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Chapters6and7CHEM101PostLesson.pptxChapters 6 and 7: Properties of Electrons and Light
CHEM 101
Fall 2020
Dr. Lauren Genova
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Chapter 4
By the end of this chapter, you will be able to:
Calculate related light-wave properties such as frequency, wavelength, and energy
Recognize the wave-particle duality of light and matter
Compare and contrast the Bohr model of the hydrogen atom with the quantum mechanical description of electrons in an atom
Describe the four quantum numbers that form the basis for specifying the state of an electron in an atom
Determine the electron configuration of an atom
Characterize an element as paramagnetic or diamagnetic
Describe and explain the observed trends in atomic size, ionization energy, electron affinity, and electronegativity of the elements
Predict the reactivity trends for metals vs. non-metals
Chapters 6 and 7: Learning Objectives
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Electronic Structure of Atoms
Electrons are the ultimate determinant of the properties and reactivity of atoms.
Understanding how electrons behave and what influences them is critical to understanding chemistry.
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Chapter 6
Late 1800s-early 1900s saw an increased understanding of atomic structure.
To better understand the configuration of electrons around an atom’s nucleus, scientists exposed atoms to various forms of electromagnetic energy.
J.J. Thomson “Plum Pudding Model,” 1904
Bohr Model
1913
Electronic Structure of Atomic Models
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Chapter 6
Continuous range of radiant energy (also called electromagnetic radiation)
The Electromagnetic Spectrum
Longer wavelength
Lower frequency
Lower energy
Shorter wavelength
Higher frequency
Higher energy
Visible light = the portion of the electromagnetic spectrum that the human eye can view!
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Chapter 6
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Electrons, as we will see, can behave as both waves and particles. And so, let’s start with waves: namely, by reviewing the electromagnetic spectrum.
Wavelengths of visible light (nm)
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Chapter 6
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Scientist Spotlight Dr. Felix S. Alfonso Postdoctoral Scholar, Department of Chemistry and Chemical Biology, Cornell University (Ithaca, NY) Topic: Light Favorite books: The Little Prince by Antoine de Saint-Exupéry and The Alchemist by Paulo Coelho
Email: [email protected]
Wavelength (λ): the distance from crest to crest or trough to trough.
Typical units: m to nm
Frequency (υ): the number of wave cycles passing a stationary point of reference per unit of time
1 wave cycle = 1 complete wavelength
Typical unit: Hz (s-1)
υ = Greek letter nu (pronounced as “new”)
Wave Properties of Light
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Chapter 6
8
Wavelength (λ) and frequency (υ) are related by:
λυ = c
where c is the speed of light:
c = 2.998 × 108 m/s
λυ = c = 2.998 × 108 m/s
Wave Properties and Light
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Chapter 6
Wavelength and frequency are inversely proportional.
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One of my favorite radio stations broadcasts at a frequency of 104.5 MHz (1 MHz = 106 Hz). What is the wavelength (in m) of these radio waves?
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Example: Wavelength and Frequency
https://i.iheart.com/v3/re/assets.brands/5c6f2321fda9bbc152bc32e6
Double check your work! (2.869 m)(1.045 × 108 s−1) = 2.998 × 108 m/s ✓
Chapter 6
1.) CD players use lasers which emit red light with a wavelength of 685 nm. What is the frequency of this light?
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Practice: Wavelength and Frequency
Chapter 6
4.38 X 1014 s-1 (or Hz)
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1.) CD players use lasers which emit red light with a wavelength of 685 nm. What is the frequency of this light?
Approach:
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Practice: Wavelength and Frequency
Chapter 6
4.38 X 1014 s-1 (or Hz)
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Irradiating matter with light can cause electrons to be ejected from a material.
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Electronic Structure of Atoms
Chapter 6
Light is a carrier of energy
Energy of a moving electron = kinetic energy (KE)
Unit of energy: Joules
++
++
+++
+++
The energy of the incoming light is equal to the energy of the ejected electron
Different elements require different energies due to differing numbers of protons and electrons
What happens if we use different colors (frequencies) of light?
