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In Figure 15.28, the logarithm of Er(t) versus the

Materials Science and Engineering: An Introduction | 9th Edition | ISBN: 9781118324578 | Authors: William Callister ISBN: 9781118324578 140

Solution for problem 15.7 Chapter 15

Materials Science and Engineering: An Introduction | 9th Edition

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Materials Science and Engineering: An Introduction | 9th Edition | ISBN: 9781118324578 | Authors: William Callister

Materials Science and Engineering: An Introduction | 9th Edition

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Problem 15.7

In Figure 15.28, the logarithm of Er(t) versus the logarithm of time is plotted for PMMA at a variety of temperatures. Plot logEr(10) versus temperature and then estimate its Tg.

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C117 Chapter 10 Notes- Chemical Bonding II: Molecular Shapes, Valence Bond Theory, and Molecular Orbital Theory 2-19-16  Molecular structure determines properties o Small structural changes cause large property changes o Structure includes:  Skeletal atom arrangement  Type of bonding between atoms  Shape of molecule- described in terms of distances and angles in 3 dimensions o Lewis bonding theory predicts shapes of molecules  Lewis Theory and Geometry o There regions of in molecules- electron groups − o Some regions result from shared bond pairs of valence between nuclei o Other regions from unshared lone pairs on single nucleus − o Lewis theory says these regions of groups should repel each other- negative charge  groups around central atom most stable when they are as far apart as possible  Basis for VSEPR theory- Valence Shell Electron Pair Repulsion  Resulting geometric arrangement allows predictions of molecular shapes and bond angles  The VSEPR Method o Draw Lewis structure o Determine number of groups around central atom − o Determine molecular geometry using VSEPR- find shape such that all pairs as spread out as possible  Electron group geometry considers all domains- lone and bonding pairs; one single/double/triple bond makes one region  Molecular geometry/shape considers only spatial arrangements of atoms  Electron Geometry − o We will consider up to 6 regions around central atom (can have 2-6) o These arrangements result in different geometries o For molecules with resonance, does not matter which structure you use; geometry at central atom will be same o 2 groups- linear (180°) o 3 groups- trigonal planar (120°) o 4 groups- tetrahedral (109.5°) o 5 groups- trigonal bipyramidal (120° and 90°)  Axial positions- above and below central atom  Equatorial positions- same base plan as central atom  Equatorial atoms less crowded  Lone pairs occupy equatorial plane first (up to 3 pairs) o 6 groups- octahedral (90°)  All positions equivalent −  Molecular/ geometry can be different  Molecular Geometry o Bond and lone pairs determine shape- where is central atom, is number of pendant atoms, is number of lone pairs on o First determine geometry; remove lone pairs to get molecular # Groups # Bonding # Lone # Pendant Approx. Bond Angles Molecular Shape = + Pairs Pairs Atoms 2 2 0 2 Linear 180 1 1 1 Linear 180 3 3 0 3 Trigonal Planar 120 2 1 2 Bent < 120 1 2 1 Linear 180 109.5 4 4 0 4 Tetrahedral 3 1 3 Trigonal Pyramidal 107.5 2 2 2 Bent 104.5 5 5 0 5 Trigonal Bipyramidal 90 and 120 4 1 4 Seesaw < 90 and < 120 3 2 3 T-shaped < 90 2 3 2 Linear 180 6 6 0 6 Octahedral 90 < 90 5 1 5 Square Pyramidal 4 2 4 Square Planar 90 3 3 3 T-shaped < 90  Bond Angles o If molecule contains bond pairs only, ideal bond angles correct o Angles change if any atom replaced by lone pair  Lone pairs occupy more −pace on central atom; their electron density is only on central atom, rather than on 2 atoms like bonding groups- pull harder o Relative sizes of repulsive forces:  