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Instant NotesPhysical Chemistry
The INSTANT NOTES seriesSeries editorB.D.HamesSchool of Biochemistry and Molecular Biology, University of Leeds,Leeds, UKAnimal BiologyEcologyMicrobiologyGeneticsChemistry for BiologistsImmunologyBiochemistry 2nd editionMolecular Biology 2nd editionNeuroscienceForthcoming titlesPsychologyDevelopmental BiologyPlant BiologyThe INSTANT NOTES Chemistry seriesConsulting editor: Howard StanburyOrganic ChemistryInorganic ChemistryPhysical ChemistryForthcoming titleAnalytical ChemistryInstant Notes
Physical ChemistryA.G.Whittaker, A.R.Mount & M.R.HealDepartment of Chemistry,University of Edinburgh, Edinburgh, UK BIOS Scientific Publishers Limited, 2000
First published 2000This edition published in the Taylor & Francis e-Library, 2005.“To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection ofthousands of eBooks please go to http://www.ebookstore.tandf.co.uk/.”All rights reserved. No part of this book may be reproduced or transmitted, in any form or by anymeans, without permission.A CIP catalogue record for this book is available from the British Library.ISBN 0-203-00992-4 Master e-book ISBNISBN 1 85996 194 0 (Print Edition)BIOS Scientific Publishers Ltd 9 Newtec Place, Magdalen Road, Oxford OX4 1RE, UK Tel. 44 (0)1865 726286. Fax 44 (0)1865 246823 World Wide Web home page:http://www.bios.co.uk/Published in the United States of America, its dependent territories and Canada by Springer-VerlagNew York Inc., 175 Fifth Avenue, New York, NY 10010–7858, in association with BIOSScientific Publishers Ltd.Published in Hong Kong, Taiwan, Cambodia, Korea, The Philippines, Brunei, Laos and Macauonly by Springer-Verlag Hong Kong Ltd, Unit 1702, Tower 1, Enterprise Square, 9 Sheung YuetRoad, Kowloon Bay, Kowloon, Hong Kong, in association with BIOS Scientific Publishers Ltd.Consulting Editor: Howard StanburyProduction Editor: Paul Barlass
CONTENTSAbbreviationsixPrefacexiSection A— States of matter1A1 Perfect gases2A2 Molecular behavior in perfect gases6A3 Non-ideal gases11A4 Liquids16A5 Crystalline solids22A6 Diffraction by solids28Section B— Thermodynamics34B1 The first law35B2 Enthalpy41B3 Thermochemistry44B4 Entropy50B5 Entropy and change55B6 Free energy59Section C— Equilibria64C1 Fundamentals of equilibria65C2 Fundamentals of acids and bases74
C3 Further acids and bases77C4 Acid-base titrations82C5 Solubility87Section D— Solutions91D1 Non-electrolyte solutions92D2 Solutions95D3 Colligative properties97D4 Phase equilibria102D5 Phase diagrams of mixtures107Section E— Ionic solutions118E1 Ions in aqueous solution119E2 Thermodynamics of ions in solution123E3 Electrochemical cells127E4 Electrochemical thermodynamics131E5 Electrochemistry and ion concentration137E6 Macroscopic aspects of ionic motion142E7 Molecular aspects of ionic motion147E8 The motion of larger ions150Section F— Kinetics154F1 Empirical approaches to kinetics155F2 Rate law determination162F3 Energetics and mechanisms171F4 Formulation of rate laws179F5 Rate laws in action184
F6 The kinetics of real systemsSection G— Quantum nature of the physical world192201G1 Nuclear structure202G2 Applications of nuclear structure207G3 Quantization of energy and particle-wave duality212G4 The wave nature of matter220G5 The structure of the hydrogen atom231G6 Many-electron atoms240G7 Chemical and structural effects of quantization246G8 Statistical thermodynamics252Section H— Bonding259H1 Elementary valence theory260H2 Valence bond theory265H3 Molecular orbital theory of diatomic molecules I271H4 Molecular orbital theory of diatomic molecules II277H5 Strong solid phase interactions287H6 Weak intermolecular interactions295Section I— Spectroscopy303I1 General features of spectroscopy304I2 Practical aspects of spectroscopy309I3 Rotational spectroscopy315I4 Vibrational spectroscopy322I5 Applied vibrational spectroscopy327I6 Electronic spectroscopy333
I7 Photochemistry in the real world339I8 Magnetic resonance spectroscopy344Appendix— mathematical relations353Further reading356Index360
ABBREVIATIONSamuatomic mass unitBqBecquerelCiCurieDdebyeemfelectromotive forceESenzyme-substrate complexESRelectron spin resonanceggeradeGygrayLCAOlinear combination of atomicorbitalsLHleft handLHSleft hand sideNMRnuclear magnetic resonanceppmparts per millionRHright handRHSright hand sidermsroot mean squareTMStetramethylsilaneuungerade
