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### Chapter 6 – Electron Structure of Atoms

Homework:

1, 2, 9, 10, 14, 15, 17, 19, 21, 23, 25, 28, 30, 31, 34, 35, 37, 39, 41, 42, 44, 47, 49, 51, 53, 55, 57, 58, 59, 60, 61, 62, 64, 66, 67, 70, 71, 72, 74, 78, 90

6.1 – The Wave Nature of Light

- The light that we see is called visible light, and is an example of electromagnetic radiation.
- Also called radiant energy, because it can carry energy through space.
- Lots of types of electromagnetic radiation
- visible light, radio waves, infrared, etc.
- All types of electromagnetic radiation travel through a vacuum at a speed of 3.0x108 m/s
- The speed of light (c)

Waves

- Waves are periodic
- Which means they repeat in regular intervals
- The distance between two adjacent peaks (or troughs) in a wave is called the wavelength (λ, lambda)
- The number of complete wavelengths (or cycles) that pass a certain point each second is called its frequency (ν, nu)

The relationship between speed, wavelength and frequency can be expressed as

- c = νλ
- Since all electromagnetic waves move at the speed of light, a high wavelength means a low frequency, and vice versa.

Frequency expressed in the units of cycles/second (s-1)

- Also known as hertz (Hz)
- Wavelength is in meters

Example

- The yellow light given off by a sodium vapor lamp has a wavelength of 590 nm. What is the frequency of this radiation?

Workspace

- c = λν
- Remember nm = 10-9 m

Example

- A laser used in eye surgery to fuse detached retinas produces radiation with a wavelength of 640.0 nm.
- What is the frequency of this radiation?

6.2 – Quantized Energy and Photons

- There are 3 phenomena we will deal with that are not explained by the wave model of light.
- The emission of light from hot objects
- Called blackbody radiation
- The emission of electrons from metals on which light shines
- Called the photoelectric effect
- The emission of light from electronically excited gas atoms
- Called the emission spectra

Hot Objects and the Quantization of Energy

- When solid objects are heated, they emit radiation
- The red glow of an electric burner
- White light of a tungsten lightbulb
- The wavelengths of the light radiated depends on the temperature
- Red-hot object cooler than a white-hot object
- In the late 19th century, physics could not explain these behaviors.

In 1900, Max Planck figured out how to handle this.

- Assumed that energy can be either released or absorbed by atoms only in discrete “chunks” of some minimum size.
- He called the smallest quantity of energy that can be emitted or absorbed as electromagnetic radiation a quantum.

Planck proposed that the energy E, of a single quantum equals a constant times the frequency of the radiation

- E = hv
- h = Planck’s constant, 6.626x10-34 J-s
- According to Planck’s theory, matter is allowed to emit and absorb energy only in whole number multiples of hv
- Such as hv, 2hv, 3hv and so on

If the quantity of energy emitted is 3hv, we say the atom emitted 3 quanta of energy.

- Because the energies can be released only in specific amounts, we say that the allowed energies are quantized
- Which means their values are restricted to certain amounts

Does this seem odd?

- If so, think of stairs
- You can only step on the stairs, not between them.
- Energy can only come in certain amounts, not between those amounts.
- Why don’t we notice this with more things?
- Why do energy changes seem continuous, rather than quantized?
- Because Planck’s constant is so small, so on a larger scale, it seems continuous, but only when you look at energy on a very small scale does it appear to be quantized.

The Photoelectric Effect and Photons

- In 1905, Albert Einstein used Planck’s quantum theory to explain the photoelectric effect.
- When photons of sufficiently high energy strike a metal surface, electrons are emitted from the metal.
- For each metal, there is a minimum frequency of light
- If we go below that frequency, no electrons will be emitted.

