1. Human beings have a natural perception and appreciation for symmetry.
|
|
- Joella Gibbs
- 7 years ago
- Views:
Transcription
1 I. SYMMETRY ELEMENTS AND OPERATIONS A. Introduction 1. Human beings have a natural perception and appreciation for symmetry. a. Most people tend to value symmetry in their visual perception of the world. b. Example: Cars and buildings tend to be very symmetrical. 2. Usually we just naturally make symmetry evaluations but never really stop to ask why something is symmetrical or what symmetry properties it has. 3. In this chapter we will develop tools with which to evaluate symmetry and then link this symmetry to a mathematical tool called group theory which will be very valuable in illustrating bonding and spectroscopic properties of molecules. B. Symmetry Elements 1. Definition: a line, point, or plane with respect to which one or more symmetry operations may be performed. 2. The symmetry elements we will be concerned with are: Name Type Symbol Rotation Axis line C n Mirror Plane plane σ Inversion Center point i Improper Rotation Axis line S n C. Symmetry Operations 1. Definition: The movement of a molecule relative to some symmetry element such that every atom in the molecule after the operation is performed coincides with an equivalent atom (or itself) of the molecule before the operation. 2. If a particular symmetry operation can be performed on a molecule, then the molecule is said to posess that symmetry element. Symmetry Page 1
2 3. The symmetry operations: Symmetry Element Symmetry Operation Actual Operation Performed C n C n m Clockwise rotation about the n-fold axis by 2pm/n radians Reflection through the σ v, σ d, σ h σ v, σ d, σ h symmetry plane i i Inversion through the center of symmetry Clockwise rotation about the n-fold axis by 2pm/n radians S n S n m followed by reflection in a plane perpendicular to the axis --- E ne tes: a. The C n and S n axes generate n-1 operations while s and i only generate one. b. σ d and σ v are mirror planes coincident with the principle rotation axis (highest n value) of the molecule and σ h is perpendicular to that axis. 4. Example: Identify the symmetry elements in the BF 3 molecule. F a F c B F b a. Rotation relative to an axis perpendicular to the plane of the molecule through the B atom by 120 will move F a to F b, F b to F c, F c to F a and B into itself. Also rotation by 240 will move F a to F c, F c to F b, F b to F a and again B into itself so BF 3 has a C 3 axis. b. Rotation relative to an axis through F a and B by 180 will exchange F c and F b while F a and B go into themselves. Likewise such axes exist through F b and B as well as F c and B. Thus, BF 3 has three C 2 axes perpendicular to the C 3 axis. Symmetry Page 2
3 c. A mirror plane through F a and B will also exchange F b and F c while F a and B are reflected into themselves. Such mirror planes also exist through F b and B as well as F c and B. Thus, BF 3 has three mirror planes designated as σ v since they are coincident with the C 3 axis.te that the C 3 axis is the principle axis since it has the highest n value. d. BF 3 also contains a mirror plane designated as σ h which is the plane of the molecule such that all atoms are reflected into themselves. e. Finally, if the molecule is rotated by 120 and reflected in the plane of the molecule all atoms go into themselves or equivalent atoms (in this case, since the molecule is planer, same result as C 3 ). Thus, the molecule also posesses an S 3 axis. II. POINT GROUPS A. The Stereographic Projection 1. At this time we will introduce a handy tool for working with symmetry operations called the stereographic projection. te that this is not in the textbook. 2. The stereographic projection is constructed around a circle with the principle axis (generally the z axis) perpendicular to the surface of the paper. Axes of different n value are indicated by symbols as shown. Principle Rotation Axis (z axis) Symbols: n=2 n=3 n=4 Other symbols used on the diagram: = mirror plane = point above the plane of the paper X = point below the plane of the paper Symmetry Page 3
