Quantum Mechanical Model of Atoms PDF

## Atomic Structure. The Periodic Table. Based on McGraw-Hill Ryerson: Chemistry 12 ### The Standard Model of Fundamental Particles and Interactions | | **Fermions** (spin = 1/2, 3/2, 5/2...) | **Bosons** (spin = 0, 1, 2...) | | :------------ | :------------------------------------...

## Atomic Structure. The Periodic Table. Based on McGraw-Hill Ryerson: Chemistry 12 ### The Standard Model of Fundamental Particles and Interactions | | **Fermions** (spin = 1/2, 3/2, 5/2...) | **Bosons** (spin = 0, 1, 2...) | | :------------ | :------------------------------------ | :---------------------------- | | **Matter Constituents** | | | | **Leptons** | | **Force Carriers** | | *Flavor* | *Approx. Mass (GeV/c²)* | *Mass (GeV/c²)* | | *electron* | *<1x10⁻⁸* | *0* | | *νe neutrino* | *0* | *0* | | *muon* | *0.000511* | *unified electroweak* | | *νμ neutrino* | *<0.0002* | *W⁻* | | *tau* | *0.106* | *W⁺* | | *ντ neutrino* | *<0.02* | *Z⁰* | | **Quarks** | *Approx. Mass (GeV/c²)* | *strong (color)* | | *Flavor* | *Electric Charge* | *Gluon* | | *u up* | *0.003* | *0* | | *d down* | *0.006* | *0* | | *c charm* | *1.3* | *0* | | *s strange* | *0.1* | *0* | | *t top* | *175* | *0* | | *b bottom* | *4.3* | *0* | **Spin** is the intrinsic angular momentum of particles. It is given in units of ħ, which is the quantum unit of angular momentum, where ħ = h/2π = 6.58x10⁻²⁵ GeV s = 1.05x10⁻³⁴ J s. **Electric Charges** are given in units of the proton's charge. In SI units, the electric charge of the proton is 1.60x10⁻¹⁹ coulombs. The energy unit of particle physics is the electronvolt (eV), the energy gained by one electron in crossing a potential difference of one volt. Masses are given in GeV/c² (remember E=mc²), where 1 GeV = 10⁹ eV = 1.60x10⁻¹⁰ joule. The mass of the proton is 0.938 GeV/c² = 1.67x10⁻²⁷ kg. #### Properties of Fundamental Particles **Structure within the Atom** * **Quark** - Size < 10⁻¹⁹ m * **Nucleus** - Size 10⁻¹⁴ m * **Atom** - Size 10⁻¹⁰ m **Baryons** are fermionic hadrons. There are about 120 types of baryons. | Symbol | Name | Quark Content | Electric Charge | Mass (GeV/c²) | Spin | | :------ | :----------- | :------------- | :--------------- | :------------- | :--- | | p | proton | uud | +1 | 0.938 | 1/2 | | p | anti-proton | ūūd | -1 | 0.938 | 1/2 | | n | neutron | udd | 0 | 0.940 | 1/2 | | Λ | lambda | uds | 0 | 1.116 | 1/2 | | Ω | omega | sss | -1 | 1.672 | 3/2 | **Mesons** are bosonic hadrons. There are about 140 types of mesons. | Symbol | Name | Quark Content | Electric Charge | Mass (GeV/c²) | Spin | | :------ | :---------------- | :------------- | :--------------- | :------------- | :--- | | π⁰ | B-zero (pion) | ud | 0 | 0.140 | 0 | | K⁻ | kaon | sū | -1 | 0.494 | 0 | | ρ⁺ | rho | ud | +1 | 0.770 | 1 | | B⁰ | B-zero | db | 0 | 5.279 | 0 | | ηc | eta-c | cc | 0 | 2.980 | 0 | **Properties of Interactions** | Interaction | Property | Acts On | Mediating Particles | Strength Relative to Electromagnetic (for two u quarks at 3x10⁻¹⁷ m) | | :---------- | :------------------------ | :---------- | :---------------------- | :----------------------------------------------------------- | | Gravitational | Mass-Energy | All | Graviton | 10⁻⁴¹ | | Weak | Flavor | Quarks, Leptons | W⁺, W⁻, Z⁰ | 10⁻⁷ | | Electromagnetic | Electric Charge | Electrically charged | photon | 1 | | Strong | Color Charge | Quarks, Gluons | Gluons | 25 | | Residual Strong | Not applicable to hadrons | Hadrons | Mesons | 20 (from 60) | **The Particle Adventure** - Visit the award-winning web feature The Particle Adventure at http://ParticleAdventure.org ### Atomic Models and Properties of Atoms Historically, scientists have used their knowledge of atomic properties to develop and refine atomic models. Today, this knowledge is applied to various research techniques. * **Scientists can now determine color patterns of ancient bird feathers by identifying elements present in fossils of the birds.** ### Developing a Nuclear Model of the Atom 1. Dalton's model (1808) 2. Thomson's model (1904) 3. Rutherford's model (1911) 4. Bohr's model (1913) 5. Quantum Mechanical model (present) ### Reviewing the Atomic Models of Dalton and Thomson #### John Dalton's model of the atom: * Marked the beginning of a new way of explaining matter. * Matter was described as being composed of small, indivisible spheres, which Dalton called *atoms*. * *Dalton envisioned atoms as hard, solid spheres.* #### J.J. Thomson's model of the atom: * Incorporated his discovery of the *electron*, using cathode ray tubes. * An atom is a positively charged spherical mass with negatively charged *electrons* embedded within. * *Thomson's "plum pudding" model of the atom.