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Atomic View of Electron Removal
Energy is required to overcome attraction between negative electron and positive nucleus
Chapter 6
For low-frequency light, no electrons are emitted, regardless of the intensity of the light.
Electrons are emitted by a given metal above a specific threshold frequency, ν0 .
For light frequency greater than ν0, the number of electrons emitted increases with the intensity of light.
For light frequencies above ν0, KE of electrons increases linearly with the frequency of light.
e-
e-
e-
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The Photoelectric Effect (Einstein, 1905)
Charged metal plate
Chapter 6
The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. Electrons emitted in this manner are called photoelectrons. This phenomenon is commonly studied in electronic physics and in fields of chemistry such as quantum chemistry and electrochemistry. (https://en.wikipedia.org/wiki/Photoelectric_effect)
0- Here we have a charged metal plate onto which we will shine light with a variety of frequencies.
1- When low-frequency light is directed onto a charged metal surface, no electrons are ejected.
2- When the intensity of the low-frequency light is increased, the result is the same.
3- These experiments showed that for low-frequency light, no electrons are emitted regardless of the intensity of the light.
4- When light with a high enough frequency is used, we see that electrons are ejected from the charged metal surface. This minimum frequency is called the threshold frequency.
5- Electrons are emitted by a given metal above a specific threshold frequency, ν0 .
6- When the intensity of this light is increased, we observe a larger number of ejected electrons.
7- For light frequency greater than ν0, number of electrons emitted increases with the intensity of light.
8- If the light used has a frequency higher than the threshold frequency, we observe the electrons with a higher kinetic energy.
9- For light frequencies above ν0, KE of electrons increases linearly with frequency of light.
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Along with other experiments involving light, the observation of the photoelectric effect led to the concept of wave-particle duality
Light is both a wave and a particle
Individual light “packets” are photons with energy determined by frequency
A photon can be considered a light particle, though it has no mass
Equation for calculating the energy of a single photon is:
Planck’s constant, h = 6.63 x 10-34 J*s
E = hυ or E = hc/λ
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Photon Energy
Chapter 6
We’ll practice using this equation in a bit
The many forms of electromagnetic energy make up a continuous spectrum, where all possible wavelengths and frequencies are represented below:
Chapter 6
Continuous vs. Line Spectra
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More common example of a continuous spectrum: the “rainbow” seen when white light is separated into its component colors by a prism
Each color blends into the next; there are no gaps in the visible spectrum (i.e., we observe a continuous spectrum of λ)
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Chapter 6
Continuous vs. Line Spectra
Examples of white light: sunlight, incandescent light bulb
Dispersion, the separation of visible light into a spectrum, may be accomplished by means of a prism or a diffraction grating. Each different wavelength or frequency of visible light corresponds to a different color , so that the spectrum appears as a band of colors ranging from violet at the short-wavelength (high-frequency) end of the spectrum through indigo, blue, green, yellow, and orange, to red at the long-wavelength (low-frequency) end of the spectrum. In addition to visible light, other types of electromagnetic radiation may be spread into a spectrum according to frequency or wavelength. (https://www.infoplease.com/encyclopedia/science/physics/concepts/spectrum/continuous-and-line-spectra)
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But, if light emitted from an excited atom is passed through a prism, only a few frequencies or wavelengths of visible light are present (there are gaps in the visible light spectrum)
This type of spectrum is called a line emission spectrum because each line represents a particular λ of visible light emitted from the atom; the missing λ’s appear as dark lines or gaps
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Chapter 6
Continuous vs. Line Spectra
Let’s look at hydrogen as an example.
