Electronic Structure of the Elements

CHY113 OpenStax

Chapter 6: Electronic Structure and Periodic Properties of the Elements

Learning Objectives

6.1 Electromagnetic Energy

•  Explain the basic behavior of waves, including travelling waves and standing waves

•  Describe the wave nature of light

•  Use appropriate equations to calculate related light-wave properties such as frequency, wavelength, and energy

•  Distinguish between line and continuous emission spectra

•  Describe the particle nature of light

6.2 The Bohr Model

•  Describe the Bohr model of the hydrogen atom

•  Use the Rydberg equation to calculate energies of light emitted or absorbed by hydrogen atoms

6.3 Development of Quantum Theory

•  Extend the concept of wave–particle duality that was observed in electromagnetic radiation to matter as well

•  Understand the general idea of the quantum mechanical description of electrons in an atom, and that it uses the notion of three-dimensional wave functions, or orbitals, that define the distribution of probability to find an electron in a particular part of space

•  List and describe traits of the four quantum numbers that form the basis for completely specifying the state of an electron in an atom

6.4 Electronic Structure of Atoms (Electron Configurations)

•  Derive the predicted ground-state electron configurations of atoms

•  Identify and explain exceptions to predicted electron configurations for atoms and ions

•  Relate electron configurations to element classifications in the periodic table

6.5 Periodic Variations in Element Properties

• Describe and explain the observed trends in atomic size, ionization energy, and electron affinity of the elements

Resources

6.1 Electromagnetic Energy

•  Explain the basic behavior of waves, including travelling waves and standing waves

•  Describe the wave nature of light

•  Use appropriate equations to calculate related light-wave properties such as frequency, wavelength, and energy

•  Distinguish between line and continuous emission spectra

•  Describe the particle nature of light

A wave is an oscillation or periodic movement that can transport energy from one point in space to another. 

All waves are characterized by the following:

Wavelength (λ)

Distance between two consecutive peaks or troughs in a wave

Frequency (v)

Number of successive wavelengths that pass a given point in a unit time 

Amplitude

One-half the distance between the peaks and troughs 

A wave can be a traveling wave, like an ocean wave, or a standing wave like a guitar string

with nodes - areas of zero amplitude.

The electromagnetic spectrum is the wave description of light. 

Light has a constant velocity in a vacuum of 2.997 x 108 m/s and given the constant "c."

The wavelength (λ) of light is inversely proportional to its frequency (v) with the relationship

c = λv. The wave-nature of light can be seen in interference patterns.

The particle description of light is as wave packets called photons with energy given by

E = hv, where h is Planck's constant and equal to 6.626 x 10-34 J.s. Based on "the 

Ultraviolet Catastrophe," Planck postulated that energy was quantized, not continuous, 

and calculated Planck's constant. Einstein used Planck's quantization of energy to explain 

the the photoelectric effect, demonstrating the particle-nature of light. 

Figure 6.11 Photons with low frequencies do not have enough energy to cause electrons to be ejected

 via the photoelectric effect. For any frequency of light above the threshold frequency, the kinetic energy

 of ejected electron will increase linearly with the energy of the incoming photon.

Max Planck was awarded the Nobel prize in Physics in 1918 "for the services he rendered to the advancement of physics by his discovery of energy quanta, which is given the symbol h." 

Albert Einstein was awarded the Nobel prize in Physics in 1921 for "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect."

6.2 The Bohr Model

•  Describe the Bohr model of the hydrogen atom

•  Use the Rydberg equation to calculate energies of light emitted or absorbed by hydrogen atoms

Continuous vs. discreet spectra.

Figure 6.13 Compare the two types of emission spectra: continuous spectrum of white light (top) and the line spectra of the light from excited sodium, hydrogen, calcium, and mercury atoms.

Based on the observations of discreet spectra of elements and Planck's quantization of energy, Bohr proposed a model of the atom with quantized energy levels that fit the observed line spectrum for hydrogen.

Bohr incorporated the following into the classical mechanics description of the atom. 

• • • Planck’s ideas of quantization.

    • Einstein’s finding that light consists of photons whose energy is proportional to their frequency.

Bohr assumed that the electron orbiting the nucleus would not normally emit any radiation.

Rather the electron emits or absorbs a photon if it moved to a different orbit. 

The energy of each electronic level being equal to 

 where k is a constant (2.18 x 10-18 J) and n is called the principle quantum number.

Figure 6.14 Quantum numbers and energy levels in a hydrogen atom. 

The more negative the calculated value, the lower the energy.

• • When the electron is in this lowest energy orbit (n = 1), the atom is said to be in its ground state. 

  • If the atom receives energy from an outside source, it is possible for the electron to move to an orbit 

  • with a higher n value (excited state), which has a higher energy. 

  • When the atom absorbs energy as a photon, the electron moves from an orbit with a lower n to a higher n. 

  • When an electron falls from an orbit with a higher n to a lower n, the atoms emits energy as a photon.  

  • Since ∆E can only be discrete values, the photon absorbed or emitted can only have a wavelength 

  • with a discrete value (not continuous).