Explaining Photoelectric Effect

- To explain this, Einstein assumed that radiant energy striking the metal is acting like a stream of tiny particles (or energy packets), not like a wave.
- He called each energy packet a photon.
- Following Planck’s quantum theory, he decided each photon must have energy consistent with Planck’s equation
- Energy of photon = E = hv

How This Works

- Under the right conditions, a photon strikes a metal surface and is absorbed.
- The photon can transfer its energy to an electron in the metal
- A certain amount of energy (called the work function) is required for an electron to free itself from the metal
- If the photon has less energy than the work function, the electron can’t escape from the metal surface.
- If the photon has enough energy, electrons are emitted from the metal, the extra energy appearing as the kinetic energy of the emitted electrons.

6.3 Line Spectra and the Bohr Model

- Planck and Einstein’s work opened up the doors for understanding how electrons are arranged in atoms.
- In 1913, Danish physicist Niels Bohr came up with a theoretical explanation of line spectra

Line Spectra

- A particular source of radiant energy may emit a single wavelength
- Radiation composed of a single wavelength is called monochromatic
- Common, most radiation sources produce radiation of many wavelengths.
- When radiation from such sources is separated into its different wavelengths, a spectrum is produced.

Not all radiation sources produce a continuous spectrum.

- When different gases are placed in a tube and a high voltage is applied, the gases will emit different colors of light.
- When this light passes through a prism, only a few wavelengths are present in the resultant spetra.

The colored lines are separated by black regions

- Black regions correspond to wavelengths that are absent from the light.
- A spectrum containing radiation of only specific wavelengths is called a line spectrum

In 1885, a Swiss schoolteacher named Johann Balmer showed the wavelengths of the four (they couldn’t see others yet) visible lines of hydrogen fit a simple equation.

- Balmer’s equation was later extended to a more general use and was called the Rydberg equation

Rydberg Equation

- Allows us to calculate the wavelengths of all of the spectral lines of hydrogen
- RH = Rydberg constant = 1.096776x107 m-1
- n1 and n2 are positive integers with n2 being larger than n1
- It took over 30 years to explain this equation

Bohr’s Model

- Bohr refined Rutherford’s model of the atom, using Planck’s idea of quantized energy

Bohr’s Postulates

- Only orbits of certain radii, corresponding to certain definite energies, are permitted for the electron in a hydrogen atom
- An electron in a permitted orbit has a specific energy and is in an “allowed” energy state. An electron in an allowed energy state will not radiate energy and therefore will not spiral into the nucleus.
- Energy is emitted or absorbed by the electron only as the electron changes from one allowed energy state to another. This energy is emitted or absorbed as a photon, E = hv

Energy States of the Hydrogen Atom

- Bohr calculated the energies corresponding to each allowed orbit for the electron in the hydrogen atom
- These energies fit the following formula
- h = Planck’s constant
- c = speed of light
- RH = Rydberg constant

What about n?

- The integer n, which can be between 1 and ∞, is called the principal quantum number
- Each orbit corresponds to a different value of n
- Radius gets larger as n increases.
- So first orbit has n = 1 (closest)
- Next allowed orbit has n = 2, and so on
- The electron in a H atom can be in any allowed orbit, and the previous equation tells us the energy that the electron will have, depending on its orbit.

The energies of the electron in a hydrogen atom will always be negative

- The lower (more negative) the energy is, the more stable the atom will be.
- The energy is lowest (most negative) for n = 1
- This lowest energy state (n = 1) is called the ground state of the atom.
- When the electron is in a higher (less negative) orbit, the atom is said to be in an excited state.

What happens….

- …to the orbit radius and energy as n becomes infinitely large?
- The radius increases as n2, so we reach a point where the electron is completely separated from the nucleus.
- When n = ∞, E = 0

So the state in which the electron is removed from the nucleus is the reference, or zero-energy, state of the hydrogen atom.

- This zero-energy state is higher in energy than the states with negative energies.

In his 3rd postulate, Bohr assumed an electron could “jump” from one allowed energy state to another by absorbing or emitting photons

- The photons’ energy must meet exactly the energy difference between the states
- Or, radiant energy is emitted when the electron jumps to a lower energy level.