Lone pair/lone pair strongest  Lone pair/bond pair intermediate  Bond/bond weakest o Bond pair angles smaller than expected o Ex. Tetrahedral geometry  4 Tetrahedral molecule, 109.5° between atoms  3 Trigonal pyramidal, 107.5° between atoms; − − angle reduced due to increased repulsion between lone pair and all 3 bonding pairs  2 Bent, 104.5° between atoms; two lone pairs increase repulsion more- − − angle reduced even more o Predicting exact values difficult despite general trends  Ex. Bent  Ex. Trigonal Planar  3D Shapes on Paper o Central atom put on plane of paper o Put as many other atoms as possible in same plane with straight line o For atoms in front of plane, use solid wedge o For atoms behind plane, use hashed/dashed wedge  Multiple Central Atoms o Describe shape around each separately o Ex. Methanol  Around - tetrahedral  Around - bent o Ex. Alkanes- all carbons tetrahedral; do not lie in straight line  Molecular Polarity o Ex. - is , has low density; is , has high density  Bonding pulled towards end o Ex. 2  Each bond polar, but dipoles cancel; nonpolar molecule o Ex. 2  Each bond polar; both sets of bonding pulled towards end  Net result is polar o Dipole moment - measure of molecular polarity  Net dipole moment = sum of all dipoles  Units- Coulomb meter = Debye ( )  Common values:  To be polar:  Must have polar bonds; electronegativity difference and bond dipole moments  Must have non-symmetrical shape; vector addition  Polarity affects intermolecular forces- boiling points, solubilities  To be nonpolar:  Must be and all must be identical 0 o 2 2 0 Linear o 3 3 0 Trigonal planar o 4 4 0 Tetrahedral o 5 5 0 Triangular bipyramidal o 6 6 0 Octahedral  OR must be “divisible” into nonpolar 0 shapes  olecules polar if don’t divide into nonpolar shapes, and:  > 0 when s equivalent: o Ex. 2 2 2 bent, polar o Ex. 3 3 1 trigonal pyramidal, polar  s differ: o Ex. 2 2 4 0 tetrahedral, polar o Ex. 4 5 0 triangular bipyramidal, polar  Polarities of Molecular Shapes  Assume identical s and identical polar bonds  Linear- nonpolar  Bent- polar  Trigonal planar- nonpolar  Tetrahedral- nonpolar  Trigonal pyramidal- polar  Molecular Polarity and Solubility  Recall like dissolves like  Recall polar and nonpolar ends of soaps/detergents  Problems with Lewis Theory o Generally predicts trends in properties, but does not give good numerical predictions (ie. bond strength/length) o Gives good first approximations for bond angles, but cannot be used to get actual angle o Cannot write one correct structure when resonance structures present o Often does not predict correct magnetic behavior of molecules (2e. paramagnetic, but Lewis structure predicts diamagnetic) o Valence bond theory and molecular orbital theory more sophisticated models − −  Valence bond theory- we know occupy atomic orbitals; atoms share when atomic orbitals overlap o Bond forms when singly-occupied atomic orbitals on two atoms overlap − o 2 shared in region of overlap must be opposite spin o Formation of bond results in lower PE for system o Ex. Overlap of orbitals in 2 o o Main Ideas  Atomic orbitals (AOs) can be hybridized- form hybrid AOs that can overlap with neighboring atom’s orbitals  Types of hybrid orbitals formed postulated based on observations  Number of hybrids formed = number of AOs mixed  Hybrid orbitals used for either making single bonds or to hold lone pairs  All have somewhat similar shape  Sets of identical hybrid orbitals form similar bonds  Hybrid orbitals follow VSEPR rules; get as far away as possible  Can understand bond angles and molecular shapes  Hybridization- mixing different types of orbitals in valence shell to make new set of equal-energy (degenerate) orbitals o Prepares AOs to maximize bonding; more bonds = more full orbitals = more stability 2 3 3 3 2 o Ex. , , , , o Same type of atom can have different hybrid types in different