Chapter 42
description
Transcript of Chapter 42
Importance of the Hydrogen Atom
The hydrogen atom is the only atomic system that can be solved exactly
Much of what was learned in the twentieth century about the hydrogen atom, with its single electron, can be extended to such single-electron ions as He+ and Li2+
More Reasons the Hydrogen Atom is Important
The hydrogen atom is an ideal system for performing precision tests of theory against experiment Also for improving our understanding of atomic
structure The quantum numbers that are used to
characterize the allowed states of hydrogen can also be used to investigate more complex atoms This allows us to understand the periodic table
Final Reasons for the Importance of the Hydrogen Atom
The basic ideas about atomic structure must be well understood before we attempt to deal with the complexities of molecular structures and the electronic structure of solids
The full mathematical solution of the Schrödinger equation applied to the hydrogen atom gives a complete and beautiful description of the atom’s properties
Atomic Spectra
A discrete line spectrum is observed when a low-pressure gas is subjected to an electric discharge
Observation and analysis of these spectral lines is called emission spectroscopy
The simplest line spectrum is that for atomic hydrogen
Uniqueness of Atomic Spectra
Other atoms exhibit completely different line spectra
Because no two elements have the same line spectrum, the phenomena represents a practical and sensitive technique for identifying the elements present in unknown samples
Absorption Spectroscopy
An absorption spectrum is obtained by passing white light from a continuous source through a gas or a dilute solution of the element being analyzed
The absorption spectrum consists of a series of dark lines superimposed on the continuous spectrum of the light source
Absorption Spectrum, Example
A practical example is the continuous spectrum emitted by the sun
The radiation must pass through the cooler gases of the solar atmosphere and through the Earth’s atmosphere
Balmer Series
In 1885, Johann Balmer found an empirical equation that correctly predicted the four visible emission lines of hydrogen Hα is red, λ = 656.3 nm
Hβ is green, λ = 486.1 nm
Hγ is blue, λ = 434.1 nm
Hδ is violet, λ = 410.2 nm
Emission Spectrum of Hydrogen – Equation
The wavelengths of hydrogen’s spectral lines can be found from
RH is the Rydberg constant RH = 1.097 373 2 x 107 m-1
n is an integer, n = 3, 4, 5,… The spectral lines correspond to different values
of n
H 2 2
1 1 12
Rλ n
Other Hydrogen Series
Other series were also discovered and their wavelengths can be calculated
Lyman series:
Paschen series:
Brackett series:
H 2
1 11 2 3 4, , ,R n
λ n
H 2 2
1 1 14 5 6
3, , ,R n
λ n
H 2 2
1 1 15 6 7
4, , ,R n
λ n
Joseph John Thomson
1856 – 1940 English physicist Received Nobel Prize in
1906 Usually considered the
discoverer of the electron Worked with the deflection
of cathode rays in an electric field Opened up the field of
subatomic particles
Early Models of the Atom, Thomson’s
J. J. Thomson established the charge to mass ratio for electrons
His model of the atom A volume of positive charge Electrons embedded
throughout the volume
Rutherford’s Thin Foil Experiment
Experiments done in 1911 A beam of positively
charged alpha particles hit and are scattered from a thin foil target
Large deflections could not be explained by Thomson’s model
Early Models of the Atom, Rutherford’s
Rutherford Planetary model Based on results of thin
foil experiments Positive charge is
concentrated in the center of the atom, called the nucleus
Electrons orbit the nucleus like planets orbit the sun
Difficulties with the Rutherford Model
Atoms emit certain discrete characteristic frequencies of electromagnetic radiation The Rutherford model is unable to explain this phenomena
Rutherford’s electrons are undergoing a centripetal acceleration It should radiate electromagnetic waves of the same
frequency The radius should steadily decrease as this radiation is
given off The electron should eventually spiral into the nucleus
It doesn’t
Niels Bohr 1885 – 1962 Danish physicist An active participant in the
early development of quantum mechanics
Headed the Institute for Advanced Studies in Copenhagen
Awarded the 1922 Nobel Prize in physics For structure of atoms and
the radiation emanating from them
The Bohr Theory of Hydrogen
In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory
His model includes both classical and non-classical ideas
He applied Planck’s ideas of quantized energy levels to orbiting electrons
Bohr’s Theory, cont.