In equation form…

- ΔE = Ef – Ei = Ephoton = hv
- Substituting the previous energy equation into this one, and remembering that v = c/λ
- ni and nf are the principal quantum numbers of the initial and final state of the atom.
- If nf is smaller than ni, the electron moves closer to the nucleus, and ΔE is a negative number (atom loses energy)

Example

- An electron moves from the 3rd principal quantum number to the 1st principal quantum number. What is the ΔE of this change?
- First off: Should we expect our ΔE to be positive or negative?
- Which quantum number is nf and which is ni?
- nf = 1, ni= 3

ΔE = -2.18x10-18 (1-1/9) = -2.18x10-18(8/9)

ΔE = -1.94x10-18 J

What is the wavelength of this photon?

- λ = 1.03x10-7 m
- Remember:
- ΔE = hc/λ
- What happened to the – sign?
- Wavelength and frequency are always positive, so negative sign is ignored.

Shortcut equation

- If we solve
- for 1/λ, we find an equation remarkably similar to the Rydberg equation

Limitations

- The Bohr model explains the line spectrum of hydrogen, but not the spectra of other atoms accurately.
- But the Bohr model did bring in two important ideas into our model of the atom
- Electrons exist only in certain discrete energy levels
- Energy is involved in moving an electron from one level to another

6.4 The Wave Behavior of Matter

- Following Bohr’s model for the hydrogen atom, the dual nature of radiant energy become a familiar concept.
- Depending on the experiment, radiation appears to have either a wavelike or particle-like nature
- Louis de Broglie extended this idea dramatically

De Broglie

- If radiant energy could behave as a stream of particles, could matter behave like a wave?
- Suppose the electron orbiting a nucleus wasn’t thought of as a particle, but a wave with a particular wavelength.
- De Broglie suggested that it’s motion around a nucleus was associated with a particular wavelength.
- He went on to propose that the particular wavelength of the electron, or of any other particle depended on its mass and velocity

De Broglie’s Equation

- The quantity mv for any object is called its momentum (p)
- h is Planck’s constant
- De Broglie used the term matter waves to describe the wave properties of matter

What this means…

- Because de Broglie’s hypothesis applies to all matter, any object with mass and velocity creates a characteristic matter wave.
- However, the larger the mass, the smaller the wavelength.
- Most ordinary objects have a wavelength so small as to be completely out of the range of any possible observation.
- Electrons are so small though, that they will still have an applicable wavelength

Example

- What is the wavelength of an electron moving with a speed of 5.97x106 ms?
- The mass of an electron is 9.11x10-28 g
- Note: de Broglie’s equation works only if
- mass is in kg
- velocity is in m/s

Workspace

- h = 6.63x10-34

Use of this?

- Used in electron microscopes

The Uncertainty Principle

- Werner Heisenberg proposed that the dual nature of matter places a limitation on how precisely we can know the location and the momentum of any object.
- This only really comes into play with matter on the subatomic level
- This principal is called the uncertainty principal

6.5 Quantum Mechanics and Atomic Orbitals

- Austrian physicist Erwin Schrodinger proposed an equation (called Schrodinger’s wave equation) that incorporated both the wavelike behavior and particle-like behavior of the electron.
- Opened up field called quantum mechanics or wave mechanics
- Equation requires advanced calculus
- So we’re not going to worry about it

Idea of Schrodinger’s Equation

- Solving his equation leads to a series of mathematical functions called wave functions that describe the electron
- These wave functions are represented by the symbol Ψ (psi)
- The square of the wave function (Ψ2) provides information about the electron’s location when the electron is in an allowed energy state
- Ψ2 is called either the probability density or electron density
- Tells where an electron is most likely to be

Orbitals and Quantum Numbers

- The solution to Schrodinger’s equation for the hydrogen atom gives a set of wave functions and corresponding energies.
- These wave functions are called orbitals.
- Each orbital describes a specific distribution of electron density in space
- So each orbital really describes a characteristic energy and shape
- Orbital ≠ Orbit

Quantum Numbers

- The Bohr model had a single quantum number, n, to describe an orbit.
- The quantum mechanical model uses 3 quantum numbers to describe an orbital
- n, l and m1

