of phase), they interfere destructively (amplitudes cancel to form a darker region)
In contrast… particles do not exhibit interference
The Particle Nature of Light
Three observations involving matter and light confused physicists at the turn of the 20th century…
Blackbody Radiation and the Quantum Theory of Energy
Observation àWhen an object is heated to about 1000K, it begins to emit visible light (as you can see in the photo of the smoldering coal).
Observation à When an object is heated to about 1500K, it begins to emit light that is brighter and more orange (as you can see in the photo of the electric heating element)
Observation àWhen an object is heated to about 2000K, it emits light that is still brighter and whiter (as you can see in the light bulb filament)
Explanation àIn 1900, Max Planck (German physicist) assumed that the hot glowing object could emit (or absorb) only certain quantities of energy (E = nhv)
· E = energy of the radiation
· n = is a positive integer (1, 2, 3, and so on) quantum number
· h = Planck’s constant (6.626x10-34 J∙s)
· v = frequency
Conclusions
· hot objects emit only certain quantities of energy and that the energy must be emitted from the object’s atoms
o this means that each atom emits only certain quantities of energy
§ this would mean that each atom has only certain quantities of energy to start with
· thus, the energy of an atom is quantized
o energy is emitted in fixed quantities
· Each change in an atom’s energy occurs when the atom absorbs or emits one or more “packets”, or definite amounts, of energy.
o Each energy packet is called a quantum (fixed quantity or quanta)
§ A quantum of energy is equal to hv
· An atom changes its energy state by emitting (or absorbing) one or more quanta
· The energy of the emitted (or absorbed) radiation is equal to the difference in the atom’s energy states
o ∆Eatom = Eemitted or absorbed
o ∆E = hv
§ h = Planck’s constant
§ v = frequency
Photoelectric Effect and the Photon Theory of Light
Despite the idea of quantization, physicist still pictured energy traveling in waves…
· the wave model could not explain the second confusing observation (the flow of current when light strikes a metal)
Observation à the photoelectric effect (when monochromatic light of sufficient frequency shines on a metal plate, a current flows)
Possible conclusion à the current arises because the light transfers energy that frees electrons from the metal surface (this conclusions has two confusing features)
1. Presence of a threshold… for current to flow, the light shining on the metal must have a minimum, or threshold, frequency, and different metals have different minimum frequencies. The wave theory associates the energy of light with the amplitude (intensity), not its frequency (color). This theory predicts that an electron would break free when it absorbs enough energy from light of any color.
2. Absence of a time lag…current flows the moment light of the minimum frequency shines on the metal, regardless of the light’s intensity. (The wave theory predicts that with dim light there would be a time lag before the current flows, because the electrons would have to absorb enough energy to break free)
Explanation à the photon theory… Einstein proposed that light itself is particulate, quantized into tiny “bundles” of energy, called photons.
· Each atom changes its energy, ∆Eatom, when it absorbs or emits one photon, one particle of light, whose energy is related to its frequency, not its amplitude.
How does the photon theory explain the two features of the photoelectric effect?
· Why is there a frequency threshold?
o A beam of light consists of an enormous number of photons. The intensity (brightness) is related to the number of photons, but not the energy of each. Therefore, a photon of a certain minimum energy must be absorbed to free an electron from the surface. Since energy depends on frequency (hv), the theory predicts a threshold frequency.
· Why is there no time lag?
o An electron breaks free when it absorbs a photon of enough energy
§ It cannot break free by “saving up” energy from several photons, each having less than the minimum energy.
· Example… ping-pong photons (if one ping-pong ball doesn’t have enough energy to knock a book of a shelf, neither does a series of ping-pong balls, because the book can’t store the energy from the individual impacts. But, one baseball traveling at the same speed does have enough energy)
§ The current is weak in dim light because fewer photons of enough energy can free fewer electrons per unit time, but some current flows as soon as light of sufficient energy (frequency) strikes the metal.
