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# PhysicalChemistry CHM453

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This 67 page Class Notes was uploaded by Blanca Krajcik on Thursday October 29, 2015. The Class Notes belongs to CHM453 at Wright State University taught by PaulSeybold in Fall. Since its upload, it has received 15 views. For similar materials see /class/231123/chm453-wright-state-university in Chemistry at Wright State University.

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Date Created: 10/29/15

Statistical Thermodynamicsl Partition functions are the key to calculating the thermodynamic properties of a system In the text the following relationships are denved Thermodynamic Quantities in Terms of the Partition Function 0 for the System Ban p I 6T v aanm V V SekBT ajw 39Vk3111 A U0 kBT1n Q U DbkBT2 4 3an I1 U k39T2 0 B 73 TmQNhT NT V Note that all of the above relationships depend on finding 0 for the system The Translational Partition Function 1 0 We use the rticleinapox mgdel to find the translational energy levels Etrans h23ml2an2 nyz quot22 aqa V The partition function therefore is qtrans Z esxlkT Z e eykT Z e ezIkT qx CW 12 The tranlational energy levels are very closely spaced relative to kT so we can convert the summation into an integral 2 e exkT z j e exden After substitution and evaluation qtrans Z kah232 V where V is the volume of the container The Vibrational Partition Function 3 0 The arational energy levels are non degenerate and evenly spaced 8 018 22 g The artition function is qvib 90m 32M eZEkT e38kT 1eae2ae3a39 1rr2r3 This is the form of a geometric series 0 m Multiply both sides by r qvibr r r2 r3 qvib 1 Rearrange and solve qvib 1 1 1 r 1 e39elkT 9 This holds for each of the vibrational modes 0 The vibratignal energy levels tend to be widely spaced compared to kT Thus Clvib is often a small number close to 1 Calculation of the Internal Energy of a Monatomic Gas For a monatomic gas the internal energy is entirely translational energy 0 We have the formula qtrans 21ckah232 V and Q qNINl So InQ N nq lnN Using Stirling s approximation an N nq N N lnN U U0 RT28 anBT Only nq has a Tdependence so we have a anlaT a constants nT32 InV8T 32a lnTBT 321T Thus Utrans U U0 32llT2 1T or Utrans 32RT The constantvolume heat capacity CV is cV awam 32R Calculation of the Translational Entropy For a hydrogen atom in a 1 m3 box at 300 K the translational partition function q10 3 0 Recall that the molar entropy is Sm U UolT k an 32R k an For one mole N L Avogadro s number and k an k lnquL Using Stirling s approximation k lnQ k Llnq LlnL L And since kL R k lnQ R Inq lnL R R lnqL R 0 So the molar entropy is Sm 52R Flln21tkaIh232 VlL Substituting V RTIP M mL taking M in grams and com bining constants one obtains SmJImolK 15R lnM 25R lnT 11517R This is called the SackurTetrode Equation The Rotational Partition Func on The rotati9na energy levels of a diatomic molecule are given by the rigid rotor approx imation Erot where l is the moment of inertia t The partition function is q Zle akT and since the rotational levels are closely spaced relative to kT the sum can be replaced by an integral m qrot I 2J1exp JJ 1 h28739l32 dJ 8n2lkBTh2 using integral tables qrot 8Tl32lkBTGh2 where o is the symmetry number The Partition Function Partition functions play a central role in statistical mechanics They are the key to calglating thermodynamic functions r 0 Mathematical definition q Z gi e ikT where g the degeneracy of energy level 8i Molecules will distribute themselves over the available energy levels according to the Eoltzman listribution Physical interpretation q is a measure of the number of thermally accessible energy states Note that it is a function of the pattern of energgLIevels 8 and the temperature T For a set of discrete levels q gOeOlkT 91981kT gzeEZIKT g1ee1lkT gzeEZIKT The Total Partition Function 0 To a good approximation the different 39 forms of molecular eneLQy are independent so that we can write 8total Etrans 8rot evib eel Since q X eS kT the sum in the exponents becomes a product qtotal eEjlItrans239 e39Ejkrrot239 e39Ekrvibzquot equotC39lkre OI CItotal qtransCIrotqvibqel Thus we can calculate each of these partition functions separately and multiply them Partition Functions Depend on Energy Level Spacing and Temperature Consider some simple twolevel systems 100 RT 939100 z 0 es 0007 e391 0368 e01 0904 qz39l q1007 q1368 q1904 39 Note that if the temperature increases the partition function increases Molecules can more easily reach the higher levels Partition Function for the Oxygen Atom 3910 The oxygen atorn has three lowlying energy levels 3P0 8 2261 cmquot1 3P1 8 1574 cm1 3P2 2 0 cm1 39 Recall