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1 Electrons in Atoms

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2 Dalton’s Atomic Theory John Dalton ( ) had four theories John Dalton ( ) had four theories 1. All elements are composed of submicroscopic indivisible particles called atoms 2. Atoms of the same element are identical. The atoms of anyone element are different from those of any other element 3. Atoms of different elements can physically mix together or can chemically combine w/ one another in simple whole-number ratios to form compounds 4. Chemical reactions occur when atoms are separated, joined, or rearranged. However, atoms of one element are never changed into atoms of another elements as a result of a chemical reaction

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3 Thomson’s Atomic Model Thomson’s Atomic Model Thomson’s Atomic Model Thomson though electrons were like plums embedded in a positively charged “pudding”, so his model was called the “plum pudding” model Thomson though electrons were like plums embedded in a positively charged “pudding”, so his model was called the “plum pudding” model

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4 Thomson’s Theory Thomson stated: The atom had negatively charged electrons stuck into a lump of positively charged protons. Thomson stated: The atom had negatively charged electrons stuck into a lump of positively charged protons. Thomson never explained Thomson never explained 1. Number of protons and neutrons 2. The arrangement of the particles in the atom 3. The ease with which atoms are stripped of electrons to form ions

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5 Rutherford Model Rutherford used the Gold Foil Experiment Rutherford used the Gold Foil Experiment Rutherford proposed the following: Rutherford proposed the following: 1. Thomson model was incorrect 2. Most of the mass of the atom and all of its positive charge reside in a very small, extremely dense region, which he called the nucleus 3. Most of the total volume of the atom is empty space in which electrons move around the nucleus

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Rutherford’s Model Discovered dense positive piece at the center of the atom Discovered dense positive piece at the center of the atom Nucleus Nucleus Electrons moved around Electrons moved around Mostly empty space Mostly empty space

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7 Bohr Model Bohr changed the Rutherford model and explained how the electrons travel. Bohr changed the Rutherford model and explained how the electrons travel. Bohr explained the following in his model: Bohr explained the following in his model: 1. Electrons travel in definite orbits around the nucleus 2. Electrons are arranged in concentric circular paths or orbitals around the nucleus 3. Electrons don’t fall into the nucleus because electrons in particular path have fixed energy and don’t lose energy 4. His model was patterned after the motion of the planets around the sun. It is often called the Planetary model.

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8 Bohr Model Cont.

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Bohr’s Model Nucleus Electron Orbit Energy Levels

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10 Quantum Theory Bohr explained how electrons were moving via Quantum Theory Bohr explained how electrons were moving via Quantum Theory Key Terms: Key Terms: 1. Energy Levels- Regions around the nucleus where the electron is likely moving 2. Quantum- Amount of energy required to move an electron from one energy level to the next 3. Quantum Leap- Abrupt Change

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Bohr’s Model Increasing energy Nucleus First Second Third Fourth Fifth } Further away from the nucleus means more energy. There is no “in between” energy Energy Levels

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12 Bohr’s Model cont. Energy levels are not equally spaced. Energy levels are not equally spaced. Energy levels more closely spaced further from the nucleus Energy levels more closely spaced further from the nucleus Higher energy level occupied by an electron, the more energetic that electron is. Higher energy level occupied by an electron, the more energetic that electron is. Amount of energy gained or lost by an electron is not always the same amount. Amount of energy gained or lost by an electron is not always the same amount.

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13 Bohr Model Cont. The Bohr Model did not account for: The Bohr Model did not account for: 1. Emission spectra of atoms containing more than one electron. So comes along the next model: So comes along the next model:

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The Quantum Mechanical Model Energy is quantized. It comes in chunks. A quanta is the amount of energy needed to move from one energy level to another. Since the energy of an atom is never “in between” there must be a quantum leap in energy. Schrodinger derived an equation that described the energy and position of the electrons in an atom

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Things that are very small behave differently from things big enough to see. Things that are very small behave differently from things big enough to see. The quantum mechanical model is a mathematical solution The quantum mechanical model is a mathematical solution It is not like anything you can see. It is not like anything you can see. The Quantum Mechanical Model