4 3. Example: Use a stereographic representation to represent a structure that has a C 3 and 3 σ v 's (such as in the case of the ammonia molecule. The basic diagram begins with: Placing a point on the diagram and performing all of the operations associated with the symmetry elements produces: σv σv σv' σv" σv" σv' tice in this case, the C 3 and C 3 2 operations would lead to redundancy which is often the case and not a problem (i.e., s v followed by C 3 and C 3 2 would give same result. B. Matrices 1. In truth, the entire concept of group theory is a mathematical one and the performing of a "symmetry operation" is really a matrix multiplication on a coordinate. 2. Example: Consider the C 2 operation. C2 = Operating (multiplying) on a point (x,y,z) then gives: y x X y = z -x -y z or x Symmetry Page 4
5 3. Thus, we see that although this is a mathematical concept, we can get the same information from the more mechanical stereographic projection as from the matrix multiplication process. Hence, we will rely on the former since it is (hopefully) a conceptually easier method of representing the operations and studying their interrelationships. B. The Concept of a Point Group 1. The point group can be defined as a closed set of symmetry operations. That is, a set of operations such that if any two operations of a group are multiplied together their product must also be a member of the group. 2. Example: Consider the example above of a structure which posesses a C 3 axis and three s v planes. To look at this we will construct a multiplication table consisting of all possible products of operations. It should be pointed out that multiplying operations is the same as performing them in succession on a point. C3 σv σv' or C3 X σv = σv' The entire multiplication table is shown below. te that the order of the product can (but will not always) make a difference in the result. E C 3 C 3 2 v v' v" E E C 3 C3 2 σ v σ v ' σ v " C 3 C 3 C3 2 E σ v " σ v σ v ' C 3 2 C 3 2 E C 3 σ v ' σ v " σ v v σ v σ v ' σ v " E C 3 C3 2 v' σ v ' σ v " σ v C 3 2 E C 3 v" σ v " σ v σ v ' C 3 C3 2 E From this table, it can be seen that if the E operation is included in the set of operations that a closed group is obtained. This is, in fact, called the C 3v point group. Symmetry Page 5
6 C. Selecting a Point Group 1. There are a limited number of point groups that are important in chemistry and the one to which a molecule belongs can be selected by noting the key symmetry elements that the molecule possesses and following the chart below. 2. To determine the point group simply answer the questions in the flow chart and take the appropriate path depending on the answer. More than one axis with n>2? One of the special point groups such as Td, Oh, or Ih. C1 Rotation axis, Cn, present? σ? n C2 Cn? σh? i? Cs nσv? Cnh Ci Cn Cnv nσv? Dn σh? Dnd Dnh 3. Examples: BF 3 D 3h NH 3 C 3v H 2 C=C=CH 2 D 2d Symmetry Page 6
Symmetry and group theory
Symmetry and group theory or How to Describe the Shape of a Molecule with two or three letters Natural symmetry in plants Symmetry in animals 1 Symmetry in the human body The platonic solids Symmetry in
More informationLecture 34: Symmetry Elements
Lecture 34: Symmetry Elements The material in this lecture covers the following in Atkins. 15 Molecular Symmetry The symmetry elements of objects 15.1 Operations and symmetry elements 15.2 Symmetry classification
More informationGroup Theory and Chemistry
Group Theory and Chemistry Outline: Raman and infra-red spectroscopy Symmetry operations Point Groups and Schoenflies symbols Function space and matrix representation Reducible and irreducible representation
More informationThe Unshifted Atom-A Simpler Method of Deriving Vibrational Modes of Molecular Symmetries
Est. 1984 ORIENTAL JOURNAL OF CHEMISTRY An International Open Free Access, Peer Reviewed Research Journal www.orientjchem.org ISSN: 0970-020 X CODEN: OJCHEG 2012, Vol. 28, No. (1): Pg. 189-202 The Unshifted
More information1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
More informationGroup Theory and Molecular Symmetry
Group Theory and Molecular Symmetry Molecular Symmetry Symmetry Elements and perations Identity element E - Apply E to object and nothing happens. bject is unmoed. Rotation axis C n - Rotation of object
More information12. Finite figures. Example: Let F be the line segment determined by two points P and Q.