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Each element has its own characteristic line emission spectra
Spectral lines are called the “fingerprints of the elements” or “chemical bar codes,” because they are unique for each element and therefore can be used to identify which elements are present in matter of unknown composition! http://chemistry.bd.psu.edu/jircitano/periodic4.html
Study of the hydrogen line spectrum led physicist Niels Bohr to a model of the H atom
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Chapter 6
Line Emission Spectra
Line emission spectra of the light from excited hydrogen, mercury, and neon atoms:
In contrast to continuous spectrum of white light:
Line spectra aren’t just unique to hydrogen! – the spectra produced by other elements are also discontinuous!
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Chapter 6
Bohr and the Hydrogen Atom
https://www.atomicheritage.org/sites/default/files/Niels_Bohr_Date_Unverified_LOC_0.jpg
Bohr Model of the Atom:
In 1913, Niels Bohr reasoned electrons in an atom were restricted to certain orbits (also called energy levels) surrounding the atomic nucleus
Called orbits because electrons were thought to orbit the nucleus in a fixed path, like how the planets orbit the sun.
Electrons could not exist between these orbits (their energy is quantized)
Electrons can’t exist between energy levels, just like people can’t stand between stair steps!
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Chapter 6
Electrons Exist in Quantized Energy Levels… just like steps on a staircase!
https://slideplayer.com/slide/12448767/;https://thumbs.dreamstime.com/z/businessman-standing-stairs-gray-background-32142377.jpg
You can only stand on the steps, not in between the steps.
You could choose to stand on the first step, the 2nd step, or the 3rd step, but not on the 1½ step or 2¼ step – no such step exists!
Only certain elevations are allowed.
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Bohr Model of the Atom:
In 1913, Niels Bohr reasoned electrons in an atom were restricted to certain orbits (also called energy levels) surrounding the atomic nucleus
Called orbits because electrons were thought to orbit the nucleus in a fixed path, like how the planets orbit the sun.
Electrons could not exist between these orbits (their energy is quantized)
Energy in the form of photons could be absorbed/emitted to cause electrons to move between these discrete energy levels
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Chapter 6
Bohr and the Hydrogen Atom
https://www.atomicheritage.org/sites/default/files/Niels_Bohr_Date_Unverified_LOC_0.jpg
Energy Level: refers to the potential energy associated with an electron/shell in an atom (represented by each of the blue orbits here; designated by n)
Energy levels get closer together when further from the nucleus
Ground State: an atom or electron in its lowest energy state (consider this the “normal” state)
Excited State: an atom or electron in an energy state higher than the ground state because of added energy
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Chapter 6
Electronic States: Definitions
Electrons can move from the ground state to the excited state (absorption) and fall back down to lower energy (emission)
https://www.webassign.net/labsgraceperiod/ucscgencheml1/lab_6/manual.html Reason for E levels getting closer together: shielding! (n=1 is shielding the nucleus. As the energy levels increase, the pull on the electrons to the nucleus decreases gradually; therefore, they are closer when n= 3,4,5...)
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Chapter 6
Absorption vs. Emission
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n=3
n=2
n=1
n=3
n=2
n=1
excited states
ground state
Absorption: when electrons gain energy and jump from the ground state to a higher energy level (i.e., an excited state)
– This energy is from a photon!
Emission: when electrons lose energy and jump from an excited state to a lower energy level
– The difference in energy is emitted as a photon (aka EM radiation aka light!)
Electrons can move from the ground state to the excited state (absorption) and fall back down to lower energy (emission)
photon (lower E)
photon (higher E)
photon
https://www.webassign.net/labsgraceperiod/ucscgencheml1/lab_6/manual.html
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Which electron transition would emit a photon with the greatest energy?
A. Starting at n=1 and jumping to n=5
B. Starting at n=1 and jumping to n=6
C. Starting at n=5 and jumping to n=1
D. Starting at n=6 and jumping to n=1
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Chapter 6
Predicting Photon Emission
Chapter 2
Chapter 2
Poll Question #1
Which electron transition would emit a photon with the greatest energy?