that q Z gi eSikT The degeneracies of these levels are given by 9 2 1 where J is the subscript in the term symbol At T 25 C kT 2072 cm391 Thus the parti tion function q is q 5902072 3915742072 1 e 2261l2072 51 30468 1 0336 6739 System Partition Functions 0 So far we have discussed the molecular partition function g 0 We can also discuss the partition function g Q for an assembly of N identical molecules We call 0 the systemartition function We must determine whether the particles in the assembly are distinguishable or indistin guishable Crystalthe molecules can be distinguished by their positions 0 qN Gasthe molecules cannot be distinguished Q qNN The N accounts for the per mutations of the N molecules 0 We shall be mainly concerned with gases and thus with indistinguishable molecules What are Enzymes Biomolecules Proteins amino acids Nucleic acids nucleotides DNA RNA Carbohydrates sugars Lipids Word from the Greek en zyme in yeast Enzymes are normally proteins although some RNAs have been found to have catalytic abilities 39 A typical mammalian cell has about 10 000 different enzymes 0 Protein structure Primary sequence of amino acids Secondary alphahelix or beta sheet Tertiary folding of secondary structure Quaternary combining of subunits Molecular weights Ribonuclease 13000 most are 50000200000 Enzyme Terminology etc 0 Some terms Substrate the molecule acted upon Turnover number the number of molecules reacted per unit time per enzyme molecule range from lt 1 to gt 105 5quot Active site the region of the enzyme involved in catalysis 0 Classification According to tvne of reaction catalvzgd Hydrolytic Oxidationreduction According to location in cell Fixed in cell wall eg in mitochondria Freely soluble in cytoplasm According to shape most are globular spherical or eggshaped 0 Some enzymes are highly specific in their action others are more general Urease urea H20 9 CO2 2NH3 but won t work on substituted ureas Pepsin hydrolyzes peptide bonds near aromatic amino acids but won t work on D amino acids only L Enzyme Kinetics In 1913 Leonor Michaelis and Maud L Menten proposed a mechanism for enzyme action k1 ka ES ES gtPE k2 Michaelis was a Germanborn American biochemist He was at the Univ of Berlin when the mechanism was proposed 0 Application of the steadystate approxima tion to the enzymesubstrate complex ES gives dESdt k1ES k2ES k3ES 0 E5 k1E1Sk2 kg The total enzyme concentration E0 consists of free enzyme E and bound enzyme ES E0 E ES 581 lemonsk2 k3 k1 81 The reaction rate is v k3ES Rearranging gives V kaEgS vmgxs k2k3k1 S KM S This is the MichaelisMenten equation The MichaelisMenten Equa on The MichaelisMenten equation V vmaxs KM g contains two important constants Vmax the maximum rate k3Eo KNI the Michaelis constant kk3k1 0 There are two simple ways to interpret the Michaelis constant KM is the substrate concentration at which the rate is half the maximum rate the dissociation constant of the enzyme substrate complex The Order of an Enzyme Catalyzed reaction 0 There are two limiting cases for the MichaelisMenten equation V VmaxISI KM 5 At low substrate concentration KIVI gtgt S so V z VmaxKMS and the reaction is first order in S At verv hidh substrate concentrations S gtgt KM and V z Vmax The reaction is zeroth order The rate in this case is limited by the availability of the enzyme E0 Values of KM and the Turnover Number For most enzymes KM lies between 10391 and 10395 M KM values of some enzymes Enzyme Substrate KM Chymotrypsin AcetylLtryptophanamide 5 x 10 3 M Lysozyme HexaNacetylglucosamine 6 x 10 6 M BGalactosidase Lactose 4 X 10 3 M Threonine deaminase Threonine 5 x 10 3 M Carbonic anhydrase C02 8 x 10 3 M Fenicillinase Benzylpenicillin 5 X 10quot5 M Pyruvate carboxylase Pyruvate 4 x 10 M HCO 1 x 10 3 M ATP 6 X 10 5 M Arginine tRNA synthetase Arginine 3 X 10 5 M tRNA 4 X 10397 M ATP 3 x 10 4 M The ratio VN gives the Maximum turnover numbers of some max enzymes fraction of enzyme sites filled with substrate 0 The turnover number is given by k3 It tells the number of substrate molecules converted to product per unit time when the enzyme is fully saturated with substrate Turnover number per second Enzyme Carbonic anhydrase 600000 3Ketosteroid isomerase 280000 Acetylcholinesterase 25000 Penicillinase 2000 Lactate dehydrogenase 1000 Chymotrypsin 100 DNA polymerase I 15 Tryptophan synthetase 2 Lysozyme 05 Figure 2513 Atkins PHYSCAL CHEMISTRY fifth edition 1994 P W Atkins W H Freeman and Company T128 LineweaverBurk Plots Taking reciprocals of the MichaelisMenten equation gives the LinemverBurk equation 1Iv KMNmax13 1V max 1N Slope KMVmax Intercept on V xaxis Intercept on y aXlS 1 max 18 0 Another way to plot the data is an Eadie