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Erwin Schrödinger: We can describe the electron mathematically, using quantum mechanics (wave mechanics). Erwin Schrödinger: We can describe the electron mathematically, using quantum mechanics (wave mechanics). Schrödinger developed a wave equation to describe the hydrogen atom. Schrödinger developed a wave equation to describe the hydrogen atom. An acceptable solution to Schrödinger’s wave equation is called a wave function. An acceptable solution to Schrödinger’s wave equation is called a wave function. A wave function represents an energy state of the atom. A wave function represents an energy state of the atom. Wave Functions

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Schrödinger’s Equation The wave function is a F(x, y, z) The wave function is a F(x, y, z) Actually F(r,θ,φ) Actually F(r,θ,φ) Solutions to the equation are called orbitals. Solutions to the equation are called orbitals. These are not Bohr orbits. These are not Bohr orbits. Each solution is tied to a certain energy Each solution is tied to a certain energy These are the energy levels These are the energy levels Animation

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Schrödinger’s Equation 18

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Has energy levels for electrons. Has energy levels for electrons. Orbits are not circular. Orbits are not circular. It can only tell us the probability of finding an electron a certain distance from the nucleus. It can only tell us the probability of finding an electron a certain distance from the nucleus. The Quantum Mechanical Model

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The atom is found inside a blurry “electron cloud” The atom is found inside a blurry “electron cloud” A area where there is a chance of finding an electron. A area where there is a chance of finding an electron. Draw a line at 90 % Draw a line at 90 % The Quantum Mechanical Model

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Werner Heisenberg: We can’t know exactly where a moving particle is AND exactly how fast it is moving at the same time. The Uncertainty Principle The photon that will enter the microscope, so that we might “see” the electron … … has enough momentum to deflect the electron. The act of measurement has interfered with the electron’s motion.

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A wave function doesn’t tell us where the electron is. The uncertainty principle tells us that we can’t know where the electron is. A wave function doesn’t tell us where the electron is. The uncertainty principle tells us that we can’t know where the electron is. However, the square of a wave function gives the probability of finding an electron at a given location in an atom. However, the square of a wave function gives the probability of finding an electron at a given location in an atom. Analogy: We can’t tell where a single leaf from a tree will fall. But (by viewing all the leaves under the tree) we can describe where a leaf is most likely to fall. Analogy: We can’t tell where a single leaf from a tree will fall. But (by viewing all the leaves under the tree) we can describe where a leaf is most likely to fall. The Uncertainty Principle

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23 Atomic Orbitals There are the region of space which there is a high probability of finding an electron There are the region of space which there is a high probability of finding an electron Within each energy level the complex math of Schrodinger’s equation describes several shapes. Within each energy level the complex math of Schrodinger’s equation describes several shapes. These are called atomic orbitals These are called atomic orbitals Quantum Numbers- numbers that specify the properties of atomic orbitals and their electrons Quantum Numbers- numbers that specify the properties of atomic orbitals and their electrons

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24 Quantum Numbers There are 4 types of Quantum Numbers There are 4 types of Quantum Numbers 1. Principal – distance from the nucleus 2. Angular Momentum- Orbital Shape 3. Magnetic- Orbital position with respect to the X, Y, & Z axes. 4. Spin- Has only two values (+1/2 or –1/2) and is needed to specify 1 of 2 positional orientations of an electron

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25 Principal Quantum Number Symbolized by the letter N, indicates the main energy levels surrounding the nucleus Symbolized by the letter N, indicates the main energy levels surrounding the nucleus There are 7 principal quantum numbers There are 7 principal quantum numbers A.K.A. – Shells A.K.A. – Shells Value of N is a whole number ex. 1,2,3 ect.. Value of N is a whole number ex. 1,2,3 ect.. Main Energy Level – N=1; closest to the nucleus or ground state Main Energy Level – N=1; closest to the nucleus or ground state Ground State- state of the lowest energy of the atom. Ground State- state of the lowest energy of the atom. As N increases, the distance from the nucleus increases and the energy increases As N increases, the distance from the nucleus increases and the energy increases

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26 Angular Momentum Quantum Number Indicates the shape of the orbital. Indicates the shape of the orbital. Within each main energy level beyond the first, orbitals with different shapes occupy different regions Within each main energy level beyond the first, orbitals with different shapes occupy different regions A.K.A. – Sublevels or Subshells A.K.A. – Sublevels or Subshells The number of sublevels = Value of the Principal Quantum Number The number of sublevels = Value of the Principal Quantum Number

