Quantum-Mechanical Model of the Atom Lecture Notes PDF

The Quantum–Mechanical Model of the Atom Chapter 7.1 – 7.7 (excl. PIB) CHMA10 - Quantum Mechanical Model of the Atom 1  Module Learning Goals By the end of this module, YOU will be able to 1. explain the need for the development of the quantum mechanical...

The Quantum–Mechanical Model of the Atom Chapter 7.1 – 7.7 (excl. PIB) CHMA10 - Quantum Mechanical Model of the Atom 1  Module Learning Goals By the end of this module, YOU will be able to 1. explain the need for the development of the quantum mechanical model of the atom and the key scientists who made major contributions to its development. 2. describe the evidence for the wave/particle duality of electrons and photons. 3. describe the electronic configuration of an atom or ion using the four quantum numbers. 4. recognize how the quantum mechanical model of the atom is reflected in how the periodic table is organized. 5. use Hund’s rule and the Aufbau principle to write electron configurations for atoms and ions. CHMA10 - Quantum Mechanical Model of the Atom 2 Views of the Matter is Changing Image from https://openstax.org/book Image from s/chemistry-atoms-first- https://upload.wikimedia 2e/pages/2-2-evolution-.org/wikipedia/commons of-atomic-theory Image from /d/d4/John_Dalton_by_C https://miro.medium.com/max harles_Turner.jpg /600/1*UaaGMVWDGJgzF97ge https://en.wikipedia.org/wik i/Democritus pKkWw.jpeg Image from https://upload.wikimedia.o rg/wikipedia/commons/6/6 e/Ernest_Rutherford_LOC.j pg 400s BCE 1600s 1800s 1909 1911 Democritus Age of Alchemy John Dalton J.J. Thompson Ernest Rutherford tiny indivisible birth of atomic theory discovered the published his discovery of that all matter is particles called atoms Main goal of trying to electron the proton and the idea of change lead to gold created of small, the nucleus of an atom make up all matter. indivisible particles Metals such as gold, salt, sulfur and silver, tin, lead mercury Elements made up of atoms of that type CHMA10 - Quantum Mechanical Model of the Atom 3  Modern Atomic Theory and Sub-Atomic Particles In 1904, J.J. Thompson discovered the electron using more modern techniques and equipment Electron was negatively charged and very small Image from https://openstax.org/books/chemistry-atoms-first-2e/pages/2-2- evolution-of-atomic-theory In 1911, Ernest Rutherford, published his discovery of the proton and the idea of the nucleus of an atom Actually graduate students Ernest Marsden and Hans Geiger did the experiment Proton much larger than electron and positively Image from charged https://upload.wikimedia.o rg/wikipedia/commons/6/6 Neutron discovery came a bit later…similar e/Ernest_Rutherford_LOC.j pg mass to proton but no charge Lecture 1- Properties of Matter, Atomic Structure and 4 Introduction to Elements Views of the Matter is Changing Image from https://openstax.org/book Image from s/chemistry-atoms-first- https://upload.wikimedia 2e/pages/2-2-evolution-.org/wikipedia/commons of-atomic-theory /d/d4/John_Dalton_by_C harles_Turner.jpg https://en.wikipedia.org/wik i/Democritus Image from https://upload.wikimedia.o rg/wikipedia/commons/6/6 e/Ernest_Rutherford_LOC.j pg 400s BCE 1800s 1909 1911 Dalton Thompson Atoms are hard Rutherford Electrons distributed inside a indivisible spheres Most of the atom is empty positive mass like in a “plum space, mass inside atom, pudding” CHMA10 - Quantum Mechanical Model of the Atom electrons surrounding nucleus 5  The Nuclear Model of the Atom (Pre-1927) Atom is made up of 3 particles: Protons: +1 charge; located in nucleus Neutrons: 0 charge; located in nucleus electrons: -1 charge; traveling around the nucleus Electrons are