Hofstee plot V k3 v EoS KM KME0 U Sometimes k3 is EMS written as km the turnover number LEnzyme Inhibition 0 Inhibition of enzymes is important and serves as a control mechanism in biological systems 0 Many drugs and toxic agents act by inhibiting enzymes Enzyme inhibitioncan be reversible or irreversible In the latter case the inhibitor often covalently binds to the enzyme Nerve gases poison the enzyme acetylcholinesterase which is needed for nerve impulse transmission 0 ln reversible inhibition the inhibitor is in rapid equilibrium with the enzyme Competi tive inhibitors usually resemble the substrate and bind to the active site Noncon39fetitive inhibitors attach elsewhere and decrease Vmax Competitive Substrate inhibitor Substrate 3 Noncompetitive V I inhibitor Competitive and Noncompetitive Inhibitors 0 Competitive inhibitors usually resemble the substrate Often the product of a reaction acts A asnhbtoro smce It resembles the reactant COO COO O 0 CH2 CHz ill OPOSZ Hg o CH2 coo 39 H CllI OH H c o PO32 COO HzC OPO3Z39 HZC OP032 succinate Malonate 13Diphosphogiycerate 23Diphosphoglycerate A competitive inhibitor A noncompetitive doesn t change Vmax inhibitor reduces Vmax Qompetitive Inhibitor Noncompetitive Inhibitor EES gtEP No inhibitor 1V present No inhibitor present 5 E fES gtEF ii i 539 s ESI 181 151 A double reciprocal plot of enzyme A doublereciprocal plot of enzyme kinetics in the presence and kinetics in the presence AAA and absence 00 of a competitive absence 000 of a noncompetitive inhibitor Vmax is unaltered inhibitor KM is unaltered by the non whereas KM is increased competitive inhibitor whereas Vmu is decreased Acetylcholinesterase Acetylcholine is an important neurotransmitter H3C39 acetyl choline Acetvlcholinesterase is an enzyme that breaks down acetylcholine to acetate ion CH3COO39 and choline HOCHZCH2NCH33 This switches off the nerve signal and allows the excitability of the postsynaptic nerve membrane to be restored Acetylcholinesterase has a very high turnover number 25000 sec1 which allows rapid restoration of the nerve The active site of acetylcholinesterase contains a negative group within a hydro phobic pocket A serine residue within the active site forms a covalent bond with the acetate portion of acetylcholine Choline is released and water then reacts with the acetyIenzyme to release acetate Inhibitors 39of Acetylcholinesterase Some poisons and nerve gases block the active site of the enzyme causing paralysis Different aliph ammonium ions inhibit ACase in different ways as seen in LineweaverBurk DILE lvxioI 39 20 39o 1 Competitive 15 3 Inhlbltlon LineweaverBurk plots for I I v n u l 39 O 2 the lnl39ubl10 by methyl 0 39 oooV lrieLhylammonium ion o i l 4 i Noncompetitive V 0 l I l I 1 0 o 4 Inhibition O o 3 2 LincWCaVer Burk plots for the O o r g o z 1 inhibition by dimethyldi i bu 17 IE 00 00 9 ammonium ion oO On Probability John Stuart Mill 18061873 English philosopher A System of Logic 1843 We must remember that the probability of an event is not a quality of the event itself but a mere name for the degree of ground which we or someone else have for expecting it Every event is in itself certain not probable if we knew all we should know positively that it will happen or positively that it will not But its probability to us means the degree of expectation of its occurrence which we are warranted by our present evidence Pierre Simon Laplace 17491827 French mathematician and astronomer 1814 The theory of chances consists in reducing all events of the same kind to a certain number of cases equally possible that is such that we are equally undecided as to their existence and in determining the number of these cases which are favorable to the event of which the probability is sought The ratio of that number to the number of all possible cases is the measure of the probability which is a fraction having for its numerator the number of cases favorable to the event and for its denominator the number of all cases which are possible What Can We Know 0 In 1812 Pierre Simon Laplace 17491827 the French mathematician and astronomer stated Theori Analytique de Probabilit Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings atoms and molecules which compose it if moreover this intell igence were vast enough to submit these data to analysis it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom to it nothing would be uncertain and the future as the past would be present to its eyes 0 There are two reasons this won t work Heisenberg s uncertainty principle prevents exact knowledge of position and momentum The calculations are too vast to be performed by a finite computer Back to Statistical Mechanics TlLermc ynamics lt gt