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27 Magnetic Quantum Number Indicates the orientation of a orbital about the nucleus Indicates the orientation of a orbital about the nucleus There are 4 types of orbital orientation There are 4 types of orbital orientation I. S Orbital II. P Orbital III. D Orbital IV. F Orbital

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28 Spin Quantum Number Has only two possible values: +1/2 or –1/2. These values indicate two possible states of an electron in an orbital Has only two possible values: +1/2 or –1/2. These values indicate two possible states of an electron in an orbital Spin Quantum # is significant because each single orbital can hold no more than two electrons, which must have opposite spin. Spin Quantum # is significant because each single orbital can hold no more than two electrons, which must have opposite spin.

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1 s orbital for every energy level 1 s orbital for every energy level Spherical shaped Spherical shaped Each s orbital can hold 2 electrons Each s orbital can hold 2 electrons Called the 1s, 2s, 3s, etc.. orbitals. Called the 1s, 2s, 3s, etc.. orbitals. S orbitals

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P orbitals Start at the second energy level Start at the second energy level 3 different directions 3 different directions 3 different shapes 3 different shapes Each can hold 2 electrons Each can hold 2 electrons

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P Orbitals

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D orbitals Start at the third energy level Start at the third energy level 5 different shapes 5 different shapes Each can hold 2 electrons Each can hold 2 electrons

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F orbitals Start at the fourth energy level Start at the fourth energy level Have seven different shapes Have seven different shapes 2 electrons per shape 2 electrons per shape

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F orbitals

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Summary s p d f # of shapes Max electrons Starts at energy level

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By Energy Level First Energy Level First Energy Level only s orbital only s orbital only 2 electrons only 2 electrons 1s 2 1s 2 Second Energy Level s and p orbitals are available 2 in s, 6 in p 2s 2 2p 6 8 total electrons

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By Energy Level Third energy level Third energy level s, p, and d orbitals s, p, and d orbitals 2 in s, 6 in p, and 10 in d 2 in s, 6 in p, and 10 in d 3s 2 3p 6 3d 10 3s 2 3p 6 3d total electrons 18 total electrons Fourth energy level s,p,d, and f orbitals 2 in s, 6 in p, 10 in d, ahd 14 in f 4s 2 4p 6 4d 10 4f total electrons

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By Energy Level Any more than the fourth and not all the orbitals will fill up. Any more than the fourth and not all the orbitals will fill up. You simply run out of electrons You simply run out of electrons The orbitals do not fill up in a neat order. The energy levels overlap Lowest energy fill first.

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39 Question for You How many principal quantum numbers are there? How many principal quantum numbers are there? What is the maximum number of electrons that can fill the 3 rd energy level? What is the maximum number of electrons that can fill the 3 rd energy level? How many orbitals are in the sublevel F? How many orbitals are in the sublevel F? What is the total number of orbitals for the 3 rd main energy level? What is the total number of orbitals for the 3 rd main energy level?

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40 Electron Configuration The way electrons are arranged in atoms The way electrons are arranged in atoms There are three rules which help dictate how electrons are arranged in the atoms. There are three rules which help dictate how electrons are arranged in the atoms. 1) Aufbau Principle- electrons occupy the orbitals of the lowest energy first 2) Hund’s Rule- Orbitals of equal energy are each occupied by one electron before any one orbital is occupied by a second electron. All electrons in a single occupied orbital must have the same spin.

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41 Electron Configuration cont. Pauli Exclusion Principle- No two electrons may occupy any given orbital without having opposite spin. No two electrons in the same atom can have the same set of four quantum numbers. Pauli Exclusion Principle- No two electrons may occupy any given orbital without having opposite spin. No two electrons in the same atom can have the same set of four quantum numbers. Let’s determine electron configuration. Let’s determine electron configuration. Let’s start with Phosphorus. Let’s start with Phosphorus. Need to account for all 15 electrons Need to account for all 15 electrons

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Electron Configurations Distribution of all electrons in an atom Consist of Number denoting the energy level

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Electron Configurations Distribution of all electrons in an atom Consist of Number denoting the energy level Letter denoting the type of orbital

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Electron Configurations Distribution of all electrons in an atom. Consist of Number denoting the energy level. Letter denoting the type of orbital. Superscript denoting the number of electrons in those orbitals.