Principal Quantum Number, n

- Can have positive integral values
- As n increases, the orbital becomes larger, and the electron spends more time farther from the nucleus
- A higher value n also mans more energy, and is less tightly bound to the nucleus
- For hydrogen atom,
- En = -(2.18x10-18)(1/n2), as per the Bohr model

Azimuthal Quantum Number, l

- Can have integral values from 0 to n – 1(for each value of n).
- This quantum number defines the shape of the orbital
- The value of l for a particular orbital is generally designated by the letters s, p, d, and f, corresponding to l values of 0, 1, 2, and 3 respectively

Magnetic Quantum Number, ml

- Can have integral values between -l and l, including zero.
- This quantum number describes the orientation of the orbital in space

Because the value of n can be any positive integer, there can be an infinite number of orbitals for the hydrogen atom.

- The electron in a hydrogen atom is described by only one of these orbitals at any given time
- We say the electron occupies a certain orbital
- The remaining orbitals are unoccupied
- Generally, we will only be concerned with the orbitals of the hydrogen atom with small values of n.

The collection of orbitals with the same value of n is called an electron shell.

- All the orbitals that have n = 3 are said to be in the 3rd shell.
- Also, each set of orbitals that have the same n and l values are called a subshell.
- Each subshell is designated by a number (the value of n) and a letter (s, p, d, f, corresponding to the value of l).
- So the orbitals that have a n = 3, and l = 2 are called 3d orbitals and are in the 3d subshell.

General Observations

- The shell with the principal quantum number n has exactly n subshells
- So first shell (n = 1) has only 1 subshell, the 1s (l = 0)
- Second shell (n = 2) has 2 subshells, the 2s (l=0) and 2p (l=1)

Each subshell has a specific number of orbitals

- Each orbital corresponds to a different allowed value of ml
- For a given value of l, there are 2l +1 allowed values of ml
- Ranging from –l to l
- So each s (l =0) subshell has 1 orbital
- So each p (l = 1) subshell has 3 orbitals

The total number of orbitals in a shell is n2, where n is the principal quantum number of the shell

- The resulting number of orbitals for the shells (1, 4, 9, 16) is related to a pattern seen in the periodic table
- We see that the number of elements in the rows of the periodic table (2, 8, 18, 32) equal twice these numbers.

Orbital Energy Diagrams

- pg. 233, Figure 6.17
- Each box represents an orbital
- Orbitals of the same subshell (such as 2p) are grouped together.
- When the electron occupies the lowest-energy orbital, the hydrogen atom is said to be in its ground state.
- When the electron occupies any other orbital, the atom is in an excited state.
- Can be in a higher-energy orbital by absorption of a photon of the right energy

Example

- What is the designation for the subshell of n = 5 and l = 1
- 5p
- How many orbitals are in this subshell?
- 3 (2l +1)
- What are the values of ml for each of these orbitals
- 1, 0, -1

6.6 – Representations of Orbitals

- We’re going to look at how to picture orbitals.

The “S” Orbital

- The 1s orbital is the lowest energy orbital of the hydrogen atom.
- The electron density for the 1s orbital shows it is spherical.
- All other s orbitals are spherical

Difference Between “S” Orbitals

- All s orbitals are spherical
- (a) shows probability to find electron highest around 1 Angstrom from nucleus, and drops off quickly after that.

(b) 2 peaks for probability, and zero probability at 1 Angstrom

- Where probability = 0, is called a node
- Larger range of where the electron could be
- More spread out area
- (c) has 2 nodes, and is even more spread out.

What does this mean?

- The radial probability functions tell us that as n increases, there is also an increase in the most likely distance from the nucleus to find an electron.
- In other words, the size of the orbital increases with n, just like in the Bohr model.

The “p” Orbitals

- Density for a 2p orbital
- Can see electron density concentrated in 2 regions on either side of the nucleus.
- Each region is called a lobe

Beginning with n = 2, each shell has 3 p orbitals.