PREFACEPhysical chemistry is an unexpected shock to many university students. From the semiempirical approaches of the school laboratory, first year undergraduates suddenly findthemselves propelled into an unexpected quagmire of definitions and equations. Worsestill, although the applicability of the subject is sometimes obvious, studying the behaviorof a particle in an infinitely deep well can seem nothing short of farcical on firstapproach.In any scientific discipline, a fundamental understanding is more important thanlearning lists, but this is probably more true in physical chemistry than in other branchesof chemistry. Let’s be clear from the outset—understanding is the key to physicalchemistry, but the maelstrom of mathematics often clouds the student’s ability to create acomprehensible mental model of the subject.As the authors of this text, we therefore found ourselves in a paradoxical situation—writing a book containing lists of facts on a subject which isn’t primarily about lists offacts. So although this book is primarily a revision text we did not wish it to be merely anencyclopedia of equations and definitions. In order that the conceptual content of thebook is given sufficient weight to aid understanding, we have limited the extent of themathematical treatments to the minimum required of a student. The rigorous argumentswhich underpin much of physical chemistry are left for other authors to tackle, with ourown recommendations for further reading included in the bibliography.Since our primary aim has been to produce a quick reference and revision text for allfirst and second year degree students whose studies include physical chemistry, we haverecognized that different aspects of the subject are useful in different fields of study. AsNMR spectroscopy is to a biochemist’s protein study, so is band theory to the solid statechemist, and thermodynamics to the chemical engineer. With this in mind, we havedrawn not just on our own teaching experiences, but have consulted with colleagues inthe life sciences and in other physical sciences. The rigor of the central themes has notbeen diluted, but the content hopefully reflects the range of scientists for whom physicalchemistry is an important supplement to their main interests.In organizing the layout of the book, we have aimed to introduce the various aspectsof physical chemistry in an order that gives the opportunity for continuous reading fromfront to back with the minimum of cross-referencing. Thus we start with the basicproperties of matter which allows us then to discuss thermodynamics. Thermodynamicsleads naturally into equilibria, solutions and then kinetics. The final sections on bondingand spectroscopy likewise follow on from the foundations laid down in the section onquantum mechanics. The background to a range of important techniques is included inthe appropriate sections, and once again this reflects the wide application of the subjectmatter as with, for example, electrophoresis and electro-osmosis.Whatever your background in coming to this book, our objective has been to use ourown perspectives of physical chemistry to aid your insight of the subject. Physical
chemistry is not the monster that it seems at first, if for no other reason than because alittle understanding goes a long way.We hope that this text contributes to helping you reach the level of understanding youneed. Understanding the world around you really is one of the thrills of science.Finally, we thank Kate, Sue and Janet for all their patience during the preparation ofthis book.M.R.Heal, A.R.Mount, A.G.Whittaker
Section A—States of matter
A1PERFECT GASESKey NotesA gas is a fluid which has no intrinsic shape, and which expandsindefinitely to fill any container in which it is held.The physical properties of a perfect gas are completely describedby the amount of substance of which it is comprised, itstemperature, its pressure and the volume which it occupies. Thesefour parameters are not independent, and the relations betweenthem are expressed in the gas laws. The three historical gaslaws—Boyle’s law, Charles’ law and Avogadro’s principle—arespecific cases of the perfect gas equation of state, which isusually quoted in the form pV nRT, where R is the gas constant.The pressure exerted by each component in a gaseous mixture isknown as the partial pressure, and is the pressure which thatcomponent would exert were it alone in that volume. For aperfect gas, the partial pressure, px, for nx moles of eachcomponent x is given by px nxRT/V.Dalton’s law states that ‘the total pressure exerted by a mixture ofideal gases in a volume is equal to the arithmetic sum of thepartial pressures’. The quantity nA/ntotal is known as the molefraction of component A, and denoted xA. It directly relates thepartial pressure, pA, of a component A, to the total pressurethrough the expression pA xAPtotal.Related topicsMolecular behavior in perfect gases (A2) Non-ideal gases (A3)GasesA gas is a fluid which has no resistance to change of shape, and will expand indefinitelyto fill any container in which it is held. The molecules or atoms which make up a gasinteract only weakly with one another. They move rapidly, and collide randomly andchaotically with one another.The physical properties of an ideal gas are completely described by four parameterswhich, with their respective SI units are: the amount of substance of which it is comprised, n, in moles; the temperature of the gas, T, in Kelvin; the pressure of the gas, p, in Pascal; the volume occupied by the gas, V, in m3.