A. Starting at n=1 and jumping to n=5
B. Starting at n=1 and jumping to n=6
C. Starting at n=5 and jumping to n=1
D. Starting at n=6 and jumping to n=1
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Chapter 6
Predicting Photon Emission
Chapter 2
Chapter 2
Poll Question #1
Absorption (low high energy)
Emission (high low energy)
Which electron transition would emit a photon with the greatest energy?
A. Starting at n=1 and jumping to n=5
B. Starting at n=1 and jumping to n=6
C. Starting at n=5 and jumping to n=1
D. Starting at n=6 and jumping to n=1
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Chapter 6
Predicting Photon Emission
Chapter 2
Chapter 2
Poll Question #1
Absorption (low high energy)
Emission (high low energy)
Which electron transition would emit a photon with the greatest energy?
A. Starting at n=1 and jumping to n=5
B. Starting at n=1 and jumping to n=6
C. Starting at n=5 and jumping to n=1
D. Starting at n=6 and jumping to n=1
Chapter 6
Predicting Photon Emission
Chapter 2
Chapter 2
Poll Question #1
n=6 is at a higher energy level than n=5, so the energy difference between n = 6 1 is greater than the energy difference from n = 5 1
https://courses.lumenlearning.com/cheminter/chapter/spectral-lines-of-hydrogen/
Energy
Chapter 6
Predicting Photon Emission
Chapter 2
Chapter 2
Poll Question #1
n=6 is at a higher energy level than n=5, so the energy difference between n = 6 1 is greater than the energy difference from n = 5 1
https://courses.lumenlearning.com/cheminter/chapter/spectral-lines-of-hydrogen/
Energy
Which electron transition would emit a photon with the greatest energy?
A. Starting at n=1 and jumping to n=5
B. Starting at n=1 and jumping to n=6
C. Starting at n=5 and jumping to n=1
D. Starting at n=6 and jumping to n=1
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Chapter 6
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The Bohr Model and Line Emission Spectrum of Hydrogen
The relationship between the energy level change of a single electron and the energy of the photon emitted can be expressed as:
RE= Rydberg constant
These equations will give you magnitude of photon properties
But you must figure out whether energy (photon) is absorbed or emitted based on the sign of the energy level change
Absorbed = positive value of ΔE; emitted = negative value of ΔE
or
New constant developed from RE, h, and c to be more convenient
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Chapter 6
Rydberg Equation to Predict Energy from n
2a.) What is the wavelength (in nm) of the line in the spectrum of hydrogen that represents the movement of an electron from Bohr orbit n = 2 to n = 4?
The answer should correspond to a wavelength in the visible region, and so it is consistent with expectations!
Chapter 6
Example: The Rydberg Equation
initial
final
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(no need to memorize wavelengths and colors in this course)
Chapter 6
Example: The Rydberg Equation
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Or, working backwards from what we’ve already solved ( ):
2b.) What is the energy (in J) of this line?
2a.) What is the wavelength (in nm) of the line in the spectrum of hydrogen that represents the movement of an electron from Bohr orbit n = 2 to n = 4?
initial
final
✅
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Chapter 6
Example: The Rydberg Equation
2c.) Was this energy emitted or absorbed?
ΔE is POSITIVE, so this energy was absorbed! (you can also tell because n goes from 2 4 (i.e., electron is jumping to an excited state; needs to absorb energy to get there)
2a.) What is the wavelength (in nm) of the line in the spectrum of hydrogen that represents the movement of an electron from Bohr orbit n = 2 to n = 4?
initial
final
2b.) What is the energy (in J) of this line?
✅
✅
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The spectra produced by other elements are also discontinuous! However, spectra for elements other than H are more difficult to predict because there are multiple electrons. This is where the Bohr model falls apart.
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Chapter 6
Emission Spectra of Other Elements
the spectra produced by other elements are also discontinuous!
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Strengths:
Accurately predicts emission spectrum of single electrons within atoms (e.g., H) or ions (e.g., He+, Li2+, Be3+, etc.)