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An electron configuration describes the distribution of electrons among the various orbitals in the atom. An electron configuration describes the distribution of electrons among the various orbitals in the atom. Electron configuration is represented in two ways. Electron configuration is represented in two ways. Electron Configurations The spdf notation uses numbers to designate a principal shell and letters (s, p, d, f) to identify a subshell; a superscript indicates the number of electrons in a designated subshell.

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Orbital Diagrams Each box represents one orbital. Each box represents one orbital. Half-arrows represent the electrons. Half-arrows represent the electrons. The direction of the arrow represents the spin of the electron. The direction of the arrow represents the spin of the electron.

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Hund’s Rule “For degenerate orbitals, the lowest energy is attained when the number of electrons with the same spin is maximized.”

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Exceptions to Electron Configuration

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Orbitals fill in order Lowest energy to higher energy. Lowest energy to higher energy. Adding electrons can change the energy of the orbital. Adding electrons can change the energy of the orbital. Half filled orbitals have a lower energy. Half filled orbitals have a lower energy. Makes them more stable. Makes them more stable. Changes the filling order Changes the filling order

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Write these electron configurations Titanium - 22 electrons Titanium - 22 electrons 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 2 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 2 Vanadium - 23 electrons 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 3 Vanadium - 23 electrons 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 3 Chromium - 24 electrons Chromium - 24 electrons 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 4 is expected 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 4 is expected But this is wrong!! But this is wrong!!

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Chromium is actually 1s 2 2s 2 2p 6 3s 2 3p 6 4s 1 3d 5 1s 2 2s 2 2p 6 3s 2 3p 6 4s 1 3d 5 Why? Why? This gives us two half filled orbitals. This gives us two half filled orbitals. Slightly lower in energy. Slightly lower in energy. The same principal applies to copper. The same principal applies to copper.

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Copper’s electron configuration Copper has 29 electrons so we expect Copper has 29 electrons so we expect 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 9 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 9 But the actual configuration is But the actual configuration is 1s 2 2s 2 2p 6 3s 2 3p 6 4s 1 3d 10 1s 2 2s 2 2p 6 3s 2 3p 6 4s 1 3d 10 This gives one filled orbital and one half filled orbital. This gives one filled orbital and one half filled orbital. Remember these exceptions Remember these exceptions

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54 Shortcuts for Electron Configuration There are two short handed methods of writing the electron configuration. There are two short handed methods of writing the electron configuration. The 1 st method is called the outer-level configuration. That tells you the outer-most configuration for that element. The 1 st method is called the outer-level configuration. That tells you the outer-most configuration for that element. The 2 nd method is called the Noble Gas Notation. This tells you the complete notation using Noble Gases. The 2 nd method is called the Noble Gas Notation. This tells you the complete notation using Noble Gases. Let’s start with outer-level notation!!! Let’s start with outer-level notation!!!

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Exceptions to the Aufbau Principle Half-filled d subshell plus half-filled s subshell has slightly lower in energy than s 2 d 4. Filled d subshell plus half-filled s subshell has slightly lower in energy than s 2 d 9. More exceptions occur farther down the periodic table. They aren’t always predictable, because energy levels get closer together.

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1s11s11s11s1 1s 2 2s 1 1s 2 2s 2 2p 6 3s 1 1s 2 2s 2 2p 6 3s 2 3p 6 4s 1 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 1 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2 4d 10 5p 6 6s 1 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2 4d 10 5p 6 6s 2 4f 14 5d 10 6p 6 7s 1 H 1 Li 3 Na 11 K 19 Rb 37 Cs 55 Fr 87

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He 2 Ne 10 Ar 18 Kr 36 Xe 54 Rn 86 1s21s21s21s2 1s 2 2s 2 2p 6 1s 2 2s 2 2p 6 3s 2 3p 6 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2 4d 10 5p 6 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2 4d 10 5p 6 6s 2 4f 14 5d 10 6p 6

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Alkali metals all end in s 1 Alkali metals all end in s 1 Alkaline earth metals all end in s 2 Alkaline earth metals all end in s 2 really have to include He but it fits better later. really have to include He but it fits better later. He has the properties of the noble gases. He has the properties of the noble gases. s2s2 s1s1 S- block

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The P-block p1p1 p2p2 p3p3 p4p4 p5p5 p6p6