12. Finite figures We now look at examples of symmetry sets for some finite figures, F, in the plane. By finite we mean any figure that can be contained in some circle of finite radius. Since the symmetry
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of
More informationC 3 axis (z) y- axis
Point Group Symmetry E It is assumed that the reader has previously learned, in undergraduate inorganic or physical chemistry classes, how symmetry arises in molecular shapes and structures and what symmetry
More informationTWO-DIMENSIONAL TRANSFORMATION
CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization
More informationChapter 18 Symmetry. Symmetry of Shapes in a Plane 18.1. then unfold
Chapter 18 Symmetry Symmetry is of interest in many areas, for example, art, design in general, and even the study of molecules. This chapter begins with a look at two types of symmetry of two-dimensional
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationMohr s Circle. Academic Resource Center
Mohr s Circle Academic Resource Center Introduction The transformation equations for plane stress can be represented in graphical form by a plot known as Mohr s Circle. This graphical representation is
More informationIn part I of this two-part series we present salient. Practical Group Theory and Raman Spectroscopy, Part I: Normal Vibrational Modes
ELECTRONICALLY REPRINTED FROM FEBRUARY 2014 Molecular Spectroscopy Workbench Practical Group Theory and Raman Spectroscopy, Part I: Normal Vibrational Modes Group theory is an important component for understanding
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationMolecular Symmetry 1
Molecular Symmetry 1 I. WHAT IS SYMMETRY AND WHY IT IS IMPORTANT? Some object are more symmetrical than others. A sphere is more symmetrical than a cube because it looks the same after rotation through
More informationMOLECULAR SYMMETRY, GROUP THEORY, & APPLICATIONS
1 MOLECULAR SYMMETRY, GROUP THEORY, & APPLICATIONS Lecturer: Claire Vallance (CRL office G9, phone 75179, e-mail claire.vallance@chem.ox.ac.uk) These are the lecture notes for the second year general chemistry
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More information2 Session Two - Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More information5.04 Principles of Inorganic Chemistry II
MIT OpenourseWare http://ocw.mit.edu 5.4 Principles of Inorganic hemistry II Fall 8 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.4, Principles of
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationPrimary Curriculum 2014
Primary Curriculum 2014 Suggested Key Objectives for Mathematics at Key Stages 1 and 2 Year 1 Maths Key Objectives Taken from the National Curriculum 1 Count to and across 100, forwards and backwards,
More informationAn introduction to molecular symmetry
44 An introduction to molecular symmetry 4.2 igure 4.1 or answer 4.2: the principal axis of rotation, and the two mirror pianes in H^O. (a) E is the identity operator. It effectively identifies the molecular
More informationAlgebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass
Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More information8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes
Solution by Inverse Matrix Method 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix algebra allows us
More informationMolecular Models in Biology
Molecular Models in Biology Objectives: After this lab a student will be able to: 1) Understand the properties of atoms that give rise to bonds. 2) Understand how and why atoms form ions. 3) Model covalent,
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationLecture L3 - Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationChapter 9. Chemical reactivity of molecules depends on the nature of the bonds between the atoms as well on its 3D structure
Chapter 9 Molecular Geometry & Bonding Theories I) Molecular Geometry (Shapes) Chemical reactivity of molecules depends on the nature of the bonds between the atoms as well on its 3D structure Molecular
More informationSolving simultaneous equations using the inverse matrix
Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationFactoring Patterns in the Gaussian Plane
Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood
More informationLCAO-MO Correlation Diagrams