much less massive than protons and neutrons Protons and neutrons are about 2000 times more massive than electrons! This model was a good start…but it would ultimately be replaced…let’s see why and how! CHMA10 - Quantum Mechanical Model of the Atom 6  The Solvay Conference, 1927 In 1927, some of the greatest scientific minds of the 20 th century met at a conference to discuss, clarify, and organize what was, at the time, some very new and very weird ideas … CHMA10 - Quantum Mechanical Model of the Atom 7  Light: Particle or Wave? Is light a: a) Particle b) Wave c) Both CHMA10 - Quantum Mechanical Model of the Atom 8  The Nature of Light Maxwell (1873) proposed that visible light consists of electromagnetic waves electromagnetic waves are composed of both an oscillating magnetic and electric field Describe waves using amplitude, wavelength (λ) and frequency (ν) using the equation 𝐜 𝛎= 𝐰𝐡𝐞𝐫𝐞 𝐜 = 𝟑 × 𝟏𝟎𝟖 𝐦 (𝐬𝐩𝐞𝐞𝐝 𝐨𝐟 𝐥𝐢𝐠𝐡𝐭) 𝛌 𝐬 CHMA10 - Quantum Mechanical Model of the Atom 9 Some Important Features of Light as a Wave CHMA10 - Quantum Mechanical Model of the Atom 10  Electromagnetic Radiation Spectrum Electromagnetic radiation is defined as energy that is emitted and transmitted in the form of electromagnetic waves. CHMA10 - Quantum Mechanical Model of the Atom 11  Properties of Waves: Interference When two waves are in phase, they When two waves are out of phase, interfere with each other to give they interfere with each other to constructive interference give destructive interference CHMA10 - Quantum Mechanical Model of the Atom 12  Properties of Waves: Diffraction Diffraction is the bending of waves A beam of particles does not diffract around obstacles and openings when passed through a small If light passed through a small opening opening, it diffracts and hence it we can classify it as a wave CHMA10 - Quantum Mechanical Model of the Atom 13  Nature of Light: Wave Evidence evidence for the wave nature of light comes from diffraction patterns when a double slit is used! Waves interact to produce light and dark spots due to constructive and destructive interference so must be a wave! up until ~1900 light was COURSE LINK You will study thought of only as waves waves in more detail if you are taking PHYA10 or PHYA21 CHMA10 - Quantum Mechanical Model of the Atom 14  Blackbody Radiation A blackbody is a theoretical object that absorbs all radiation that falls on it and re-emits it with a broad range of frequencies intensity of radiation escaping a blackbody varies with frequency of the radiation frequency shifts to higher values as the temperature increases Majority falls in visible range Image from https://ian-r-rose.github.io/articles/dimensional_analysis/images/solar_spectrum.svg CHMA10 - Quantum Mechanical Model of the Atom 15  Blackbody Radiation and the UV Catastrophe classical physics correctly matched experimental data for IR region but did not match the measured intensity at short wavelengths The inconsistency between observations and predictions based on classical physics is called the Ultraviolet Catastrophe (Rayleigh–Jeans catastrophe) How to solve this problem? Image from https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/Black_body.svg/600px- Planck provided the solution to the problem and provides the Black_body.svg.png correct radiation at all frequencies (The Rayleigh–Jeans law is its low-frequency limit) CHMA10 - Quantum Mechanical Model of the Atom 16  The Birth of Quantum Mechanics In 1900, Max Planck (1858-1947) set out to solve the Ultraviolet Catastrophe Recall that a blackbody is made of atoms that vibrate Vibrations generate the light that we see More kinetic energy (higher the temperature) means brighter light Planck’s big idea was to postulate that the energy that is emitted or absorbed (in the form of light aka electromagnetic radiation) exists ONLY in discrete energy bundles called quanta energy is not continuous but quantized CHMA10 - Quantum Mechanical Model of the Atom 17  The Planck Constant and Energy Quanta Plank developed an equation that predicts the energy of blackbody radiation Energy is proportional to its frequency! 