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Transition Metals -d block d1d1 d2d2 d3d3 s1d5s1d5 d5d5 d6d6 d7d7 d8d8 s 1 d 10 d 10

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F - block inner transition elements inner transition elements

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Each row (or period) is the energy level for s and p orbitals. Each row (or period) is the energy level for s and p orbitals

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D orbitals fill up after previous energy level so first d is 3d even though it’s in row 4. D orbitals fill up after previous energy level so first d is 3d even though it’s in row d

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f orbitals start filling at 4f f orbitals start filling at 4f f 5f

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66 Summary Outer-Level Configuration

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Writing Electron configurations the easy way Yes there is a shorthand

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Electron Configurations repeat The shape of the periodic table is a representation of this repetition. The shape of the periodic table is a representation of this repetition. When we get to the end of the column the outermost energy level is full. When we get to the end of the column the outermost energy level is full. This is the basis for our shorthand. This is the basis for our shorthand.

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The Shorthand Write the symbol of the noble gas before the element. Write the symbol of the noble gas before the element. Then the rest of the electrons. Then the rest of the electrons. Aluminum - full configuration. Aluminum - full configuration. 1s 2 2s 2 2p 6 3s 2 3p 1 1s 2 2s 2 2p 6 3s 2 3p 1 Ne is 1s 2 2s 2 2p 6 Ne is 1s 2 2s 2 2p 6 so Al is [Ne] 3s 2 3p 1 so Al is [Ne] 3s 2 3p 1

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More examples Ge = 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 2 Ge = 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 2 Ge = [Ar] 4s 2 3d 10 4p 2 Ge = [Ar] 4s 2 3d 10 4p 2 Hf=1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2 4d 10 5p 6 6s 2 4f 14 5d 2 Hf=1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2 4d 10 5p 6 6s 2 4f 14 5d 2 Hf=[Xe]6s 2 4f 14 5d 2 Hf=[Xe]6s 2 4f 14 5d 2

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The Shorthand Again Sn- 50 electrons The noble gas before it is Kr [ Kr ] Takes care of 36 Next 5s 2 5s 2 Then 4d 10 4d 10 Finally 5p 2 5p 2

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Quantum Numbers Solving the wave equation gives a set of wave functions, or orbitals, and their corresponding energies. Solving the wave equation gives a set of wave functions, or orbitals, and their corresponding energies. Each orbital describes a spatial distribution of electron density. Each orbital describes a spatial distribution of electron density. An orbital is described by a set of three quantum numbers. An orbital is described by a set of three quantum numbers.

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The Wave-like Electron Louis deBroglie The electron propagates through space as an energy wave. To understand the atom, one must understand the behavior of electromagnetic waves.

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The Quantum Mechanical Model A totally new approach A totally new approach De Broglie said matter could be like a wave. De Broglie said matter could be like a wave. De Broglie said they were like standing waves. De Broglie said they were like standing waves. The vibrations of a stringed instrument The vibrations of a stringed instrument

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What’s possible? You can only have a standing wave if you have complete waves. You can only have a standing wave if you have complete waves. There are only certain allowed waves. There are only certain allowed waves. In the atom there are certain allowed waves called electrons. In the atom there are certain allowed waves called electrons Erwin Schroedinger described the wave function of the electron 1925 Erwin Schroedinger described the wave function of the electron Much math, but what is important are the solutions Much math, but what is important are the solutions

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Schrödinger’s Equation The wave function is a F(x, y, z) The wave function is a F(x, y, z) Actually F(r,θ,φ) Actually F(r,θ,φ) Solutions to the equation are called orbitals. Solutions to the equation are called orbitals. These are not Bohr orbits. These are not Bohr orbits. Each solution is tied to a certain energy Each solution is tied to a certain energy These are the energy levels These are the energy levels Animation

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What does the wave Function mean? nothing. nothing. it is not possible to visually map it. it is not possible to visually map it. The square of the function is the probability of finding an electron near a particular spot. The square of the function is the probability of finding an electron near a particular spot. best way to visualize it is by mapping the places where the electron is likely to be found. best way to visualize it is by mapping the places where the electron is likely to be found.