Provided early model of electron organization in the atom
Limitations:
Does not explain spectra of multi-electron atoms
Requires that electrons act only as predictable particles, which later experiments disproved
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Chapter 6
The Bohr Model Evaluation
A new model was needed to describe more complex atoms.
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We’ve established that light has wave-particle duality…. but could a particle in motion, such as electron, also behave as a wave?
Louis de Broglie (French grad student in physics, 1924) sought to answer this question! Let’s walk through his thought process and derivation:
Assuming that particles and waves have the same traits, de Broglie hypothesized that the energies for each of these systems would be equal.
And so, by combining the equation for the energy of waves (E = hυ) with Einstein’s equation famous relationship relating matter and energy (E = mc2), de Broglie derived the following equation:
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Chapter 6
Are Electrons Particles or Waves?
Givens: E = hυ, E = mc2
Assumption: these energies are equal.
Therefore, E = E
hυ = mc2
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(Skipped in 2019)
de Broglie’s derivation, continued:
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Chapter 6
Are Electrons Particles or Waves?
Givens: E = hυ, E = mc2. Assumption: these energies are equal. Therefore, E = E
hυ = mc2
By rearranging the equation c = λυ, we know
By substituting this value for υ into the equation hυ = mc2, we obtain:
Because real particles do not travel at the speed of light, de Broglie substituted velocity (v) for the speed of light (c).
By solving for λ, we obtain:
note: velocity (v) is NOT the same as frequency (υ)
And so, de Broglie was able to show that matter (represented by “m” in the equation) indeed has wave properties (represented by λ)!
(No, I won’t ask you to derive this on an exam )
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(Skipped in 2019)
If electrons can behave as both particles AND waves simultaneously, what happens to Bohr’s model?
Where Bohr had postulated the electron as being a particle orbiting the nucleus in quantized orbits, de Broglie argued that Bohr’s assumption of quantization can be explained if the electron is considered not solely as a particle, but rather as a circular standing wave of matter such that only an integer number of wavelengths could fit exactly within the orbit.
Bohr model (electrons as particles)
Electrons as matter waves
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Chapter 6
Electrons as Matter Waves
Standing waves = good model of the wave behavior of electrons
Each red dot = a node (point in which wave amplitude = 0)
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Chapter 6
Standing Waves
For an electron in a circular orbit, only whole-number multiples of wavelengths are allowed. Any fractional number of wavelengths would result in cancellation of the wave due to destructive interference.
Just like the Bohr model of the atom, where only certain orbits were possible, the idea of quantization still remains here (where only certain discrete wavelengths are allowed!)
node
Standing wave – oscillates up and down between the nodes but doesn’t “travel”
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In 1926, Austrian physicist Erwin Schrödinger expanded on this idea of electrons’ wave-particle duality by describing electrons as 3-D standing waves of matter, or wavefunctions, represented by Ψ (Greek letter psi).
– This led to the creation of a brand new field: quantum mechanics!
His equation states that the energy of an electron (EΨ, right-side of equation) needs to incorporate energy of motion, energy due to distance from nucleus, and energy from wave properties (left-side of equation)
This is the model of the atom that is still accepted today!
– Can be applied to atoms with more than one electron
The solutions to Schrödinger’s wave equations for electrons in atoms are very complex, but we can graph them to see what they look like!
Chapter 6
The Schrödinger Equation
Newton’s laws don’t work well when you attempt to describe things as small as atoms and molecules.
YIKES. A more complete mathematical description is encompassed in quantum mechanics. This involves a set of postulates that are not intuitive (e.g. energy is quantized??) You don’t experience this in your world. Would it make sense if you could drive your car only 50 mph or 60 mph, but no values in between???) Scientists with a background in physics and mathematics developed a framework that appears to explain the observed phenomena. We will take this in good faith and not attempt to gain a deeper understanding that requires a fundamentally better mathematical understanding. Instead, we will highlight a few of the important findings and use them to make sense of some aspects of general chemistry. A more in-depth description comes in physical chemistry and beyond.