This model is now considered obsolete It has been replaced by a probabilistic
quantum-mechanical theory The model can still be used to develop ideas
of energy quantization and angular momentum quantization as applied to atomic-sized systems
Bohr’s Assumptions for Hydrogen, 1
The electron moves in circular orbits around the proton under the electric force of attraction The Coulomb force
produces the centripetal acceleration
Bohr’s Assumptions, 2
Only certain electron orbits are stable These are the orbits in which the atom does not
emit energy in the form of electromagnetic radiation
Therefore, the energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion
This representation claims the centripetally accelerated electron does not emit energy and therefore does not eventually spiral into the nucleus
Bohr’s Assumptions, 3
Radiation is emitted by the atom when the electron makes a transition from a more energetic initial state to a lower-energy orbit The transition cannot be treated classically The frequency emitted in the transition is related to the
change in the atom’s energy The frequency is independent of frequency of the electron’s
orbital motion The frequency of the emitted radiation is given by Ei – Ef = hƒ
If a photon is absorbed, the electron moves to a higher energy level
Bohr’s Assumptions, 4
The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum
The allowed orbits are those for which the electron’s orbital angular momentum about the nucleus is quantized and equal to an integral multiple of h
Mathematics of Bohr’s Assumptions and Results
Electron’s orbital angular momentummevr = nħ where n = 1, 2, 3,…
The total energy of the atom is
The total energy can also be expressed as
Note, the total energy is negative, indicating a bound electron-proton system
221
2 e e
eE K U m v k
r
2
2ek e
Er
Bohr Radius
The radii of the Bohr orbits are quantized
This shows that the radii of the allowed orbits have discrete values—they are quantized When n = 1, the orbit has the smallest radius, called
the Bohr radius, ao
ao = 0.052 9 nm
2 2
21 2 3, , ,n
e e
nr n
m k e
Radii and Energy of Orbits
A general expression for the radius of any orbit in a hydrogen atom is rn = n2ao
The energy of any orbit is
This becomes
En = - 13.606 eV / n2
2
2
11 2 3,
2, ,e
no
k eE n
a n
Specific Energy Levels
Only energies satisfying the previous equation are allowed
The lowest energy state is called the ground state This corresponds to n = 1 with E = –13.606 eV
The ionization energy is the energy needed to completely remove the electron from the ground state in the atom The ionization energy for hydrogen is 13.6 eV
Energy Level Diagram
Quantum numbers are given on the left and energies on the right
The uppermost level, E = 0, represents the
state for which the electron is removed from the atom Adding more energy than
this amount ionizes the atom
Active Figures 42.7 and 42.8
Use the active figure to choose initial and final energy levels
Observe the transition in both figures
PLAYACTIVE FIGURE
Frequency of Emitted Photons
The frequency of the photon emitted when the electron makes a transition from an outer orbit to an inner orbit is
It is convenient to look at the wavelength instead
2
2 2
1 1
2ƒ i f e
o f i
E E k e
h a h n n
Wavelength of Emitted Photons
The wavelengths are found by
The value of RH from Bohr’s analysis is in excellent agreement with the experimental value
2
2 2 2 2
1 1 1 1 1
2
ƒ eH
o f i f i
k eR
λ c a hc n n n n
Extension to Other Atoms
Bohr extended his model for hydrogen to other elements in which all but one electron had been removed
Z is the atomic number of the element and is the number of protons in the nucleus
2
2 2
21 2 3
2, , ,
on
en
o
ar n
Z
k e ZE n
a n
Difficulties with the Bohr Model
Improved spectroscopic techniques found that many of the spectral lines of hydrogen were not single lines Each “line” was actually a group of lines spaced
very close together Certain single spectral lines split into three
closely spaced lines when the atoms were placed in a magnetic field
Bohr’s Correspondence Principle
Bohr’s correspondence principle states that quantum physics agrees with classical physics when the differences between quantized levels become vanishingly small Similar to having Newtonian mechanics be a
special case of relativistic mechanics when v << c
The Quantum Model of the Hydrogen Atom
The potential energy function for the hydrogen atom is
ke is the Coulomb constant r is the radial distance from the proton to the
electron The proton is situated at r = 0
2
( ) e
eU r k
r
Quantum Model, cont.