* * **Rutherford's experimental observations required a new atomic model.** #### **What is Cathode Ray?** * Cathode rays (also called an electron beam or *e-beam*) are streams of electrons observed in vacuum tubes. * If an evacuated glass tube is equipped with two electrodes and a voltage is applied, the glass opposite of the negative electrode is observed to glow. * This is due to electrons emitted from and traveling perpendicular to the cathode (the electrode connected to the negative terminal of the voltage supply). They were first observed in 1869 by German physicist Johann Hittorf. ### Rutherford's Experiments with Alpha Particles * Alpha particles were aimed at gold foil and the scattering of the alpha particles was monitored. * **Expectation (Thomson's model):** Particles pass through or some slightly deflected * **Observation:** Some particles deflected at large angles. ### Rutherford's Atomic Model * **Rutherford's model of the atom:** * Large deflections of particles proposed to be due to the presence of an electric field at the center of the atom. * An atom has a positively charged nucleus at the center, with electrons in motion surrounding the nucleus. * *The nuclear, or planetary model, of the atom.* ### The Limitations of Rutherford's Atomic Model Based on the understanding of physics at the time, for an electron in motion around a central core: * Radiation must be emitted, so it was expected that a continuous spectrum of light energy was being given off. * The electron would lose energy, and its orbit would decrease until it spiraled into the nucleus, destroying the atom. ### Rethinking Atomic Structure Based on the Nature of Energy * Light is one form of electromagnetic radiation, which travels through space as waves. * **Electromagnetic waves:** * Have frequency, wavelength, and amplitude. * Interact with matter in discrete particles called *photons*. ### Atomic Spectra * When atoms are excited due to the absorption of energy, they emit light as they lose energy and return to a non-excited state. * Atoms of each element emit light of particular wavelengths called a line spectrum or *emission spectrum*. * **Each element has a characteristic line spectrum.** ### The Bohr Model of the Hydrogen Atom Niels Bohr set out to explain the stability of the nuclear model of the atom. In this model, electrons: * Are in circular orbits. * Can only exist in certain "allowed" orbits or energy levels (the energy of electrons is quantized). * Do not radiate energy while in one orbit. * Can jump between orbits by gaining or losing a specific amount of energy. ### Bohr's Atomic Model Explains the Line Spectrum of Hydrogen * Calculated wavelengths of the possible energies of photons that could be emitted from an excited hydrogen atom (transitions from n=6, 5, 4, and 3 to n=2) corresponded with hydrogen's visible line spectrum. **Limitations** * Could only explain single-electron systems (H, He⁺, Li*²*⁺) ### Review - The Nuclear Model of the Atom * Building on and extending from the work of scientists who came before them as well as their peers, Ernest Rutherford and Niels Bohr developed atomic models that, today, are referred to as nuclear and planetary models of the atom. **Key Terms:** * Electromagnetic radiation * Emission spectrum or line spectrum * Frequency * Nuclear model * Photon * Quantum **Key Concepts:** * Historically, the development of and modifications to the atomic model have been the result of experimental evidence and new ideas about the nature of matter and energy. * **Thomson's discovery of the electron disproved Dalton's model of the atom as a solid, indivisible sphere.** Instead, Thomson proposed that the atom existed as a positively charged sphere, with enough negatively charged electrons embedded in it to balance the overall charge on the atom. * **Observations and inferences made by Rutherford led to a nuclear model of the hydrogen atom.** This model is still the most commonly depicted, with negatively charged electrons orbiting around a central, positively charged nucleus. * **Bohr's model of the atom refined the Rutherford model, incorporating the concept that, like light, electrons in atoms could have only certain amounts of energy,** and therefore, could exist in only specific orbits around the nucleus. Each allowed orbit had a specific amount of energy and a specific radius. ### The Quantum Mechanical Model of the Atom Today's quantum mechanical model of the atom incorporates the wave properties of electrons. * **Wave functions, initially described by Erwin Schrödinger, represent a region in space around a nucleus where an electron will be found.