molecules  Particular kind that occurs is one that yields lowest energy for molecule  Number and type of AOs combined determines shape/orientation of hybrid orbitals o Electron Geometry Remaining Orbitals Mixed # and Hybrid Type Unhybridized Geometry + 2 + Linear 2 + + 3 Trigonal Planar 4 3 None Tetrahedral + + + + + + + 5 + + + Trigonal Bipyramidal 3 2 + + + + + 6 + + Octahedral  Expanded-octet molecules hypervalent- hybridization includes orbitals 3 o Orbitals  One + ,,= four 3  Intuitively gives tetrahedral shape  Ex. forms 4 orbitals −  Each orbital, 109.5° apart, holds 1  4 equivalent covalent bonds form, ie. 4   Ex. 3  Recall hybrid orbitals can hold lone pairs  Ex. 2 o Orbitals 2  One + three  remains unused  Each hybrid 120° apart; trigonal planar shape  Ex. Boron compounds (ie. ,3 , 3tc.)  3 equivalent covalent bonds o Orbitals  One + two along -axis  and not used  Each 180° apart; linear shape  Ex. Beryllium compounds (ie. , , etc.) 2 2  2 equivalent covalent bonds 3 o Orbitals  Atom with 5 groups  Uses empty orbitals from valence shell  −  Trigonal bipyramidal geometry; 120° and 90° bond angles  Ex. 5 o Orbitals −  Atom with 6 groups  Octahedral geometry  Ex. 6  Predicting Hybridization and Bonding Scheme o Draw Lewis structure − o Use VSEPR to predict geometry of central atom o Select hybridization that matches geometry  Sigma Bonds and Pi Bonds o Different ways for orbitals to overlap o Sigma () bond- 2 half-filled orbitals combine end-to-end horizontally; found only in single bonds  Can be standard or hybrid orbitals  to , to , hybrid to hybrid, to hybrid, etc.  Overlap of orbitals along internuclear axis  Always first to form  Have free rotation o Pi () bond- 2 half-filled orbitals combine end-to-end vertically; found in double and triple bonds  Must be parallel to each other and perpendicular to axis connecting two bonding nuclei  Between unhybridized orbitals  Forms second and third in multiple bonds  Weaker than bonds  Prevent rotation about bond axes: Molecule C/C Bonding C/C Rotation Ethane ( − ) Yes 3 3 Ethene ( 2 ) 2 , No Ethyne ( = ) ,, No  Non-rotating double bonds allow cis-trans isomerism to occur o In double bond: one bond, one bond o In triple bond: one bond, two bonds o Ex. Carbon molecules with multiple bonds  Tetrahedral centers- 4 ,3 2,6etc. 2  Trigonal planar centers- 2, 2,4etc.  Linear centers- 2 2, etc. o Ex. Ethene 2  Each C atom has head-to-head overlap of orbitals; bond  Each C also has sideways overlap of orbitals; bond −  Unhybridized orbitals each have 1  Each C has 2 orbitals that bond to H atoms o Molecules with Triple Bonds  Ex. 22(acetylene); hybridized   One bond, two bonds  Leaves two unhybridized orbitals on each ; each contains single to make bonds  and overlap to make bond between carbons and between C and H   Problems with Valence Bond (VB) Theory o Predicts many properties better than Lewis theory (ie. bonding schemes, bond strengths, bond lengths, bond rigidity)  Predicts many properties incorrectly (ie. magnetism o2 ) o Presumes electrons are localized in orbitals on atoms in molecule; does not account for delocalization  Molecular Orbital Theory o Valence atomic orbitals (AOs) combine to form molecular orbitals (MOs)- called linear combination of atomic orbitals (LCAO)  # AOs mixed = # MOs formed −  Valence fill MOs in molecules  Ex. atom has 1 valence orbital;2 has 2 MOs  Ex. has 4 valence orbitals (2 and 22; has 8 MOs o MOs can extend over entire molecules; not confined to pairs of atoms o Filling Molecular Orbitals  Arrange MOs in order of increasing energy; relative energies of MOs deduced from experiments  Each MO can hold up to 2 ; in filled MO, have opposite spins by Pauli Exclusion −  Electrons occupy lowest-energy orbitals first; use all valence to fill from “the bottom up” by Aufbau principle −  If there are equal-energy orbitals, 1 goes in each orbital