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Chapter 6
Plotting 3-D Electron Probability Clouds
Step 2: Function is squared (ψ2)
– Gives probability function
Ψ
Ψ2
These 3-D regions in which electrons are most likely to be found are called orbitals
Step 1: Wavefunction (Ψ) for a given energy is solved
Step 3: Ψ2 function is plotted such that it reflects the probability of finding an electron in a given space (example: 90% probability)
Darker regions show increased electron density…aka, increased probability of finding an electron!)
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A region in 3-D space in which electrons in atoms are most likely to be found
Centered at the nucleus of an atom (at the origin of these plots!)
Note: we can only discuss the probability of finding an electron at a specific location because any attempt to measure its exact location causes it to move
Heisenberg uncertainty principle: “It is impossible to know both the position and the momentum of a small particle simultaneously.”
In the Schrödinger equation, we can know the energy of an electron precisely, but not its precise location
Called orbitals because they are similar to (but not the same as!) the “orbits” for electrons in the Bohr model of the atom
3 variables (called quantum numbers) are used to describe an orbital: n, l and ml
This information is contained within Schrödinger’s equation!
Chapter 6
Orbitals
=
location of
nucleus
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n, related to radius
(principle quantum number)
Radial function
Angular functions
ml, projection of angular momentum in 3D
(magnetic quantum number)
l, related to angle
(angular momentum quantum number)
Chapter 6
A Closer Look into the Schrödinger Equation
Newton’s laws don’t work well when you attempt to describe things as small as atoms and molecules.
YIKES. A more complete mathematical description is encompassed in quantum mechanics. This involves a set of postulates that are not intuitive (e.g. energy is quantized??) You don’t experience this in your world. Would it make sense if you could drive your car only 50 mph or 60 mph, but no values in between???) Scientists with a background in physics and mathematics developed a framework that appears to explain the observed phenomena. We will take this in good faith and not attempt to gain a deeper understanding that requires a fundamentally better mathematical understanding. Instead, we will highlight a few of the important findings and use them to make sense of some aspects of general chemistry. A more in-depth description comes in physical chemistry and beyond.
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Before we go any further into orbitals and quantum numbers, I know this was a LOT of information to handle.
Chapter 6
So, let’s review the key take-away points!
https://www.wgi.org/the-pride-of-the-netherlands-brings-m-c-escher/escher-penrose-500/
Please don’t panic if these concepts are hard to wrap your head around – they ARE! Particle physics is a very complex field filled with apparent contradictions… it’s analogous to Escher’s staircase (where the top of the staircase is actually the bottom… just as wave–particle duality ascribes two seemingly contradictory traits to a single object!)
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Chapter 6
Bohr vs. Schrödinger’s Quantum Mechanical Model of the Atom
| Bohr Model | Quantum Mechanical Model (model still accepted today!) |
| 2-D model of the atom | 3-D model of the atom (an expansion of Bohr’s 2-D model) |
| Can only describe atoms with one electron | Can describe more complex atoms (i.e., those with more than one electron) |
| Energy is quantized | Energy is quantized |
| Electron treated as a particle (classical mechanics) | Electron treated as a matter wave (quantum mechanics) |
| Electron energy and location are known precisely (note: location could be known precisely because the electrons were thought to orbit the nucleus in a fixed path) | Electron energy is known precisely, but location is not (Heisenberg uncertainty principle). We can only predict the probable location (orbital) of an electron by the Schrödinger equation. |
| Electrons are in energy levels (orbits, n) | Electrons are in energy levels (n), sublevels (l) and sub-sub-levels (ml), all of which are needed to describe an orbital (electron cloud) |
Wavefunctions’ energies in Schrödinger’s equations depend on the values of 3 quantum numbers: n, l and ml
These quantum numbers, along with a 4th quantum number ms (spin), are needed to completely describe the probable “location” (orbital) of each electron in an atom.