LCAO-MO Correlation Diagrams (Linear Combination of Atomic Orbitals to yield Molecular Orbitals) For (Second Row) Homonuclear Diatomic Molecules (X 2 ) - the following LCAO-MO s are generated: LCAO MO
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationLMB Crystallography Course, 2013. Crystals, Symmetry and Space Groups Andrew Leslie
LMB Crystallography Course, 2013 Crystals, Symmetry and Space Groups Andrew Leslie Many of the slides were kindly provided by Erhard Hohenester (Imperial College), several other illustrations are from
More informationINVERSION AND PROBLEM OF TANGENT SPHERES
South Bohemia Mathematical Letters Volume 18, (2010), No. 1, 55-62. INVERSION AND PROBLEM OF TANGENT SPHERES Abstract. The locus of centers of circles tangent to two given circles in plane is known to
More informationCovalent Bonding and Molecular Geometry
Name Section # Date of Experiment Covalent Bonding and Molecular Geometry When atoms combine to form molecules (this also includes complex ions) by forming covalent bonds, the relative positions of the
More informationPrentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)
New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct
More informationLesson 3. Chemical Bonding. Molecular Orbital Theory
Lesson 3 Chemical Bonding Molecular Orbital Theory 1 Why Do Bonds Form? An energy diagram shows that a bond forms between two atoms if the overall energy of the system is lowered when the two atoms approach
More informationLecture Notes on Pitch-Class Set Theory. Topic 4: Inversion. John Paul Ito
Lecture Notes on Pitch-Class Set Theory Topic 4: Inversion John Paul Ito Inversion We have already seen in the notes on set classes that while in tonal theory, to invert a chord is to take the lowest note
More informationCommon Lay-up Terms and Conditions
Mid-Plane: Centerline of the lay-up. Plane forming the mid-line of the laminate. Symmetry: A laminate is symmetric when the plies above the mid-plane are a mirror image of those below the mid-plane. Symmetrical
More informationElements in the periodic table are indicated by SYMBOLS. To the left of the symbol we find the atomic mass (A) at the upper corner, and the atomic num
. ATOMIC STRUCTURE FUNDAMENTALS LEARNING OBJECTIVES To review the basics concepts of atomic structure that have direct relevance to the fundamental concepts of organic chemistry. This material is essential
More informationMATHS LEVEL DESCRIPTORS
MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and
More information12.510 Introduction to Seismology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 04/30/2008 Today s
More informationIntroduction to Fractions, Equivalent and Simplifying (1-2 days)
Introduction to Fractions, Equivalent and Simplifying (1-2 days) 1. Fraction 2. Numerator 3. Denominator 4. Equivalent 5. Simplest form Real World Examples: 1. Fractions in general, why and where we use
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationModule 3: Correlation and Covariance
Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis
More informationVisualizing Molecular Orbitals: A MacSpartan Pro Experience
Introduction Name(s) Visualizing Molecular Orbitals: A MacSpartan Pro Experience In class we have discussed Lewis structures, resonance, VSEPR, hybridization and molecular orbitals. These concepts are
More informationPhysics 112 Homework 5 (solutions) (2004 Fall) Solutions to Homework Questions 5
Solutions to Homework Questions 5 Chapt19, Problem-2: (a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic fields in Figure P19.2, as shown. (b) Repeat
More informationMolecular Models Experiment #1
Molecular Models Experiment #1 Objective: To become familiar with the 3-dimensional structure of organic molecules, especially the tetrahedral structure of alkyl carbon atoms and the planar structure of
More informationph. Weak acids. A. Introduction
ph. Weak acids. A. Introduction... 1 B. Weak acids: overview... 1 C. Weak acids: an example; finding K a... 2 D. Given K a, calculate ph... 3 E. A variety of weak acids... 5 F. So where do strong acids
More informationHybrid Molecular Orbitals
Hybrid Molecular Orbitals Last time you learned how to construct molecule orbital diagrams for simple molecules based on the symmetry of the atomic orbitals. Molecular orbitals extend over the entire molecule
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More information2. Spin Chemistry and the Vector Model