The formal procedure to solve the hydrogen atom is to substitute U(r) into the Schrödinger equation, find the appropriate solutions to the equations, and apply boundary conditions
Because it is a three-dimensional problem, it is easier to solve if the rectangular coordinates are converted to spherical polar coordinates
Quantum Model, final
ψ(x, y, z) is converted to ψ(r, θ, φ)
Then, the space variables can be separated:
ψ(r, θ, φ) = R(r), ƒ(θ), g(φ) When the full set of
boundary conditions are applied, we are led to three different quantum numbers for each allowed state
Quantum Numbers, General
The three different quantum numbers are restricted to integer values
They correspond to three degrees of freedom Three space dimensions
Principal Quantum Number
The first quantum number is associated with the radial function R(r) It is called the principal quantum number It is symbolized by n
The potential energy function depends only on the radial coordinate r
The energies of the allowed states in the hydrogen atom are the same En values found from the Bohr theory
Orbital and Orbital Magnetic Quantum Numbers
The orbital quantum number is symbolized by ℓ It is associated with the orbital angular
momentum of the electron It is an integer
The orbital magnetic quantum number is symbolized by mℓ It is also associated with the angular orbital
momentum of the electron and is an integer
Quantum Numbers, Summary of Allowed Values
The values of n can range from 1 to The values of ℓ can range from 0 to n - 1 The values of mℓ can range from –ℓ to ℓ Example:
If n = 1, then only ℓ = 0 and mℓ = 0 are permitted If n = 2, then ℓ = 0 or 1
If ℓ = 0 then mℓ = 0
If ℓ = 1 then mℓ may be –1, 0, or 1
Shells
Historically, all states having the same principle quantum number are said to form a shell Shells are identified by letters K, L, M,…
All states having the same values of n and ℓ are said to form a subshell The letters s, p, d, f, g, h, .. are used to designate
the subshells for which ℓ = 0, 1, 2, 3,…
Wave Functions for Hydrogen
The simplest wave function for hydrogen is the one that describes the 1s state and is designated ψ1s(r)
As ψ1s(r) approaches zero, r approaches and is normalized as presented
ψ1s(r) is also spherically symmetric This symmetry exists for all s states
1 3
1( ) or a
s
o
ψ r eπa
Probability Density
The probability density for the 1s state is
The radial probability density function P(r) is the probability per unit radial length of finding the electron in a spherical shell of radius r and thickness dr
2 21 3
1or a
so
ψ eπa
Radial Probability Density
A spherical shell of radius r and thickness dr has a volume of 4πr2 dr
The radial probability function is
P(r) = 4πr2 ψ2
P(r) for 1s State of Hydrogen
The radial probability density function for the hydrogen atom in its ground state is
The peak indicates the most probable location
The peak occurs at the Bohr radius
22
1 3
4( ) or a
so
rP r e
a
P(r) for 1s State of Hydrogen, cont.
The average value of r for the ground state of hydrogen is 3/2 ao
The graph shows asymmetry, with much more area to the right of the peak
According to quantum mechanics, the atom has no sharply defined boundary as suggested by the Bohr theory
Electron Clouds
The charge of the electron is extended throughout a diffuse region of space, commonly called an electron cloud
This shows the probability density as a function of position in the xy plane
The darkest area, r = ao, corresponds to the most probable region
Wave Function of the 2s state
The next-simplest wave function for the hydrogen atom is for the 2s state n = 2; ℓ = 0
The normalized wave function is
ψ2s depends only on r and is spherically symmetric
32
22
1 1( ) 2
4 2or a
so o
rψ r e
a aπ
Comparison of 1s and 2s States
The plot of the radial probability density for the 2s state has two peaks
The highest value of P corresponds to the most probable value In this case, r 5ao
Active Figure 42.12
Use the active figure to choose values of r/ao
Find the probability that the electron is located between two values
PLAYACTIVE FIGURE
Physical Interpretation of ℓ
The magnitude of the angular momentum of an electron moving in a circle of radius r is
L = mevr The direction of is perpendicular to the
plane of the circle The direction is given by the right hand rule
In the Bohr model, the angular momentum of the electron is restricted to multiples of
L
Physical Interpretation of ℓ, cont.