Pauli exclusion principle: Since two electrons cannot occupy the same space at the same time, no two electrons in an atom can have the same four quantum numbers.
Therefore, each electron has a unique “postal address” to describe its location (as designated by its unique four quantum numbers)
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Chapter 6
Quantum Numbers: “Electron Addresses”
No two cities have the same zip code, and no two electrons have the same “zip code” in an atom!
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Principal quantum number, n
Related to “energy level” or “shell number”
Higher n values are further from nucleus
Describes the size of the orbital
Angular momentum quantum number, l
Possible values depend on n
Describes the shape of the orbital
Magnetic quantum number, ml
Possible values depend on l
Describes the orientation (direction) of the orbital in space
Chapter 6
The Quantum Numbers n, l, and ml
n is still defined as energy levels, similar to Bohr’s model
n follows same rules as for Bohr’s model: lowest level is n = 1 and counts up in whole-number increments (n = 1, 2, 3, 4, 5, 6…)
n is also called the shell number
Shells of an atom can be thought of as concentric circles radiating out from the nucleus
The electrons that belong to a specific shell are most likely to be found within the corresponding circular area
n describes the size of the orbital
As the value of n increases, the energy of the electron increases, and the distance of the electron from the atomic nucleus increases (just like the Bohr model!)
50
Chapter 6
Principal Quantum Number, n
Each l value tells us the shape of the orbital (or which subshell in which the electron is located)
Subshells are s (l = 0), p (l = 1), d (l = 2), f (l = 3), etc.
For a given n, possible l values are 0 through n-1
Again scales by integers: l = 0, 1, 2, 3, … n-1
Example: n = 4 means that l can equal 0, 1, 2, or 3
l=0
l=1
l=2
l=3
s orbital
p orbitals
d orbitals
f orbitals
For n=4,
l can equal:
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Chapter 6
Angular Momentum Quantum Number, l
| l | shape |
| 0 | s |
| 1 | p |
| 2 | d |
| 3 | f |
The 1st energy level (n=1, where l =0) has only 1 type of subshell, or orbital: s (l =0)
The 2nd energy level (n=2, where l =0,1) has 2 types of subshells, or orbitals: s (l =0), p (l =1)
In the 3rd energy level (n=3, where l =0,1,2), there are 3 types of subshells, or orbitals: s (l =0), p (l =1), d (l =2)
And so on…
l=0
l=1
l=2
l=3
s orbital
p orbitals
d orbitals
f orbitals
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Chapter 6
Another way to think about l
| l | shape |
| 0 | s |
| 1 | p |
| 2 | d |
| 3 | f |
For n=4,
l can equal:
ml=0
ml= –1, 0 or 1
ml= –2, –1, 0, 1 or 2
ml= –3, –2, –1, 0,
1, 2 or 3
s orbital
p orbitals
d orbitals
f orbitals
l=0
l=1
l=2
l=3
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Chapter 6
Magnetic Quantum Number, ml
The magnetic quantum number, ml, tells about the arrangement of an orbital in space, or the orientation around the nucleus.
ml has whole-number values spanning from ml = –l to +l (intervals of 1).
E.g., if l = 2 (defined as a d orbital), ml = –2, –1, 0, 1, or 2.
p orbitals
d orbitals
f orbitals
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Chapter 6
Magnetic Quantum Number, ml
The number of allowed values of ml for a particular subshell (l) reveal the total number of possible arrangements (orientations):
1 arrangement for an s type (l = 0) orbital (ml =0)
3 arrangements for a p type (l = 1) orbital (ml = –1, 0, 1)
5 arrangements for a d type (l = 2) orbital (ml = –2, –1, 0, 1, 2)
7 arrangements for a f type (l = 3) orbital (ml = –3, –2, –1, 0, 1, 2, 3)
And so on…
s orbital
ml=0
ml= –1, 0 or 1
ml= –2, –1, 0, 1 or 2
ml= –3, –2, –1, 0,
1, 2 or 3
l=0
l=1
l=2
l=3
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Chapter 6
Part of 4.) Subshell Quantum Numbers: Summary of the Last Few Slides
| ℓ | Type of subshell | mℓ ( ±ℓ ) | Number of arrangements (orientations) |
| 0 | s | 0 | 1 |
| 1 | p | –1, 0, 1 | 3 |
| 2 | d | –2, –1, 0, 1, 2 | 5 |
| 3 | f | –3, –2, –1, 0, 1, 2, 3 | 7 |
Let’s talk about the SHAPES of each of these orbitals, or subshells!