Mistical Mechanics macroscopic microscopic bulk moles molecules n1n2PVT microstates Basic premise In the limit of very large numbers of particles or trials statistical averages become very strongly peaked JL 1 Low N Moderate N High N History Maxwell Boltzmann 18601900 Gibbs 1902 Einstein 19021904 Postulate The average value of a macroscopic quantity equals the average value of that property in the ensemble Microstates and Macrostates Microstate a specific microscopic arrangement of the particles of a system Macrostate a state of the system on the macroscopic scale usually defined by giving the pressure volume temperature Ensemble a collection containing a very large number of molecules g a very large number of identical systems Law of equal probability each microstate 39of system is equally probable Law of eg uipartition of energy the energy of a system is equally divided among the available degrees of freedom a classical law that needs modification due to quantum laws 39 Ergodic theorem the space average over many particles and the time average for a single particle for a property are the same Distributions A distribution is a ggeral way of distribut ing molecules over available energy levels A microstate is a specific arrangement of the molecules over the energy levels Because all microstates are assumed to be many probable the distribution with the most corresponding microstates will be the most probable distribution Example Consider a set of equallyspaced energy levels with three molecules and two quanta of energy 8 2 a b c 8 1 8 0 o o b C a C a b A distribution Microstates There are three microstates corresponding to the distribution shown Example of Distributions Consider four molecules and equally spaced energy levels If the total energy is 0 as might occur at T 0 there is iust one possible distribution with one microstate Note that the order of the molecules in a b c d an energy level does not matter 0 Now consider the same four molecules but with E 3 quanta of energy There are three pgssible distributions A is the most probable The number of microstates for a 39 distribution is Q N n1n2ng 39 O o o where N is the total QA 12 93 4 SEC 4 number of particles and the ni are those in each energy level 0A 4 12 and PA 06 211 20 Molecular StructureProperty Relationships One of the most basic concepts ofchemistry is that the structure of a compoundqeomet ric and electronicdetermines its bronerties We seek a mathematical relationship P fS where P is the property and S represents the structure But how can we describe the elusive idea of M13 in mathematical terms 0 We can define structural indexes numbers that descr me certain features of the structure For exmle the number of carbon atoms the number of terminal methyl groups etc A popular technique is multiple linear rgressm using indexes x1 x2 P a1x1 a2x2 where the a are regression coefficients It Forces Between Charged Species The potential energy V between two charges q1 and q2 separated by a distance r is Note that e is the dielectric constant of the solvent or environment This reduces the energy of interaction Water eg has an unusually high a value of 80 A classical force is given by the negative gradient of its potential F VV awar here F Al e 41teoe r2 The aove inversesquare dependence of the force on the distance is Coulomb s law after the French scientist Charles de Coulomb 17361806 who made the first measurements The LennardJones Potential Originated by the English physicist Sir John LennardJones Born John Jones he took the name LennardJones upon marrying Kathleen Lennard in 1925 0 The formula accounts for both repulsive forces approximated as AIM and attractive forces approximated as Blr5 VLJ 4 or12 or 5 where e is the well depth and o is the rvalue at which V 0 Potential energy van der Waals Forces 0 The Dutch physicist Johannes Diderik van der Waals 18371923 first explained the behaviors of real gases assuming intermolec ular attractive forces and finite volumes He won the Nobel Prize for Physics in 1910 The attractive forces are named in his honor There are three main types of attractive dimlar forces between neutral molecules Mnent dipolepermanent dipole or Keesom forces The energy of interaction is v 24422 1 45 mo M 180 r6 F M lM on forces a permanent dipole induces a dipole in a neighboring molecule Vina 1733 r16 a pinnrbilnt Liam gm rsion forces A fleeting charge asymmary induces a transient dipole vdis 39 C a1ggl 41ceo2 r6 where C depends on the ionization potentials Hydrogen Bonds These involve a tugofwar by two electro negative atoms struggling over a hydrogen atom eg NHo It can be viewed in terms of lone pairs of electrons competinq for the positivecharcLed QI OtOI39l quotD