Sample Problem 7.2 p. 267
Calculating the Energy of Radiation from its Wavelength
∆E = hv
Step 1
Convert wavelength to meters
1.20 cm / 1 m / = 0.0120 m100 cm
Step 2
Calculate frequency from wavelength
Speed(c) = frequency(v) x wavelength(λ)
3.00x108 m/s = frequency(v) x 0.0120 m
frequency(v) = 2.50x1010 s-1
Step 3
Calculate energy of one photon
∆E = hv
∆E = (6.626x10-34 J∙s) (2.50x1010 s-1)
∆E = 1.66x10-23 J
Follow-Up Problem 7.2 p. 267
(a)
Speed(c) = frequency(v) x wavelength(λ)
3.00x108 m/s = frequency(v) x 1x10-8 m
frequency(v) = 3x1016 s-1
∆E = hv
∆E = (6.626x10-34 J∙s) (3x1016 s-1)
∆E = 2x10-17 J
(b)
Speed(c) = frequency(v) x wavelength(λ)
3.00x108 m/s = frequency(v) x 5x10-7 m
frequency(v) = 6x1014 s-1
∆E = hv
∆E = (6.626x10-34 J∙s) (6x1014 s-1)
∆E = 4x10-19 J
(c)
Speed(c) = frequency(v) x wavelength(λ)
3.00x108 m/s = frequency(v) x 1x10-4 m
frequency(v) = 3x1012 s-1
∆E = hv
∆E = (6.626x10-34 J∙s) (3x1012 s-1)
∆E = 2x10-21 J
As wavelength increases (frequency decreases) energy decreases
Atomic Spectra
The third confusing observation about matter and energy involved the light emitted when an element is vaporized and then excited electrically.
Line Spectra and the Rydberg Equation
When light from electrically excited gaseous atoms passes through a slit and is refracted by a prism, it does not create a continuous spectrum or rainbow, as sunlight does.
· Instead, it creates a line spectrum, a series of lines at specific frequencies separated by black spaces.
o Each line spectrum is characteristic of the element producing it. (Figure 7.8 below)
Features of the Rydberg Equation
Johannes Rydberg (Swedish physicist) developed a relationship, called the Rydberg equation that predicted the position and wavelength of any line in a given series.
· 1/λ = R (1/n21 - 1/n22)
o λ = wavelength
o n1 and n2 are positive integers with n2 > n1
o R = Rydberg constant (1.096776x107 m-1)
Problems with Rutherford’s Nuclear Model
Almost as soon as Rutherford proposed his nuclear model, a major problem arose…
· A positive nucleus and a negative electron attract each other, and for them to stay apart, the kinetic energy of the electron’s motion must counterbalance the potential energy of the attraction.
o Laws of classical physics say that a negative particle moving in a curved path around a positive particle must emit radiation and thus lose energy.
§ If electrons behave in this way, they would spiral into the nucleus and atoms would collapse!
§ Laws of classical physics would also suggest that the emitted radiation would create a continuous spectrum, not a line spectrum.
The Bohr Model of the Hydrogen Atom
Niels Bohr (Danish physicist) suggested a model for the hydrogen atom that did predict the existence of line spectra…
· Postulates of the model…
o The hydrogen atom has only certain energy levels
§ Bohr referred to these as stationary states
· Each state is associated with a fixed circular orbit around the nucleus
o The higher the energy level, the further the orbit is from the nucleus
o The atom does not radiate energy while in one of its stationary states
§ Although this violates classical physics, the atom does not change energy while the electron moves within an orbit
o The atom changes to another stationary state (the electron moves to another orbit) only by absorbing or emitting a photon. The energy of the photon (hv) equals the difference in the energies of the two states.
§ Ephoton = ∆Eatom = Efinal – Einitial = hv
Features of the Model
Quantum numbers and the electron orbit… the quantum number n is a positive integer (1, 2, 3, …) associated with radius of an electron orbit, which is directly related to the electron’s energy. (the lower the n value, the smaller the radius of the orbit, and the lower the energy level
Ground state… when the electron is in its first orbit (n=1), it is closest to the nucleus, and the hydrogen atom is in its lowest energy level, called the ground state
Excited states… if the electron is in any orbit farther from the nucleus, the atom is in its excited state. With the electron in the second orbit (n=2), the atom is in the first excited state; when it is in the third orbit (n=3), the atom is in its second excited state, and so forth.
Absorption… if a hydrogen atom absorbs a photon whose energy equals the difference between lower and higher energy levels, the electron moves to the outer (higher energy) orbit.
Emission… if a hydrogen atom in a higher energy level (electron in a farther orbit) returns to a lower energy level (electron in closer orbit), the atom emits a photon whose energy equals the difference between the two levels.
How the Model Explains Line Spectra
A spectral line results because a photon of specific energy (and thus frequency) is emitted. The emission occurs when the electron moves to an orbit closer to the nucleus as the atom’s energy changes from a higher state to a lower one.
· Key point…an atomic spectrum is not continuous because the atom’s energy is not continuous, but rather has only certain states.