- So there are three 2p orbitals, three 3p orbitals, and so forth
- For each value of n, the three p orbitals have the same shape, but differ in spatial orientation
- Usually labeled as the px, py and pz orbitals
- Like s orbitals, p orbitals increase in size with increase n values.

The “d” and “f” Orbitals

- When n is 3 or greater, we get five equivalent d orbitals (l = 2)

When n is 4 or greater, we get seven equivalent f orbitals (l = 3)

- VERY COMPLICATED SHAPES

6.7 – Many-Electron Atoms

- So far, we’ve only looked at hydrogen (1 electron)
- Unfortunately, most atoms have more than 1 electron
- To look at electronic structure of these many-electron atoms, we need to consider how the electrons populate available orbitals

Orbitals and Energy

- Many-electron atoms have the same general shapes as the corresponding hydrogen orbitals.
- However, the presence of more than 1 electron changes the energies of the orbitals.
- In hydrogen, energy depended only on its principal quantum number (n)
- This is not the case in many-electron atoms

In many electron atoms, for a given value of n, the energy of an orbital increases with increasing values of l

- So 3s < 3p < 3d
- Energy remains the same in the orbitals within a given subshell
- Each of the three 2p orbitals have the same energy
- Figure 6.24, pg. 238 for example

Electron Spin

- Electrons have a property called electron spin
- This causes an each electron to behave as if it were a tiny sphere spinning on its own axis
- Electron spin is quantized, and is given a new quantum number
- ms
- Only has 2 possible values, +½ and -½
- This spinning charge produces a magnetic field
- 2 oppositely directed spins gives us 2 oppositely charged magnetic fields.

Pauli Exclusion Principle

- No 2 electrons in an atom can have the same set of quantum numbers n, l, mland ms
- For a given orbital (1s, 2pz and so forth), the values of n, l and ml are fixed
- So if we want to put more than one electron in an orbital, we must assign them different msvalues.
- Because there are only 2 such values, an orbital can only hold 2 electrons, and they must have opposite spin.

6.8 Electron Configurations

- We can now consider the arrangement of electrons in atoms
- The way in which electrons are distributed among the various orbitals of an atom are called its electron configuration
- The most stable electron configuration (ground state) is that in which the electrons are in the lowest possible energy states.
- If there were no restrictions on where electrons could go, they would all crowd into the 1s orbital
- But, the Pauli Exclusion Principle tells us that there can be only 2 electrons in any given orbital

The orbitals get filled in order of increasing energy, with no more than 2 electrons per orbital

- Consider Lithium
- 3 electrons
- The 1s orbital can hold 2 electrons, so the third one goes into the next lowest energy orbital, the 2s

1s

2s

Orbital Diagram- We can represent electron configuration by writing the symbol for the occupied subshell and adding a superscript to indicate the number of electrons
- Lithium example
- 1s22s1

Electrons having opposite spins are said to be paired when they are in the same orbital (↑↓)

- An unpaired electron is one not with a partner of opposite spin within the same orbital.
- In the lithium atom, the 2 electrons in the 1s are paired, but the electron in the 2s orbital is unpaired.

1s

2s

2p

Hund’s Rule- When placing electrons in orbitals, we follow energy trends
- Fill lowest energy levels first
- But we run into problems once we get to Boron

When going to carbon, where does the 6th electron go?

- We know it has to go into the 2p orbital, but which one?
- Question is answered by Hund’s rule
- For orbitals of the same energy and type (called degenerate orbitals), the lowest energy is attained when the number of electrons with the same spin is maximized.
- Basically, this means that you will put 1 electron in each of the 2p orbitals, all with the same spin, before you start creating paired electrons.

Hund’s rule is based (partially) on the fact that electrons repel each other.

- By occupying different orbitals, the electrons remain as far from possible from each other, minimizing electron-electron replsions.

Example

- Write the electron configuration for phosphorus, element 15
- Also, how many unpaired electrons does phosphorus have?