𝐄 = 𝐡𝝂 where h = Planck’s Constant = 6.626 x 10-34 J·s and 𝝂 is the frequency in Hertz, s-1 Using his new equation, Planck was able to derive an equation that could be used to explain experimental observations Recall that classic physics could not accomplish this and so UV Catastrophe was no longer a problem! Note that Planck’s constant is extremely small…so what? This means that the gradations between allowed values is so tiny that we can’t actually detect them using a measuring apparatus in the lab! Energy is continuous on the macroscale CHMA10 - Quantum Mechanical Model of the Atom 18  Worked Example: Using The Planck Equation What is the energy of a photon of green light with wavelength 5200 Å? We are given a wavelength (λ = 5200, 1 Å = 1.0 x 10-10 m) so we need to convert this into frequency knowing that the speed of light is constant (c = 300,000,000 m/s) given by the formula 𝐜 𝛎= 𝛌 𝟑𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎 𝐦/𝐬 𝟏Å 𝟏𝟒 −𝟏 𝛎= × −𝟏𝟎 = 𝟓. 𝟕𝟕 𝐱 𝟏𝟎 𝐬 𝟓𝟐𝟎𝟎 Å 𝟏. 𝟎 × 𝟏𝟎 𝐦 Now we can use Plank’s Formula: 𝐄 = 𝐡𝛎 = 6.626 x 10−34 J·s × 𝟓. 𝟕𝟕 𝐱 𝟏𝟎𝟏𝟒 𝐬 −𝟏 = 𝟑. 𝟖𝟐 × 𝟏𝟎−𝟏𝟗 𝐉 CHMA10 - Quantum Mechanical Model of the Atom 19  Worked Example: Using The Planck Equation What is the energy of a photon of green light with wavelength 5200 Å? We are given a wavelength (λ = 5200, 1 Å = 1.0 x 10-10 m) so we need to convert this into frequency knowing that the speed of light is constant (c = 300,000,000 m/s) given by the formula 𝐜 𝛎= 𝛌 𝟑𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎 𝐦/𝐬 𝟏Å 𝟏𝟒 −𝟏 𝛎= × −𝟏𝟎 = 𝟓. 𝟕𝟕 𝐱 𝟏𝟎 𝐬 𝟓𝟐𝟎𝟎 Å 𝟏. 𝟎 × 𝟏𝟎 𝐦 Now we can use Plank’s Formula: 𝐄 = 𝐡𝛎 = 6.626 x 10−34 J·s × 𝟓. 𝟕𝟕 𝐱 𝟏𝟎𝟏𝟒 𝐬 −𝟏 = 𝟑. 𝟖𝟐 × 𝟏𝟎−𝟏𝟗 𝐉 CHMA10 - Quantum Mechanical Model of the Atom 20  Worked Example: Using The Planck Equation What is the energy of a photon of green light with wavelength 5200 Å? We are given a wavelength (λ = 5200, 1 Å = 1.0 x 10-10 m) so we need to convert this into frequency knowing that the speed of light is constant (c = 300,000,000 m/s) given by the formula 𝐜 𝛎= 𝛌 𝟑𝟎𝟎,𝟎𝟎𝟎,𝟎𝟎𝟎 𝐦/𝐬 𝟏Å 𝛎= × = 𝟓. 𝟕𝟕 𝐱 𝟏𝟎𝟏𝟒 𝐬 −𝟏 frequency 𝟓𝟐𝟎𝟎 Å 𝟏.𝟎 ×𝟏𝟎−𝟏𝟎 𝐦 Now we can use Plank’s Formula: 𝐄 = 𝐡𝛎 = 6.626 x 10−34 J·s × 𝟓. 𝟕𝟕 𝐱 𝟏𝟎𝟏𝟒 𝐬 −𝟏 = 𝟑. 𝟖𝟐 × 𝟏𝟎−𝟏𝟗 𝐉 energy CHMA10 - Quantum Mechanical Model of the Atom 21  Worked Example: Using The Planck Equation What is the energy of a photon of green light with wavelength 5200 Å? We are given a wavelength (λ = 5200, 1 Å = 1.0 x 10-10 m) so we need to convert this into frequency knowing that the speed of light is constant (c = 300,000,000 m/s) given by the formula 𝐜 𝛎= 𝛌 𝟑𝟎𝟎,𝟎𝟎𝟎,𝟎𝟎𝟎 𝐦/𝐬 𝟏Å 𝛎= × = 𝟓. 𝟕𝟕 𝐱 𝟏𝟎𝟏𝟒 𝐬 −𝟏 frequency 𝟓𝟐𝟎𝟎 Å 𝟏.𝟎 ×𝟏𝟎−𝟏𝟎 𝐦 Now we can use Plank’s Formula: 𝐄 = 𝐡𝛎 = 6.626 x 10−34 J·s × 𝟓. 𝟕𝟕 𝐱 𝟏𝟎𝟏𝟒 𝐬 −𝟏 = 𝟑. 𝟖𝟐 × 𝟏𝟎−𝟏𝟗 𝐉 energy CHMA10 - Quantum Mechanical Model of the Atom 22  Nature of Light: The Photoelectric Effect Another problem that was stumping classical physics was the photoelectric effect When light shines on some metal surfaces, then electrons are emitted from the metal that can be detected The kinetic energy of these electrons can be measured in various Introduction Photoelectric effect conditions Photoelectric effect High vs. low frequency light Einstein, Nobel Prize 1921 Photoelectric effect High vs. low amplitude (intensity) light Albert Einstein