2. Spin Chemistry and the Vector Model The story of magnetic resonance spectroscopy and intersystem crossing is essentially a choreography of the twisting motion which causes reorientation or rephasing
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationDCI for Electronegativity. Data Table:
DCI for Electronegativity Data Table: Substance Ionic/covalent EN value EN Value EN NaCl ionic (Na) 0.9 (Cl) 3.0 2.1 KBr (K) 0.8 (Br) 2.8 MgO (Mg) 1.2 (O) 3.5 HCl (H) 2.1 (Cl) 3.0 HF (H) 2.1 (F) 4.0 Cl
More informationEXPERIMENT O-6. Michelson Interferometer. Abstract. References. Pre-Lab
EXPERIMENT O-6 Michelson Interferometer Abstract A Michelson interferometer, constructed by the student, is used to measure the wavelength of He-Ne laser light and the index of refraction of a flat transparent
More information1 of 79 Erik Eberhardt UBC Geological Engineering EOSC 433
Stress & Strain: A review xx yz zz zx zy xy xz yx yy xx yy zz 1 of 79 Erik Eberhardt UBC Geological Engineering EOSC 433 Disclaimer before beginning your problem assignment: Pick up and compare any set
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationMolecular-Orbital Theory
Molecular-Orbital Theory 1 Introduction Orbitals in molecules are not necessarily localized on atoms or between atoms as suggested in the valence bond theory. Molecular orbitals can also be formed the
More informationUnit 9. Unit 10. Unit 11. Unit 12. Introduction Busy Ant Maths Year 2 Medium-Term Plans. Number - Geometry - Position & direction
Busy Ant Maths Year Medium-Term Plans Unit 9 Geometry - Position & direction Unit 0 ( Temperature) Unit Statistics Unit Fractions (time) 8 Busy Ant Maths Year Medium-Term Plans Introduction Unit Geometry
More informationDNA Worksheet BIOL 1107L DNA
Worksheet BIOL 1107L Name Day/Time Refer to Chapter 5 and Chapter 16 (Figs. 16.5, 16.7, 16.8 and figure embedded in text on p. 310) in your textbook, Biology, 9th Ed, for information on and its structure
More informationUnit 3 (Review of) Language of Stress/Strain Analysis
Unit 3 (Review of) Language of Stress/Strain Analysis Readings: B, M, P A.2, A.3, A.6 Rivello 2.1, 2.2 T & G Ch. 1 (especially 1.7) Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering
More informationGeometric Optics Converging Lenses and Mirrors Physics Lab IV
Objective Geometric Optics Converging Lenses and Mirrors Physics Lab IV In this set of lab exercises, the basic properties geometric optics concerning converging lenses and mirrors will be explored. The
More informationTheoretical Methods Laboratory (Second Year) Molecular Symmetry
Theoretical Methods Laboratory (Second Year) DEPARTMENT OF CHEMISTRY Molecular Symmetry (Staff contacts: M A Robb, M J Bearpark) A template/guide to the write-up required and points for discussion are
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationFRICTION, WORK, AND THE INCLINED PLANE
FRICTION, WORK, AND THE INCLINED PLANE Objective: To measure the coefficient of static and inetic friction between a bloc and an inclined plane and to examine the relationship between the plane s angle
More informationGeometries and Valence Bond Theory Worksheet
Geometries and Valence Bond Theory Worksheet Also do Chapter 10 textbook problems: 33, 35, 47, 49, 51, 55, 57, 61, 63, 67, 83, 87. 1. Fill in the tables below for each of the species shown. a) CCl 2 2
More informationLogo Symmetry Learning Task. Unit 5
Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to
More informationNumeracy Targets. I can count at least 20 objects
Targets 1c I can read numbers up to 10 I can count up to 10 objects I can say the number names in order up to 20 I can write at least 4 numbers up to 10. When someone gives me a small number of objects
More informationEXPERIMENT 17 : Lewis Dot Structure / VSEPR Theory
EXPERIMENT 17 : Lewis Dot Structure / VSEPR Theory Materials: Molecular Model Kit INTRODUCTION Although it has recently become possible to image molecules and even atoms using a high-resolution microscope,
More informationModule 3 : Molecular Spectroscopy Lecture 13 : Rotational and Vibrational Spectroscopy
Module 3 : Molecular Spectroscopy Lecture 13 : Rotational and Vibrational Spectroscopy Objectives After studying this lecture, you will be able to Calculate the bond lengths of diatomics from the value