Condensed Electron Configurations

- Neon fills out the 2p subshell, and will have a stable configuration with eight electrons (called an octet) in its outermost occupied shell.
- The next element, Na, marks the beginning of a new row of the periodic table.
- Sodium has a single 3s electron beyond the stable configuration of neon.
- We can abbreviate the electron configuration for sodium, based upon its relationship with the stable configuration for neon.
- Na: [Ne]3s1

What did I do?

- The symbol [Ne]represents the electron configuration of the ten electrons of neon (1s22s22p6).
- Writing [Ne]3s1 helps focus attention on the outermost electrons of the atom
- Which are the ones mostly responsible for the chemical behavior

To Generalize:

- In writing the condensed electron configuration of an element, the electron configuration of the nearest noble-gas element of lower atomic number is represented by its chemical symbol in brackets
- So lithium would be
- Li: [He]2s1

We refer to the electrons represented by the symbol for a noble-gas as the noble-gas core of the atom.

- Usually, these inner-shell electrons are referred to simply as the core electrons.
- The electrons after the core electrons are called the outer-shell electrons.
- And the outer-shell electrons will include the electrons involved in chemical bonding, which we call the valence electrons.

For lighter elements

- (atomic number 30 or less) ALL of the outer-shell electrons are valence electrons.
- Things get odd in the heavier elements.

Transition Metals

- Argon ends the row started by sodium.
- Argon is 1s22s22p63s23p6
- The element following argon is potassium
- Potassium is obviously a member of the alkali metals
- It’s outermost electron occupies an “s” orbital
- But this means that the electron that has the highest energy did NOT go into the 3d orbital
- Because the 3rd shell has s, p, and d orbitals, but the 3d seems to have been skipped

So the condensed electron configuration for potassium is

- K: [Ar]4s1
- Following complete filling of the 4s orbital
- So starting with scandium and extending through zinc, electrons will be added to the five 3d orbitals until they are filled
- So, the fourth row of the periodic table is ten elements wider than the two previous rows
- These ten elements are known as either transition elements or transition metals

How to Fill Transition Metals

- Finding the electron configuration for these follows Hund’s rule.
- Electrons are added to the 3d orbitals singly until all five orbitals have 1 electron each
- Additional electrons are then put in the 3d orbitals with spin pairing until the shell is completely full.

Then…

- Once we have the 3d orbitals filled with 2 electrons each, the 4p orbitals get filled
- Until we get a full octet of outer electrons
- 4s24p6 with krypton

Lanthanides and Actinides

- The sixth row of the periodic table begins like the previous ones
- One electron in the 6s orbital for cesium, and two electrons in the 6 orbital of barium.
- However, the periodic table then has a break, and elements 57-70 placed below the periodic table
- Here is where we encounter the new set of orbitals, 4f.

Since there are seven 4f orbitals, it will take 14 electrons to fill it

- These 14 electrons come from the elements known as the lanthanide elements or the rare earth metals
- Because the 4f and 5d orbitals are very close to each other, the electron configurations of some of the lanthanides involve 5d electrons
- This causes exceptions to some of the rules
- For now, don’t memorize the exceptions…

The final row of the periodic table is the actinide elements

- Most are radioactive, and not usually found in nature
- Built by filling up the 5f orbitals

6.9 – Electron Configuration and the Periodic Table

- Just by looking at the periodic table, we can determine the electron configuration of any element.

How to Use This

- Find the element you want
- Find which block it is in
- Fill up orbitals until the block you’re looking for
- So look at what should come before what you’re looking at
- Count how many elements into the row you are
- That determines how many electrons are in that specific subshell
- And remember
- s & p blocks: Period number = coefficient number
- d block: Period number -1 = coefficient number
- f block: Period number -2 = coefficient number

Examples

- Bi (atomic # 83)
- Co (atomic #27)
- Te (atomic #52)

Valence Electrons

- Again, these are the outer shell electrons used in bonding.
- For representative elements, do not consider completely full d or f subshells to be part of the valence electrons
- For transition metals, do not consider a full f subshell to be part of the valence electrons

Anomalous Electron Configuration

- Some elements have weird electron configurations.
- Most are among the f-block metals.
- We’re not going to worry about these exceptions.

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