Nobel Prize 1921 CHMA10 - Quantum Mechanical Model of the Atom 23 Nature of Light: The Photoelectric Effect and Frequency Let’s examine some data when we keep amplitude the same but change the frequency (i.e. different coloured light of same brightness): a) Light below a certain frequency would not eject electrons This is the threshold frequency, ν0 b) Light with frequency > ν0 gave the electrons more kinetic energy but did NOT increase the number of electrons Current (number of electrons per unit time) remained constant even with changing frequency Image from https://tinyurl.com/lo9qzdw CHMA10 - Quantum Mechanical Model of the Atom 24 Nature of Light: The Photoelectric Effect and Amplitude Let’s examine some data when we keep frequency the same but change the amplitude (i.e. brighter light of same colour): a) Higher amplitude means more electrons are ejected per unit time Greater current produced b) Higher amplitude light did not increase the kinetic energy of the electrons being ejected Image from https://tinyurl.com/lo9qzdw CHMA10 - Quantum Mechanical Model of the Atom 25 Photoelectric effect and solar panels CHMA10 - Quantum Mechanical Model of the Atom 26 XPS Analysis of Pigment from Mummy Artwork Pb3O4 Egyptian Mummy 2nd Century AD World Heritage Museum University of Illinois PbO2 C O 150 145 140 135 130 Binding Energy (eV) Pb Pb N Ca XPS analysis showed Na that the pigment used Cl Pb on the mummy wrapping was Pb3O4 rather than Fe2O3 500 400 300 200 100 0 Binding Energy (eV)  Quantum Mechanics: Einstein’s Nobel Prize In 1905, Einstein (1879-1955) used Planck’s ideas of quantization to explain the photoelectric effect by embracing quantization He showed that light exists as discrete packets called photons by combining the description of light as a wave with Planck’s Equation 𝒉𝒄 𝐄𝐩𝐡𝐨𝐭𝐨𝐧 = 𝒉𝝂 = = ∆𝐄𝐚𝐭𝐨𝐦 𝝀 CHMA10 - Quantum Mechanical Model of the Atom 28  Quantum Mechanics: Einstein’s Nobel Prize How does Einstein’s proposal of light as photons explain the photoelectric effect? Why does the intensity of light not increase the kinetic energy of electron? A beam of light made up of many photons Increasing the intensity means using more photons NOT more energetic photons Hence kinetic energy remains the same regardless of intensity Why is there a threshold frequency to eject electron? For a photon to eject electron, it must have a minimum energy (binding energy) Any photons below this threshold frequency won’t be able to remove an electron Energy given of a photon given by E = hν, which explains why a minimum frequency is necessary CHMA10 - Quantum Mechanical Model of the Atom 29  Worked Example: Energy of a Photon You want to heat a meal in the microwave. The wavelength of the radiation is 1.20 cm. What is the energy of one photon of microwave radiation? CHMA10 - Quantum Mechanical Model of the Atom 30  Worked Example: Energy of a Photon You want to heat a meal in the microwave. The wavelength of the radiation is 1.20 cm. What is the energy of one photon of microwave radiation? ℎ𝑐 Speed of light 𝐸 = ℎ𝜈 = 3.00 × 108 m/s 𝜆 Wavelength in m Planck’s constant 6.626 × 10−34 J ∙ s CHMA10 - Quantum Mechanical Model of the Atom 31  Worked Example: Energy of a Photon You want to heat a meal in the microwave. The wavelength of the radiation is 1.20 cm. What is the energy of one photon of microwave radiation? ℎ𝑐 𝐸 = ℎ𝜈 = 𝜆 (6.626 × 10−34 J ∙ s)(3.00 × 108 m/s) 𝐸= 10−2 m (1.20 cm) 1 cm 𝐸 = 1.66 × 10−23 J CHMA10 - Quantum Mechanical Model of the Atom 32  Atomic Spectra: Back To The Lab An emission spectrum is formed by An absorption spectrum is formed an electric current passing through a by shining a beam of white light gas in a vacuum tube (at very low