H CO Flg rine nitroqen and oxygen are the most common electronegative atoms Hydrogen bonds are individuallymuch waiter 1050 kJmol than covalent bonds 1amp1000 kJImol But collectively they exert great in uence especially in biological cmpds 0 Pauling has termed hydrogen bonds the most important structural feature for molecu lar biology 39 The Boltzmann Distribution The question How will molecules distribute themselves over the available energy levels Constraints There are N Z ni molecules and the total energy is E Z eini The distribution with the most microstates will be the most probable distribution etc 85 quot5 e4 quot4 83 quot3 82 n2 8 1 n1 Thwoblem To maximize the number of microstates 2 under the constraints of a constant number of particles N and a constant energy E The method Maximize In 2 using the method of Lagrange multipliers and Stirling s approximation Derivation of the Boltzmann Distribution We wish to maximize f In 2 under the constraints 91 N Zni 0 and 92 E 81m 0 o The Lagrange formula is a X1le9tz gz 0 an ani an Recall that Q N n1 n2 n3 an nN nn1 nn2 nn3 o The derivative of this is a mo alnN lnn1 nn2 an an anj anj anj ln nj 0 The constraints give egganj alanjmgmi E 8i Boltzmann formula derivation cont V The result is a 7L1Q917V2 92 0 ani anj an ltradition we let M 0L and k2 B Thus lnnj 0c Bej 0 so nnj 0L Bej we now raise e to the mwer on eaCh Sm 39 nj e expBei We call em A so that nj Aexp38j We need to evaluate A We know that 2 nj N so 2 AeBEi N Therefore A NZ expBei M Evaluation of 3 To evaluate 3 we combine statistical mech anics and thermodynamics We set Skan where S is the entropy Q is the statistical weight number of microstates for the distribution and k Boltzmann constant We again apply Stirling s approximation In 9 nN Inn1 nn2 lnn3l so S kNnN 2 nj nni Since nj eae ei 2 n nn Z quot106 B9 i 0LN 3E and s kN InN 0LN 3E 0 From thermodynamics BSBENN 1T From above BSBE kB so 1T k3 So 3 1kT The most probable distribution is therefore pi expl eiJQ N Z expl qkT The form of the Boltzmann Distribution In a complete treatment we must take into account the degeneracies gi of the energy levels This gives e39kT j g e I 39 N Z 398lkT The most common form is to take the ratio of the i111 and jm energy level populations quotMn gs9 e39A E kT o This distribution applies to huge numbers of molecules Note that the most probable distri bution dominates all others It is useful to define the partition functim q 2 398ikT Note that q is a function of both the energy level pattern and the temperature It will increase as T increases Y Henry Eyring 1901 1981 Born in Colonia Juarez Mexico Family moved to El Paso when Henry was 11 to avoid Mexican Revolution Studied engineering at the Univ of Arizona then worked for a mining company Received MS in metallurgy at Arizona then went to the Univ of California Berkeley Obtained PhD in 2 years Worked with Polanyi in Berlin Princeton Univ 19311946 Univ of Utah 1946 1981 Activated Complex Theory Basic Idea Reactions pass through an activated complex stage that can be treated with a combination of kinetic and thermodynamic approaches 39 Absolute reaction rate theory paper sent late to the J Chem Phys because Eyring was in an auto accident it was as finished a paper as I have ever written strict1y by accident The editor Harold Urey sent it to a reviewer who concluded that the theory was incorrect and couldn t be true The paper was rejected Hugh Taylor and Eugene Wigner persuaded Urey to reconsider and accept the paper It was published J Chem Phys 3 107 1935 and is now a classic TransitionState Thetmt Recall the distinction between thermo dynamics and kinetics Thermodynamics Kinetics Products Products Reactants Reactants One considers that the reaction proceeds over a Gibbs energy barrier as indicated by AG At the top of the barrier one finds the transition state or activated complex In the forward direction a pseudoequilib rium is established between reactants and activated complexes so that k v expAGr FIRT where v is a frequency factor Thermodvnamic Analvsis One assumes that the Gibbs energy of activation can be expanded in the usual way AG AH TASt so that k V 3331quot eAHynT One thus speaks of the entropy of activation AS and the enthalpy of activation AHi We will examine this expression more carefully when we study statistical mechanics One obtains the expression kf kBTh eAStR eAH1RT so that a plot of nkf vs 1T will show a slope equal approximately to AHtR Tranjsition State Them Consider the bimoleoular reaction A B gt C D AB T S AG thermo c o Assume the following mechanism 1 A B AB rapid equilibrium 2 AB a C D slow ratedetermining step We can express the equilibrium 1 as K AB AHB and the rate of