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Probability Distance from nucleus

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Sum of all Probabilities Distance from nucleus

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Defining the size The nodal surface. The nodal surface. The size that encloses 90% to the total electron probability. The size that encloses 90% to the total electron probability. NOT at a certain distance, but a most likely distance. NOT at a certain distance, but a most likely distance. For the first solution it is a a sphere. For the first solution it is a a sphere.

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© 2009, Prentice-Hall, Inc. Quantum Mechanics Erwin Schrödinger developed a mathematical treatment into which both the wave and particle nature of matter could be incorporated. Erwin Schrödinger developed a mathematical treatment into which both the wave and particle nature of matter could be incorporated. It is known as quantum mechanics. It is known as quantum mechanics.

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© 2009, Prentice-Hall, Inc. Quantum Mechanics The wave equation is designated with a lower case Greek psi ( ). The wave equation is designated with a lower case Greek psi ( ). The square of the wave equation, 2, gives a probability density map of where an electron has a certain statistical likelihood of being at any given instant in time. The square of the wave equation, 2, gives a probability density map of where an electron has a certain statistical likelihood of being at any given instant in time.

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© 2009, Prentice-Hall, Inc. Quantum Numbers Solving the wave equation gives a set of wave functions, or orbitals, and their corresponding energies. Solving the wave equation gives a set of wave functions, or orbitals, and their corresponding energies. Each orbital describes a spatial distribution of electron density. Each orbital describes a spatial distribution of electron density. An orbital is described by a set of three quantum numbers. An orbital is described by a set of three quantum numbers.

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Quantum Numbers There are many solutions to Schrödinger’s equation There are many solutions to Schrödinger’s equation Each solution can be described with quantum numbers that describe some aspect of the solution. Each solution can be described with quantum numbers that describe some aspect of the solution. Principal quantum number (n) size and energy of an orbital Principal quantum number (n) size and energy of an orbital Has integer values >0 Has integer values >0

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© 2009, Prentice-Hall, Inc. s Orbitals Observing a graph of probabilities of finding an electron versus distance from the nucleus, we see that s orbitals possess n−1 nodes, or regions where there is 0 probability of finding an electron.

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© 2009, Prentice-Hall, Inc. p Orbitals The value of l for p orbitals is 1. The value of l for p orbitals is 1. They have two lobes with a node between them. They have two lobes with a node between them.

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© 2009, Prentice-Hall, Inc. d Orbitals The value of l for a d orbital is 2. Four of the five d orbitals have 4 lobes; the other resembles a p orbital with a doughnut around the center.

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Quantum numbers Angular momentum quantum number l Angular momentum quantum number l shape of the orbital shape of the orbital integer values from 0 to n-1 integer values from 0 to n-1 l = 0 is called s l = 0 is called s l = 1 is called p l = 1 is called p l =2 is called d l =2 is called d l =3 is called f l =3 is called f l =4 is called g l =4 is called g

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© 2009, Prentice-Hall, Inc. Magnetic Quantum Number (m l ) The magnetic quantum number describes the three-dimensional orientation of the orbital. The magnetic quantum number describes the three-dimensional orientation of the orbital. Allowed values of m l are integers ranging from -l to l: Allowed values of m l are integers ranging from -l to l: −l ≤ m l ≤ l. −l ≤ m l ≤ l. Therefore, on any given energy level, there can be up to 1 s orbital, 3 p orbitals, 5 d orbitals, 7 f orbitals, etc. Therefore, on any given energy level, there can be up to 1 s orbital, 3 p orbitals, 5 d orbitals, 7 f orbitals, etc.

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© 2009, Prentice-Hall, Inc. Magnetic Quantum Number (m l ) Orbitals with the same value of n form a shell. Orbitals with the same value of n form a shell. Different orbital types within a shell are subshells. Different orbital types within a shell are subshells.

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© 2009, Prentice-Hall, Inc. Spin Quantum Number, m s This led to a fourth quantum number, the spin quantum number, m s. This led to a fourth quantum number, the spin quantum number, m s. The spin quantum number has only 2 allowed values: +1/2 and −1/2. The spin quantum number has only 2 allowed values: +1/2 and −1/2.

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© 2009, Prentice-Hall, Inc. Pauli Exclusion Principle No two electrons in the same atom can have exactly the same energy. No two electrons in the same atom can have exactly the same energy. Therefore, no two electrons in the same atom can have identical sets of quantum numbers. Therefore, no two electrons in the same atom can have identical sets of quantum numbers.

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