  • This explained the line spectra for the hydrogen atom. 

Figure 6.15 The horizontal lines show the relative energy of orbits in the Bohr model of the hydrogen atom, 

and the vertical arrows depict the energy of photons absorbed (left) or emitted (right) as electrons move 

between these orbits. Transitions from and to n = 1 are called the Lyman series and their wavelengths all occur in the ultraviolet region of the electromagnetic spectrum. Transitions to and from n = 2 and above belong to the Balmer series - the four lowest energy transitions are responsible for the hydrogen line spectrum.

Use the Rydberg equation to calculate the energy/wavelength/ or frequency of the radiation associated with any transition in the hydrogen atom.

 where n1 is the quantum number for the initial state and n2 in the final state.

Niels Bohr was awarded the Nobel prize in Physics in 1922 for "for his services in the investigation of the structure of atoms and of the radiation emanating from them."

6.3 Development of Quantum Theory

•  Extend the concept of wave–particle duality that was observed in electromagnetic radiation to matter as well

•  Understand the general idea of the quantum mechanical description of electrons in an atom, and that it uses the notion of three-dimensional wave functions, or orbitals, that define the distribution of probability to find an electron in a particular part of space

•  List and describe traits of the four quantum numbers that form the basis for completely specifying the state of an electron in an atom

Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the behavior of quantum-scale objects.

h = Planck’s constant

m = particle mass

v = particle velocity

p = particle momentum

λ = de Broglie wavelength

The de Broglie wavelength is a characteristic of particles, not electromagnetic radiation.

In quantum mechanics we describe electrons as standing waves.

Some points:

It appears that while electrons are small localized particles, their motion does not follow the equations of motion implied by classical mechanics.

Instead some type of wave equation governs a probability distribution for an electron’s motion. 

Thus the wave–particle duality first observed with photons is actually a fundamental behavior intrinsic to all quantum particles.

Werner Heisenberg determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously.

The more accurately we measure the momentum of a particle, the less accurately we can determine its position at that time, and vice versa. 

Heisenberg Uncertainty Principle: It is fundamentally impossible to determine simultaneously and exactly both the momentum and the position of a particle.

∆x = uncertainty in the position of the particle 

∆px = uncertainty in the momentum of the particle in the x direction. 

Schrödinger Wave Equation is the wave equation that describes the electron.

Erwin Schrödinger extended de Broglie’s work by incorporating the de Broglie relation into a wave equation.

Today this equation is know as the Schrödinger equation.

Schrödinger properly thought of the electron in terms of a three-dimensional stationary wave, or wavefunction, represented by the Greek letter psi, ψ.

A few years later, Max Born: 

Electrons are still particles, and so the waves represented by ψ are not physical waves but, instead, are complex probability amplitudes.

The square of the magnitude of a wavefunction ∣ψ∣2 describes the probability of the quantum particle being present near a certain location in space. 

Wavefunctions can be used to determine the distribution of the electron’s density with respect to the nucleus in an atom.

In the most general form, the Schrödinger equation can be written as:

 Ĥ is the Hamiltonian operator, a set of mathematical operations representing the total energy of the quantum particle. 

 ψ is the wavefunction of the particle. 

 E is the total energy of the particle. 

Our description of the electron now has certain constraints, namely,

(i) Electrons in atoms can exist only in discrete energy levels but not between them. 

 (ii) The energy of an electron in an atom is quantized.  

 (iii) The energy can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels.

(iv) The energy levels are labeled with an n value, where n = 1, 2, 3, …. 

 (v) Generally speaking, the energy of an electron in an atom is greater for larger values of n. 

 (vi) n is referred to as the principal quantum number or shell number. 

 (vii) The principal quantum number defines the location of the energy level and is similar in concept to the Bohr model. 

The quantum numbers. Every electron in an atom possesses a unique set of 4 quantum numbers.

(i) The principal quantum number (n) is one of three quantum numbers used to characterize an orbital. 

(ii) An atomic orbital is a general region in an atom that an electron is most probable to reside. 

(iii) An atomic orbital is distinct from an orbit. 

(iv) The quantum mechanical model specifies the probability of finding an electron in a three-dimensional space around the nucleus. 

(1) The principal quantum number (n), where n = 1, 2, 3, ….

(2) The angular momentum quantum number (ℓ) is an integer that defines the shape of that orbital - called subshells.

    ℓ takes on the values, ℓ = 0, 1, 2, …, n – 1. We name the general shape of the orbital with a letter.

    For ℓ = 0: s orbital, for ℓ = 1: p orbital, for ℓ = 2: d orbital, and for ℓ = 3: f orbital.

As n increases, the number of nodes increase. For example, going from 1s to 2s to 3s, we have 0 nodes for 1s, 1 spherical node for the 2s orbital, and 2 spherical nodes for the 3s orbital.

Figure 6.21: The graphs show the probability (y axis) of finding an electron for the 1s, 2s, 3s orbitals as a function of distance from the nucleus.