Perfect gases3The four parameters are not independent, and the relations between them are expressed inthe gas laws. The gas laws are unified into a single equation of state for a gas whichfully expresses the relationships between all four properties. These relationships,however, are based on approximations to experimental observations, and only apply to aperfect gas. In what might be deemed a circular argument, a perfect gas is defined as onewhich obeys the perfect gas equation of state. In practical terms, however, adherence tothe perfect gas equation of state requires that the particles which make up the gas areinfinitesimally small, and that they interact only as if they were hard spheres, and soperfect gases do not exist. Fortunately, it is found that the behavior of most gasesapproximates to that of a perfect gas at sufficiently low pressure, with the lighter noblegases (He, Ne) showing the most ideal behavior. The greatest deviations are observedwhere strong intermolecular interactions exist, such as water and ammonia. The behaviorof non-ideal gases is explored in topic A3.The perfect gas equationsHistorically, several separate gas laws were independently developed:Boyle’s law; p.V constantat constant temperature;Charles’ law;at constant pressure;Avogadro’s principle;at constant pressure and temperature.These three laws are combined in the perfect gas equation of state (also known as theideal gas law or the perfect gas equation) which is usually quoted in the formpV nRTAs written, both sides of the ideal gas equation have the dimensions of energy where R isthe gas constant, with a value of 8.3145J K 1 mol 1. The perfect gas equation may alsobe expressed in the form pVm RT, where Vm is the molar gas volume, that is, the volumeoccupied by one mole of gas at the temperature and pressure of interest. The gas laws areillustrated graphically in Fig. 1, with lines representing Boyle’s and Charles’ lawsindicated on the perfect gas equation surface.The gas constant appears frequently in chemistry, as it is often possible to substitute for temperature, pressure or volume in an expression using the perfect gas equation—andhence the gas constant—when developing mathematical expressions.Partial pressureWhen two or more gases are mixed, it is often important to know the relationshipbetween the quantity of each gas, the pressure of each gas, and the overall pressure of themixture. If the ideal gas mixture occupies a volume, V, then the pressure exerted by eachcomponent equals the pressure which that component would exert if it were alone in thatvolume. This pressure is called the partial pressure, and is denoted as pA for component
Physical Chemistry4A, pB for component B, etc. With this definition, it follows from the perfect gas equationthat the partial pressure for each component is given by:px nxR T/Vwhere px is the partial pressure of nx moles of component x.The total pressure exerted by a mixture of ideal gases is related to the partial pressuresthrough Dalton’s law, which may be stated as,‘the total pressure exerted by a mixture of ideal gases in a volume is equal to thearithmetic sum of the partial pressures’.If a gas mixture is comprised of, for example, nA, nB, and nC moles of three ideal gases,A, B, and C, then the total pressure is given by:Ptotal pA pB pC nAR T/V nBR T/V nCR T/V (nA nB nC)R T/V ntotalR T/Vwhere ntotal is the total number of moles of gas, making this a simple restatement of theideal gas law.Fig. 1. Graphical representations ofthe ideal gas equations. (a) Boyle’slaw; (b) Charles’ law; (c) The surface
Perfect gases5representing the perfect gas equation.The locations of the lines from (a) and(b) are indicated on the surface.The partial pressure of component A divided by the total pressure is given by:pA/ptotal (nAR T/V)/(ntotalR T/V) nA/ntotalThe quantity nA/ntotal is known as the mole fraction of component A, and is denotedxA(Topic D1). The advantage of this quantity is that it is easily calculated, and allowsready calculation of partial pressures through the relation:pA xn ptotal
A2MOLECULAR BEHAVIOR IN PERFECTGASESKey NotesThe kinetic theory of gases is an attempt to describe themacroscopic properties of a gas in terms of molecular behavior.Pressure is regarded as the result of molecular impacts with thewalls of the container, and temperature is related to the averagetranslational energy of the molecules. The molecules areconsidered to be of negligible size, with no attractive forcesbetween them, travelling in straight lines, except during thecourse of collisions. Molecules undergo perfectly elasticcollisions, with the kinetic energy of the molecules beingconserved in all collisions, but being transferred betweenmolecules.The range of molecular speeds for a gas follows the Maxwelldistribution. At low temperatures, the distribution comprises anarrow peak centered at low speed, with the peak broadening andmoving to higher speeds as the temperature increases. A usefulaverage, the root mean square (rms) speed, c, is given byc (3RT/M)1/2 where M is the molar mass.According to the kinetic theory of gases, the pressure which a gasexerts is attributed to collisions of the gas molecules with thewalls of the vessel within which they are contained. The pressurefrom these collisions is given by p (nMc2)/3V, where n is thenumber of moles of gas in a volume V. Substitution for c, yieldsthe ideal gas law.Effusion is the escape of a gas through an orifice. The rate ofescape of the gas will be directly related to the root mean squarespeed of the molecules. Graham’s law of effusion relates the ratesof effusion and molecular mass or density of any two gases atconstant temperatures:
Molecular behavior in perfect gases7The mean free path, λ, is the mean distance travelled by a gasmolecule between collisions given bywhere σ is the collision cross-section of the gas molecules.The collision frequency, z, is the mean number of collisionswhich a molecule undergoes per second, and is given by:Related topicsPerfect gases (A1)Non-ideal gases (A3)The kinetic theory of gasesThe gas laws (see Topic A1) were empirically developed from experimentalobservations. The kinetic theory of gases attempts to reach this same result from amodel of the molecular nature of gases. A gas is described as a collection of particles inmotion, with the macroscopic physical properties of the gas following from this premise.Pressure is regarded as the result of molecular impacts with the walls of the container,and temperature is related to the average translational energy of the molecules.Three basic assumptions underpin the theory, and these are considered to be true ofreal systems at low pressure:1. the size of the molecules which make up the gas is negligible compared to the distancebetween them;2. there are no attractive forces between the molecules;3. the molecules travel in straight lines, except during the course of collisions. Moleculesundergo perfectly elastic collisions; i.e. the kinetic energy of the molecules isconserved in all collisions, but may be transferred between them.The speed of molecules in gasesAlthough the third premise means that the mean molecular energy is constant at constanttemperature, the energies, and hence the velocities of the molecules, will be distributedover a wide range. The distribution of molecular speeds follows the Maxwelldistribution of speeds. Mathematically, the distribution is given by:where f(s)ds is the probability of a molecule having a velocity in the range from s to s ds,N is the number and M is the molar mass of the gaseous molecules. At low temperatures,
Physical Chemistry8the distribution is narrow with a peak at low speeds, but as the temperature increases, thepeak moves to higher speeds and distribution broadens out (Fig. 1).Fig. 1. The Maxwell distribution ofspeeds for a gas, illustrating the shiftin peak position and distributionbroadening as the temperatureincreases.The most probable speed of a gas molecule is simply the maximum in the Maxwelldistribution curve, and may be obtained by differentiation of the previous expression togive:A more useful quantity in the analysis of the properties of gases is the root mean square(rms) speed, c. This is the square root of the arithmetic mean of the squares of themolecular speeds given by:The rms speed is always greater than the most probable speed. For oxygen molecules atstandard temperature, the most probable speed is 393 m s 1 and the root mean squarespeed is 482 m s 1.
Molecular behavior in perfect gases9The molecular origin of pressureIn the kinetic theory of gases, the pressure which a gas exerts is attributed to collisions ofthe gas molecules with the walls of the vessel within which they are contained. Amolecule colliding with the wall of the vessel will change its direction of travel, with acorresponding change in its momentum (the product of the mass and velocity of theparticle). The force from the walls is equal to the rate of change of momentum, and so thefaster and heavier and more dense the gas molecules, the greater the force will be. Theequation resulting from mathematical treatment of this model may be written as:where n is the number of moles of gas in a volume V (i.e. the density). This equation maybe rearranged to a similar form to that of the ideal gas law: PV n Mc2/3.Substituting for c, yields PV n m(3RT/m)/3 nRT, i.e. the ideal gas law.Alternatively, we may recognize that the value ½Mc2 represents the rms kinetic energyof the gas, Ekinetic, and rewrite the equation to obtain the kinetic equation for gases:EffusionEffusion is the escape of a gas through an orifice. The rate of escape of the gas is directlyrelated to c:where ρ is the density of the gas. For two gases at the same temperature and pressure, forexample nitrogen and hydrogen, it follows that the ratio of the velocities is given by:This is Graham’s law of effusion.Mean free pathGas particles undergo
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