According to quantum mechanics, an atom in a state whose principle quantum number is n can take on the following discrete values of the magnitude of the orbital angular momentum:
L can equal zero, which causes great difficulty when attempting to apply classical mechanics to this system
H 2 2
1 1 1
2R
λ n
Physical Interpretation of mℓ
The atom possesses an orbital angular momentum
There is a sense of rotation of the electron around the nucleus, so that a magnetic moment is present due to this angular momentum
There are distinct directions allowed for the magnetic moment vector with respect to the magnetic field vector
μ
B
Physical Interpretation of mℓ, 2
Because the magnetic moment of the atom can be related to the angular momentum vector, , the discrete direction of translates into the fact that the direction of is quantized
Therefore, Lz, the projection of along the z axis, can have only discrete values
μ
μ
L
L
L
Physical Interpretation of mℓ, 3
The orbital magnetic quantum number mℓ specifies the allowed values of the z component of orbital angular momentum
Lz = mℓ The quantization of the possible orientations
of with respect to an external magnetic field is often referred to as space quantization
L
Physical Interpretation of mℓ, 4
does not point in a specific direction Even though its z-component is fixed Knowing all the components is inconsistent with
the uncertainty principle Imagine that must lie anywhere on the
surface of a cone that makes an angle θ with the z axis
L
L
Physical Interpretation of mℓ, final
θ is also quantized Its values are specified
through
mℓ is never greater than ℓ, therefore θ can never be zero
cos
1zL m
θL
Zeeman Effect
The Zeeman effect is the splitting of spectral lines in a strong magnetic field
In this case the upper level, with ℓ = 1, splits into three different levels corresponding to the three different directions of µ
Spin Quantum Number ms
Electron spin does not come from the Schrödinger equation
Additional quantum states can be explained by requiring a fourth quantum number for each state
This fourth quantum number is the spin magnetic quantum number ms
Electron Spins
Only two directions exist for electron spins
The electron can have spin up (a) or spin down (b)
In the presence of a magnetic field, the energy of the electron is slightly different for the two spin directions and this produces doublets in spectra of certain gases
Electron Spins, cont.
The concept of a spinning electron is conceptually useful
The electron is a point particle, without any spatial extent Therefore the electron cannot be considered to be actually
spinning The experimental evidence supports the electron
having some intrinsic angular momentum that can be described by ms
Dirac showed this results from the relativistic properties of the electron
Spin Angular Momentum
The total angular momentum of a particular electron state contains both an orbital contribution and a spin contribution
Electron spin can be described by a single quantum number s, whose value can only be s = ½
The spin angular momentum of the electron never changes
L
S
Spin Angular Momentum, cont
The magnitude of the spin angular momentum is
The spin angular momentum can have two orientations relative to a z axis, specified by the spin quantum number ms = ± ½ ms = + ½ corresponds to the spin up case
ms = - ½ corresponds to the spin down case
3( 1)
2S s s
Spin Angular Momentum, final
The z component of spin angular momentum is Sz = msh = ½ h
Spin angular moment is quantized
Spin Magnetic Moment
The spin magnetic moment µspin is related to the spin angular momentum by
The z component of the spin magnetic moment can have values
Sspine
eμ
m
spin 2,ze
eμ
m
Quantum States
There are eight quantum states corresponding to n = 2 These states depend on the addition of the
possible values of ms
Table 42.3 summarizes these states
Wolfgang Pauli 1900 – 1958 Austrian physicist Important review article on
relativity At age 21
Discovery of the exclusion principle
Explanation of the connection between particle spin and statistics
Relativistic quantum electrodynamics
Neutrino hypothesis Hypotheses of nuclear spin
The Exclusion Principle
The four quantum numbers discussed so far can be used to describe all the electronic states of an atom regardless of the number of electrons in its structure
The exclusion principle states that no two electrons can ever be in the same quantum state Therefore, no two electrons in the same atom can have the
same set of quantum numbers
If the exclusion principle was not valid, an atom could radiate energy until every electron was in the lowest possible energy state and the chemical nature of the elements would be modified
Filling Subshells
The electronic structure of complex atoms can be viewed as a succession of filled levels increasing in energy
Once a subshell is filled, the next electron goes into the lowest-energy vacant state If the atom were not in the lowest-energy state
available to it, it would radiate energy until it reached this state
Orbitals
An orbital is defined as the atomic state characterized by the quantum numbers n, ℓ and mℓ
From the exclusion principle, it can be seen that only two electrons can be present in any orbital One electron will have spin up and one spin down
Each orbital is limited to two electrons, the number of electrons that can occupy the various shells is also limited
Allowed Quantum States, Example with n = 3
In general, each shell can accommodate up to 2n2 electrons
Hund’s Rule
Hund’s Rule states that when an atom has orbitals of equal energy, the order in which they are filled by electrons is such that a maximum number of electrons have unpaired spins Some exceptions to the rule occur in elements
having subshells that are close to being filled or half-filled
Periodic Table
Dmitri Mendeleev made an early attempt at finding some order among the chemical elements
He arranged the elements according to their atomic masses and chemical similarities
The first table contained many blank spaces and he stated that the gaps were there only because the elements had not yet been discovered
Periodic Table, cont.