More informationAngles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
More informationHealth Science Chemistry I CHEM-1180 Experiment No. 15 Molecular Models (Revised 05/22/2015)
(Revised 05/22/2015) Introduction In the early 1900s, the chemist G. N. Lewis proposed that bonds between atoms consist of two electrons apiece and that most atoms are able to accommodate eight electrons
More informationLevel 1 - Maths Targets TARGETS. With support, I can show my work using objects or pictures 12. I can order numbers to 10 3
Ma Data Hling: Interpreting Processing representing Ma Shape, space measures: position shape Written Mental method s Operations relationship s between them Fractio ns Number s the Ma1 Using Str Levels
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary
Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:
More informationThe elements used in commercial codes can be classified in two basic categories:
CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for
More informationPaper 1 (7404/1): Inorganic and Physical Chemistry Mark scheme
AQA Qualifications AS Chemistry Paper (7404/): Inorganic and Physical Chemistry Mark scheme 7404 Specimen paper Version 0.6 MARK SCHEME AS Chemistry Specimen paper Section A 0. s 2 2s 2 2p 6 3s 2 3p 6
More informationThe Point-Slope Form
7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More informationThe purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law.
260 17-1 I. THEORY EXPERIMENT 17 QUALITATIVE STUDY OF INDUCED EMF Along the extended central axis of a bar magnet, the magnetic field vector B r, on the side nearer the North pole, points away from this
More informationRotation Matrices and Homogeneous Transformations
Rotation Matrices and Homogeneous Transformations A coordinate frame in an n-dimensional space is defined by n mutually orthogonal unit vectors. In particular, for a two-dimensional (2D) space, i.e., n
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationAP CHEMISTRY 2009 SCORING GUIDELINES
AP CHEMISTRY 2009 SCORING GUIDELINES Question 6 (8 points) Answer the following questions related to sulfur and one of its compounds. (a) Consider the two chemical species S and S 2. (i) Write the electron
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, Three-Dimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More informationObjectives After completing this section, you should be able to:
Chapter 5 Section 1 Lesson Angle Measure Objectives After completing this section, you should be able to: Use the most common conventions to position and measure angles on the plane. Demonstrate an understanding
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationMolecule Projections
Key Definitions ü Stereochemistry refers to the chemistry in 3 dimensions (greek stereos = solid). This science was created by Pasteur (1860), van Hoff et LeBel (1874). ü Stereisomers are isomeric molecules
More informationMy Year 1 Maths Targets
My Year 1 Maths Targets Number number and place value I can count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number. I can count in multiples of twos, fives and
More informationRESONANCE, USING CURVED ARROWS AND ACID-BASE REACTIONS
RESONANCE, USING CURVED ARROWS AND ACID-BASE REACTIONS A STUDENT SHOULD BE ABLE TO: 1. Properly use curved arrows to draw resonance structures: the tail and the head of every arrow must be drawn in exactly
More informationWe emphasize Lewis electron dot structures because of their usefulness in explaining structure of covalent molecules, especially organic molecules.
Chapter 10 Bonding: Lewis electron dot structures and more Bonding is the essence of chemistry! Not just physics! Chemical bonds are the forces that hold atoms together in molecules, in ionic compounds,
More informationExcel supplement: Chapter 7 Matrix and vector algebra
Excel supplement: Chapter 7 atrix and vector algebra any models in economics lead to large systems of linear equations. These problems are particularly suited for computers. The main purpose of this chapter
More informationVolumes of Revolution
Mathematics Volumes of Revolution About this Lesson This lesson provides students with a physical method to visualize -dimensional solids and a specific procedure to sketch a solid of revolution. Students
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More information