The s subshell has a spherical shape. 

The p subshell has a dumbbell shape. 

The d and f orbitals are more complex. 

These shapes represent the three-dimensional regions where the electron is likely to be found.

        The three 2p orbitals: 2px, 2py, and 2pz. 

        The five 3d orbitals, 3dxy, 3dxz, 3dyz, 3dx2-y2, 3dz2.

(3) The magnetic quantum number, ml, specifies the orientation of the orbital in space. If an orbital has an angular momentum (ℓ ≠ 0), then this orbital can point in different directions.

The value of mℓ depends on the value of ℓ.

mℓ = –ℓ … –1, 0, +1, … + ℓ

There are 2ℓ + 1 orbitals with the same ℓ value. 

One s-orbital for ℓ = 0

Three p-orbitals for ℓ = 1

Five d-orbitals for ℓ = 2

Seven f-orbitals for ℓ = 3

In the case of a hydrogen atom, energies of all the orbitals with the same n are the same. 

This is called a degeneracy, and the energy levels with the same principal quantum number, n, are called degenerate energy levels. 

However, in atoms with more than one electron, this degeneracy is eliminated by electron–electron interactions.

Orbitals that belong to different subshells have different energies.

Orbitals within the same subshell are still degenerate and have the same energy.

        Figure 6.22: The chart shows the energies of electron orbitals in a multi-electron atom.

(4) The last quantum number is the spin quantum number, ms, describes the two possible spin states of the electron, either spin up, ms = +1/2, or spin down, ms = -1/2. 

The Pauli Exclusion Principle states that "no two electrons in the same atom can have exactly the same set of all four quantum numbers." Therefore an electron in an atom is completely described by four quantum numbers: n, l, ml, and ms.

6.4 Electronic Structure of Atoms (Electron Configurations)

•  Derive the predicted ground-state electron configurations of atoms

•  Identify and explain exceptions to predicted electron configurations for atoms and ions

•  Relate electron configurations to element classifications in the periodic table

The Aufbau Principle: The build up technique for describing the electron configuration of ground state atoms.

Figure 6.26: This diagram depicts the energy order for atomic orbitals and is useful for deriving ground-state electron configurations.

But you can read the relative energies of the orbitals right from the periodic table...

Figure 6.27: This periodic table shows the electron configuration for each subshell. By “building up” from hydrogen, this table can be used to determine the electron configuration for any atom on the periodic table.

Examples of electron configurations:

Note Hund's rule: evenly distribute subshell e-'s with same spin

Valence vs. Core Electrons:

(i) The electrons occupying the orbital(s) in the outermost shell (highest value of n) are called valence electrons. 

(ii) The electrons occupying the inner shell orbitals are called core electrons.  

(iii) The core electrons represent noble gas electron configurations.

(iv) Electron configurations can be expressed in an abbreviated format by writing the noble gas that matches the core electron configuration, along with the valence electrons.

There are some exceptions to the order of filling subshells.

Examples: Cu and Cr

There is stability associated with a half-filled or fully filled d subshell.

An electron shifts from the 4s into the 3d subshell to achieve this stability. 

Cu:

Figure 6.31: Within each period, the trend in atomic radius decreases as Z increases; for example, from K to Kr. Within each group (e.g., the alkali metals shown in purple), the trend is that atomic radius increases as Z increases.

Cations are always smaller than neutral atom and anions are always larger than neutral atoms.

First ionization energies increase as Zeff increases.

The amount of energy required to remove the most loosely bound electron from a gaseous atom in its ground state is called its first ionization energy (IE1). 

X(g) ⟶ X+(g)   +   e− IE1

The energy required to remove the second most loosely bound electron is called the second ionization energy (IE2).

X+(g) ⟶ X2+(g)   +   e− IE2

The energy required to remove the third electron is the third ionization energy, and so on.

Deviation: Oxygen (O) has a lower IE1 than nitrogen (N).

Removing one electron from O will eliminate the electron–electron repulsion caused by pairing the electrons in the 2p orbital.

This means that removing an electron from O is more energetically favorable than removing an electron from N.

The electron affinity (EA) is the energy change for the process of adding an electron to a gaseous atom to form an anion.

X(g)   +   e−   ⟶   X−(g) EA1

This process can be either endothermic or exothermic, depending on the element.

For some elements energy is released when the gaseous atom accepts an electron (negative EA).

For other elements energy is required for the gaseous atom to accept an electron (positive EA).

It becomes easier to add an electron as the effective nuclear charge of the atoms increases.

Electron affinities becomes more negative across a period. 

There are some deviations from this trend. 

The noble gases, group 18, have a completely filled shell and the incoming electron must be added to a higher energy n level.

Group 2 has a filled ns subshell, and so the next electron added goes into the higher energy np subshell. 

Group 15 has a half-filled np subshell and the next electron must be paired with an existing np electron. 

In all of these cases, the initial relative stability of the electron configuration disrupts the trend in EA.

This version of the periodic table displays the electron affinity values (in kJ/mol) for selected elements.