** This region of space is called an *atomic orbital*. * *An electron density diagram represents an atomic orbital.* * **Atomic orbitals can be visualized as "fuzzy clouds".** * The higher the density of the "cloud," the higher the probability of finding an electron at that point. * The cloud has no definite boundary. * The region where an electron will spend 90 percent of its time is depicted by drawing a circle. * *The circle does not represent a real boundary*. ### Quantum Numbers Describe Orbitals Electrons in the quantum mechanical model of the atom are described using quantum numbers. * Three quantum numbers describe the distribution of electrons in the atom, and a fourth describes the behavior of each electron. * **Symbols for the four quantum numbers:** * **n** - Principle Quantum Number * **l** - Orbital-Shape Quantum Number * **m₁** - Magnetic Quantum Number * **m₂** - Spin Quantum Number #### The Principle Quantum Number, n * Is the first quantum number. * Describes the energy level, or *shell*, of an orbital. * All orbitals with the same n value are in the same shell. * The larger the n value, the larger the size of the shell. * Values can range from *n = 1* to *n = ∞* | *n* | Shell | | :-- | :------- | | 1 | first | | 2 | second | | 3 | third | | 4 | fourth | #### The Orbital-Shape Quantum Number, l * Is the second quantum number. * Describes the shape of an orbital. * Refers to energy sublevels, or *subshells*. * Values depend on the value of n. They are positive integers from 0 to (n-1). * Each value is identified by a letter: | *l* | Orbital | Letter | | :-- | :------- | :----- | | 0 | s | s | | 1 | p | p | | 2 | d | d | | 3 | f | f | *An energy sublevel is identified by combining n with the orbital letter. For example, n=2, l=1: 2p sublevel.* #### The Magnetic Quantum Number, m₁ * Is the third quantum number. * Indicates the orientation of the orbital in space. * For a given l, there are (2l+1) values for m₁. * The total number of orbitals for an energy level is n². * *s, p, and d orbitals have characteristic shapes*. #### The Spin Quantum Number, m₂ * Is the fourth quantum number. * Specifies the orientation of the axis of electron spin. * There are two possible values: +1/2 or -1/2 #### To Summarize: | Name | Symbol | Allowed Values | Property | | :--------------- | :------ | :---------------------------------------------- | :------------------------------------------------------------------------------------------ | | Principal (shell) | n | Positive integers (1, 2, 3, etc.) | Orbital size and energy level | | Orbital-shape (subshell) | l | Integers from 0 to (n-1) | Orbital shape (l values 0, 1, 2, and 3 correspond to s, p, d, and f orbitals) | | Magnetic | m₁ | Integers from -l, to +l | Orbital orientation | | Spin | m₂ | +1/2 or -1/2 | Spin orientation | ### Identifying Electrons Using Sets of Quantum Numbers According to the *Pauli exclusion principle*: * An orbital can have a maximum of two electrons. * Two electrons in an orbital must have opposite spins. * No two electrons of an atom have the same set of four quantum numbers. | Atom | Electron | Quantum Numbers | Electron Configuration | Orbital Diagram | | :------- | :-------- | :--------------------------------------------------------------------------- | :-------------------- | :------------------------------------------------------------------------------------------------------------------------- | | hydrogen | Lone | n=1, l=0, m₁=0, m₂=+1/2 | 1s¹ | ↑ | | helium | First | n=1, l=0, m₁=0, m₂=+1/2 | 1s² | ↑↓ | | | Second | n=1, l=0, m₁=0, m₂=-1/2 | | | | lithium | First | n=1, l=0, m₁=0, m₂=+1/2 | 1s²2s¹ | ↑↓ ↑ | | | Second | n=1, l=0, m₁=0, m₂=-1/2 | | | | | Third | n=2, l=0, m₁=0, m₂=+1/2 | | | ### Writing Electron Configurations and Orbital Diagrams * **Follow the aufbau principle:** "build up" electronic configurations of atoms in order of increasing atomic number. * **Guidelines for "Filling" Orbitals:** 1. Place electrons into the orbitals in order of increasing energy level. 2. Completely fill orbitals of the same energy level before proceeding to the next orbital or series of orbitals. 3. When electrons are added to orbitals of the same energy sublevel, each orbital receives one electron before any pairing occurs. 