through a sample of gas pressure) which causes the gas to Absorption spectra indicate the emit light. wavelengths of light that have been It is also called a bright line spectrum absorbed. CHMA10 - Quantum Mechanical Model of the Atom 33  Atomic Spectra Every element has a unique spectrum we can use spectra to identify elements This can be done in the lab, stars, fireworks, etc. Remember the black body? It gave a continuous spectrum of many frequencies…why do atoms give discrete colours? CHMA10 - Quantum Mechanical Model of the Atom 34  Applying Quantum Mechanics: The Bohr Atomic Model The “Great Dane” Niels Bohr (1885-1962) took Plank’s idea of quantization and applied it to atomic spectra It was known that molecules and atoms have very specific atomic spectra (colours) Bohr aimed to link spectral data (colours it displayed) to the underlying structure of a particular compound or atom by applying quantization to electrons In 1912, he developed a mathematical model of the atom with discrete energy levels or shells Electrons release or emit light while moving between shells Predicted hydrogen spectrum exactly! 𝟏 𝐄𝐧 = −𝐑 𝐇 𝐧𝟐 n = 1,2,3…. and RH = 2.18 x 10-18 J CHMA10 - Quantum Mechanical Model of the Atom 35  Bohr Atomic Model Atoms have definite and discrete energy levels (orbital) in which an electron may exist without emitting or absorbing electromagnetic radiation electron moves in circular orbit about nucleus and its motion is governed by ordinary laws of mechanics and electrostatics RESTRICTION: angular momentum of the electron is quantized i.e. it can only have certain discrete values As the orbital radius increases, so does the energy i.e. the larger the orbital, the higher the energy level CHMA10 - Quantum Mechanical Model of the Atom 36  Bohr Atomic Model electron may move from one discrete energy level (orbit) to another, with the energy difference given by 1 1 𝛥𝐸 = −𝑅𝐻 2− 2 𝑛𝑓 𝑛𝑖 monochromatic radiation is emitted or absorbed when an electron moves energy levels and the energy of the light photons is given by ΔE = ℎ𝜈 ΔE > 0 when light is absorbed ΔE < 0 when light is emitted https://en.wikipedia.org/wiki/Bohr_model#/media/File:Bohr-atom-PAR.svg CHMA10 - Quantum Mechanical Model of the Atom 37  Ground State vs. Excited State E = h Atoms that have not absorbed any energy are said to be in the ground state The ground state is the electron configuration that such that all electrons are most stable E = h When electrons are excited to higher energy states, they release the energy absorbed to get back to the ground state Energy released in the form of light “ground state” n=1 CHMA10 - Quantum Mechanical Model of the Atom 38  Worked Example: The Bohr Atomic Model A hydrogen atom absorbs a photon of UV light and the electron is promoted to n=4. What is the change in energy of the electron? What is the wavelength of the light? To find the energy change: 1 1 1 1 ΔE = −R H 2 − = −2.18 × 10−18 J 2 − = 2.04 × 10−18 J nf n2i 4 12 To find wavelength: ΔE=hc/λ ∴ hc (6.626 ×10−34J∙s)(3.00×108 m/s) λ= = = 9.74 × 10−8 m ΔE 2.04×10−18J −8 1 nm Convert to nanometers: 9.74 × 10 m × = 97.4 nm 10−9 m CHMA10 - Quantum Mechanical Model of the Atom 39  Lecture 1 - Learning Goals By the end of this section, students will 1. Recognize why quantum mechanics was developed and recall the important people responsible for its development. 2. Be able to relate the energy of light quanta to its frequency and/or wavelength. 3. Interpret atomic emission spectra in terms of the Bohr theory of the atom; calculate energies of emitted light for given transitions (and vise versa) CHMA10 - Quantum Mechanical Model of the Atom 40 CHMA10 - Quantum Mechanical Model of the Atom 41