By noting the columns in which some missing elements should be located, he was able to make rough predictions about their chemical properties
Within 20 years of the predictions, most of the elements were discovered
The elements in the periodic table are arranged so that all those in a column have similar chemical properties
Periodic Table, Explained
The chemical behavior of an element depends on the outermost shell that contains electrons
For example, the inert gases (last column) have filled subshells and a wide energy gap occurs between the filled shell and the next available shell
Hydrogen Energy Level Diagram Revisited
The allowed values of ℓ are separated horizontally
Transitions in which ℓ does not change are very unlikely to occur and are called forbidden transitions Such transitions actually
can occur, but their probability is very low compared to allowed transitions
Selection Rules
The selection rules for allowed transitions are Δℓ = ±1 Δmℓ = 0, ±1
The angular momentum of the atom-photon system must be conserved
Therefore, the photon involved in the process must carry angular momentum The photon has angular momentum equivalent to that of a
particle with spin 1 A photon has energy, linear momentum and angular
momentum
Multielectron Atoms
For multielectron atoms, the positive nuclear charge Ze is largely shielded by the negative charge of the inner shell electrons The outer electrons interact with a net charge that
is smaller than the nuclear charge Allowed energies are
Zeff depends on n and ℓ
2eff
2
13 6.n
ZE eV
n
X-Ray Spectra
These x-rays are a result of the slowing down of high energy electrons as they strike a metal target
The kinetic energy lost can be anywhere from 0 to all of the kinetic energy of the electron
The continuous spectrum is called bremsstrahlung, the German word for “braking radiation”
X-Ray Spectra, cont.
The discrete lines are called characteristic x-rays
These are created when A bombarding electron collides with a target atom The electron removes an inner-shell electron from
orbit An electron from a higher orbit drops down to fill
the vacancy
X-Ray Spectra, final
The photon emitted during this transition has an energy equal to the energy difference between the levels
Typically, the energy is greater than 1000 eV The emitted photons have wavelengths in the
range of 0.01 nm to 1 nm
Moseley Plot
Henry G. J. Moseley plotted the values of atoms as shown
λ is the wavelength of the Kα line of each element The Kα line refers to the
photon emitted when an electron falls from the L to the K shell
From this plot, Moseley developed a periodic table in agreement with the one based on chemical properties
Stimulated Absorption
When a photon has energy hƒ equal to the difference in energy levels, it can be absorbed by the atom
This is called stimulated absorption because the photon stimulates the atom to make the upward transition
Active Figure 42.25
Use the active figure to adjust the energy difference between the states
Observe stimulated absorption
PLAYACTIVE FIGURE
Spontaneous Emission
Once an atom is in an excited state, the excited atom can make a transition to a lower energy level
Because this process happens naturally, it is known as spontaneous emission
Stimulated Emission
In addition to spontaneous emission, stimulated emission occurs
Stimulated emission may occur when the excited state is a metastable state
Stimulated Emission, cont.
A metastable state is a state whose lifetime is much longer than the typical 10-8 s
An incident photon can cause the atom to return to the ground state without being absorbed
Therefore, you have two photons with identical energy, the emitted photon and the incident photon They both are in phase and travel in the same direction
Active Figure 42.27
Use the active figure to adjust the energy difference between states
Observe the stimulated emission
PLAYACTIVE FIGURE
Lasers – Properties of Laser Light
Laser light is coherent The individual rays in a laser beam maintain a
fixed phase relationship with each other Laser light is monochromatic
The light has a very narrow range of wavelengths Laser light has a small angle of divergence
The beam spreads out very little, even over long distances
Lasers – Operation
It is equally probable that an incident photon would cause atomic transitions upward or downward Stimulated absorption or stimulated emission
If a situation can be caused where there are more electrons in excited states than in the ground state, a net emission of photons can result This condition is called population inversion
Lasers – Operation, cont.
The photons can stimulate other atoms to emit photons in a chain of similar processes
The many photons produced in this manner are the source of the intense, coherent light in a laser
Conditions for Build-Up of Photons
The system must be in a state of population inversion
The excited state of the system must be a metastable state In this case, the population inversion can be established
and stimulated emission is likely to occur before spontaneous emission
The emitted photons must be confined in the system long enough to enable them to stimulate further emission This is achieved by using reflecting mirrors
Laser Design – Schematic
The tube contains the atoms that are the active medium An external source of energy pumps the atoms to excited
states The mirrors confine the photons to the tube
Mirror 2 is only partially reflective
Energy-Level Diagram for Neon in a Helium-Neon Laser
The atoms emit 632.8-nm photons through stimulated emission
The transition is E3* to E2
* indicates a metastable state