4. When electrons are added individually to different orbitals of the same energy, the electrons must all have the same spin. ### Filling Orbitals for Periods 1 and 2 * Boxes in orbital diagrams are written and filled from left to right (increasing energy of orbitals). * For C, apply Hund's rule, and for O, apply Pauli exclusion principle. | Atomic Number (Z) | Element | Orbital Diagram | Electron Configuration | | :---------------- | :---------- | :------------------------------------------------------------------------------------------------------------ | :-------------------- | | 1 | H | ↑ | 1s¹ | | 2 | He | ↑↓ | 1s² | | 3 | Li | ↑↓ ↑ | 1s²2s¹ | | 4 | Be | ↑↓ ↑↓ | 1s²2s² | | 5 | B | ↑↓ ↑↓ ↑ | 1s²2s²2p¹ | | 6 | C | ↑↓ ↑↓ ↑↑ | 1s²2s²2p² | ### LEARNING CHECK Write the electron configurations and draw orbital diagrams for N and F. ### For nitrogen * 1s²2s²2p³ * ↑↓ ↑↓ ↑↑↑ ### For fluorine * 1s²2s²2p⁵ * ↑↓ ↑↓ ↑↓↓↑ ### Filling Orbitals for Period 3 * Follow the same guidelines as done for Period 1 and 2 elements. * Use condensed electron configuration and corresponding partial orbital diagrams. * **Full for Z=11:** 1s²2s²2p⁶3s¹ * **Condensed for Z=11:** [Ne]3s¹ * For filling orbitals for transition and Group 12 elements, keep in mind that the order of orbital energies is: * **1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < ...** ### Filling Orbitals for Period 4 Follow the same guidelines up to Z=23. After that, two exceptions are Cr [Ar]4s¹ 3d⁵ and Cu [Ar]4s¹ 3d¹⁰ | Atomic Number (Z) | Element | Orbital Diagram | Condensed Electron Configuration | | :---------------- | :---------- | :----------------------------------------------------------------------- | :----------------------------- | | 19 | K | ↑ | [Ar]4s¹ | | 20 | Ca | ↑↓ | [Ar]4s² | | 21 | Sc | ↑↓ ↑ | [Ar]4s² 3d¹ | | 22 | Ti | ↑↓ ↑↑ | [Ar]4s² 3d² | | 23 | V | ↑↓ ↑↑↑ | [Ar]4s² 3d³ | | 24 | Cr | ↑ ↑↑↑↑↑ | [Ar]4s¹ 3d⁵ | | 25 | Mn | ↑↓ ↑↑↑↑↑↑↑ | [Ar]4s² 3d⁵ | | 26 | Fe | ↑↓ ↑↑↑↑↑↑↑↑ | [Ar]4s² 3d⁶ | ### Using the Periodic Table to Predict Electron Configurations Based on the filling pattern of orbitals, the periodic table can be divided into s block, p block, d block, and f block regions. ### Group and Period Numbers Provide Patterns Elements in a group have similar electron configurations and the same number of valence electrons. Patterns include: * The last numeral of the group number is the same as the number of valence electrons for main group elements (He is an exception). * The value of n for the highest occupied orbital is the *period number*. * At a given energy level, the total number of orbitals is n² and the maximum number of electrons is 2n². ### LEARNING CHECK The condensed electron configuration for silicon is [Ne]3s²3p². Without using a periodic table, identify the group number, period number, and orbital block for silicon. ### Group 14, Period 3, p block ### Electron Configurations and Periodic Trends in Atomic Properties * Patterns of electron configurations in the periodic table are related to periodic trends. #### Atomic radius trend: * For main group elements, generally a decrease across a period, and an increase down a group. #### First Ionization Energy * First Ionization Energy is the energy required to remove the first electron from an atom. * Within a group, it generally *decreases* as you move down the group. * Within a period, it generally *increases* as you move form left to right. #### Electron Affinity * The trend is more irregular. * In combination with ionization energy, there are trends. * **Atoms high in both electron affinity and ionization energy easily form anions.** * **Atoms low in both easily form cations.** * **Atoms with very high ionization energies and very low electron affinities do not bond (noble gases).** ### Review - Electron Configurations and the Periodic Table * The quantum mechanical model of the atom explains the experimentally determined, electronic structure of the periodic table and the properties of its elements. **Key Terms:** * Atomic radius * Aufbau principle * Electron affinity * Electron configuration * Ionization energy * Hund's Rule * Orbital diagram **Key Concepts:** * For hydrogen, the relative energies of the atomic orbitals depend only on the principal quantum number, n. For many-electron atoms, other factors such as the orbital-shape quantum number, l, influence the relative energies of atomic orbitals. * Electron configuration notation and orbital diagrams are two methods commonly used to represent or describe the distribution of electrons in atoms. * Applying the Pauli exclusion principle, Hund's rule, and the aufbau principle, the electron configuration of atoms can be built up according to the position of the element in the periodic table. * The modern, quantum mechanical model of the atom enables us to understand the elements, their positions in the periodic table, and their chemical and physical properties based on their electron configuration. ### The End