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s orbitals (ℓ =0)
The shape of an s orbital is a sphere (just one possible orientation!)
Orbital available for all values of n
The number before the orbital name designates the energy level/shell number, n (e.g., 2s is an s-orbital that is in the 2nd energy level/shell)
As n increases, the size of the s orbital spheres get larger increasing energy level, increasing distance
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Chapter 6
s Orbitals
1s
3s
2s
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Chapter 6
The Three p Orbitals
p orbitals (ℓ =1) are possible in energy level n=2 or higher.
Shaped like two identical balloons tied together at the nucleus (or, like a dumbbell or hourglass!)
Direction given by third quantum number, mℓ
Possible ml values are –1, 0, 1 three values means there are three different orientations
The number before the orbital name designates the energy level/shell number, n (e.g., 2p is a p-orbital that is in the 2nd energy level)
Composite view = depiction of p-orbitals arranged at 90o angles to each other along the x, y, and z axes
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Chapter 6
The Five d Orbitals
d orbitals (ℓ =2) are possible in energy level n=3 or higher.
Possible ml values are –2, –1, 0, 1, 2 Five values means there are five different orientations
Four of them are shaped like four identical balloons tied together at the nucleus (or, like a four-leaf clover!)
The other looks like a p-orbital inside a donut
The number before the orbital name designates the energy level/shell number, n (e.g., 3d is a d-orbital that is in the 3rd energy level)
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Chapter 6
The Seven f Orbitals
f orbitals (ℓ =3) are possible in energy level n=4 or higher.
Possible ml values are –3, –2, –1, 0, 1, 2, 3 Seven values means there are seven different orientations
By far some of the most exotic-looking shapes we have seen so far! Some are shaped like eight identical balloons tied together at the nucleus (or, like an octopus); another looks like a p-orbital inside two donuts
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Chapter 6
Orbital Nodes
Just as standing waves have nodes, orbitals do, too!
In the context of orbitals, nodes are regions of zero electron density (i.e., where the probability of finding an electron is always 0).
There are two types of nodes that orbitals can have: radial nodes and angular nodes. In this course, we’ll only discuss angular nodes.
The number of angular nodes that an orbital has is equal to the orbital angular quantum number, l
# angular nodes = l
node
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Chapter 6
Number of Angular Nodes = l
Each s orbital (l = 0) has zero angular nodes
Each p orbital (l = 1) has one angular node (i.e., one angle at which the probability of finding an electron is always zero)
Each d orbital (l = 2) has two angular nodes (two angles at which the probability of finding an electron is always zero)
Each f orbital (l = 3) has three angular nodes (three angles at which the probability of finding an electron is always zero)
And so on…
https://lavelle.chem.ucla.edu/forum/viewtopic.php?t=34046; https://socratic.org/questions/the-orientation-in-space-of-an-atomic-orbital-is-associated-with-what
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| ℓ | Type of subshell (and representative sketch) | Number of angular nodes | mℓ ( ±ℓ ) | Number of arrangements (orientations) |
| 0 | s | 0 | 0 | 1 |
| 1 | p | 1 | –1, 0, 1 | 3 |
| 2 | d | 2 | –2, –1, 0, 1, 2 | 5 |
| 3 | f | 3 | –3, –2, –1, 0, 1, 2, 3 | 7 |
Chapter 6
Rest of 4.) Subshell Quantum Numbers