with same spin before pairing any due to Hund’s rule o Bonding MOs- constructive interference; more stable/lower energy than original AOs − o Antibonding (*) MOs- destructive interference; less stable; have a node between atoms (area of 0 density) o Properties  Bond order related to difference between # electrons in bonding/antibonding orbitals  Only need to consider valence  May be a fraction  Higher bond order = stronger/shorter bonds  If bond order = 0, bond is unstable compared to individual atoms, no bond forms # − #  Bond order = 2 −  Substance is paramagnetic if its MO diagram has unpaired ; if all paired, diamagnetic o Ex. Consider 2 2 molecular orbitals produced by combining 1 atomic orbitals from each 2−0  Bond order = 2 = 1 o Ex. Find the bond order of 2.  2 + 2 = 4 − 2−2  Bond order = 2 = 0; as noble gas, does not bond o Interaction of Orbitals o Molecular Orbital Diagrams  Ordered by increasing energy; 2 and 2can change for different systems  For convenience, only use left one  Ex. 2, 2, 2 , 2  All but 2 paramagnetic 2−0  2bond order = 2 = 1 6−2  2bond order = 2 = 2 6−4  2bond order = = 1 26−6  2bond order = = 0 2 o Paramagnetic behavior of 2   Predicted diamagnetic by Lewis and Valence Bond models  Unpaired in 2molecular orbital predict paramagnetism, which is correct according to experiments o Heteronuclear diatomic molecules- simple model  Same MO energy ordering as diatomic molecules  Ex. vs. − −  has 9 valence , one unpaired in 2 5−0  Bond order = 2 = 2.5  has 10, all paired 6−0  Bond order = = 3 2  When combining AOs are identical and of equal energy, contribution of each AO to MO is equal  When combining different types and different energies, AO closest in energy to MO contributes more to MO  The more electronegative an atom, the lower in energy its orbitals  Lower energy AOs contribute more to bonding MOs; higher energy to antibonding o Polyatomic molecules- when many atoms combined, AOs of all atoms combine to make set of MOs, which are delocalized over entire molecule  Gives results that better match real molecule properties better than Lewis and VB theories  Ex. Ozone o Resonance structures explain properties o Geometry of each structure asymmetric o Multiple structures needed to get symmetric bonding picture as seen in experiments  There is delocalized MO over whole molecule o Holds 2 ; picture consistent with experiment −2  Ex. 3  Has 3 resonance structures; all bonds experimentally shown to be equal  There is delocalized MO with 2  Ex. Benzene  MOs spread around entire ring  Total of 6 in 3 different orbitals (each shown in picture)  MO theory consistent with symmetric structure/equal bonds found in benzene

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Chapter 15, Problem 15.7 is Solved
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Textbook: Materials Science and Engineering: An Introduction
Edition: 9
Author: William Callister
ISBN: 9781118324578

The answer to “In Figure 15.28, the logarithm of Er(t) versus the logarithm of time is plotted for PMMA at a variety of temperatures. Plot logEr(10) versus temperature and then estimate its Tg.” is broken down into a number of easy to follow steps, and 30 words. Since the solution to 15.7 from 15 chapter was answered, more than 633 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Materials Science and Engineering: An Introduction, edition: 9. This full solution covers the following key subjects: versus, logarithm, plotted, figure, loger. This expansive textbook survival guide covers 22 chapters, and 1041 solutions. The full step-by-step solution to problem: 15.7 from chapter: 15 was answered by , our top Engineering and Tech solution expert on 11/14/17, 08:41PM. Materials Science and Engineering: An Introduction was written by and is associated to the ISBN: 9781118324578.

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