formation of products as Rate k AB Thus the reaction rate can be written Rate k K AB TransitionState Theor cont The rate expression v k K AB is the standard expression for the rate of a bimolecular elementam reaction m an elementary reaction is a reaction at the molecular level a molecular event When we write an overall chemical reaction it may proceed by a number of elementary steps This set of elementary reactions is termed the mechanism of the reaction We can picture reaction 2 as occurring along some weak vibrational coordinate of the activated complex ABt Instead of vibrating the molecule separates to form the products C and D The frequency is k v 0 As noted before we can calculate K using statistical mechanics We ll do this later A comment on the sign of A31 for the reaction A B gt AB the activated complex in a gasphase reaction is always more order ed than the reactants so A i is negative Recall 5 138 SAB SA B T Basic Ideas of the TransitionState Theory There are some basic ideas associated with the transitionstate theory 1 An equilibrium is established between the reactants A and B and the activated complex AB 2 In order to form the products P the activated complex AB breaks up along a weak vibrational coordinate 39 We can express the equilibrium constant K between reactants and AB as K ABAB 0 We can further analyze this using statistical mechanics and partition functions q K q expAeolkT quB where q is the partition function for AB and A30 is the energy difference between A B and AB Statistical Mechanics and TransitionState Theory At equilibrium the species in an equilibrium will distribute themselves in a Boltzmann Estribution over the available energy levels I A80 I V Reactants A B Activated complex AB We can separate out the contribution of the weak vibration from the partition function q qweak 1 eXP39hVkT and since hv ltlt kT exphvlkT 1 hvkT Thus qweak lehv and qit kThvq So K E q expAeolkT hv E where q is q without the weak vibration Putting it together J Let us recall the TST mechanism A B AB gt ProductsJ The rate of formation of products is thus dPdt vAB where v is the frequency rate constant at which the activated complex AB breaks up 0 From our previous work AB KAB so that dPdt vKAB 39 Using our expression for K in terms of partition functions dP 3 expA okT dt h quB The partition functions for A B and AB can be calculated using quantum mechanical methods Thus one can calculate the rate expression from first principles hence the term absolute reaction rate theory Practical Application of TST There are a number of practical details to factor in at this point especially to get the units correct One divides by NA to get moles and by V to get the rate per unit volume 0 For a bimolecular gasphase reaction we use the ideal gas equation to give V FITPquot There are many complications but for atom atom collisions the preexponential part of k is 1012 molLs For more complicated species this decreases For two nonlinear molecules it is 107 molILs 0 TST for the reaction D H2 a DH H gives 55x109 cm3mols at 450 K The experimental value is 9x109 cm3mols The difference is mainly due to tunneling which contributes a factor of 2 at this temperature James Clerk Maxwell 18311879 0 Born in Edinburgh Sco and A humble deeply religious man A child prodigy in mathematics he contrib uted an original paper to the Edinburgh Society at the age of 15 He developed the kinetic theory of ses in 18591860 showing that gas molecules have a distribution of speeds at any temperature 0 In 1873 he published his Treatise on Electricity and Magnetism proposing the fundamental equations now called Maxwell s equations relating electric and magnetic fields 0 He suggested that light is an electromag netic wave later confirmed by Hertz 1888 Maxwell s relations in thermodynamics Maxwell s Demon Invented in 1867 in a letter to the physicist Peter G Tait An imaginary being Theory of Heat 1871 let us suppose that a vessel is divided into two portions A and B by a division in which there is a small hole and that a being who can see the individual molecules opens and closes this hole so as to allow only the swifter molecules to pass from A to B and only the slower ones to pass from B to A He will thus without the expenditure of work raise the temperature of B and lower that of A in contradiction to the second law of thermo dynamics Modern theories deny the possibility of such action and depend on the information necessary to detect the molecules Stirling s Approximation The value of N increases enormoust for large N Try 70 and 100 on your calculator In 1730 the Scottish mathematician James Stirling 16921770 proposed the following approximations for large N N z NNe39N21cN lnN z NII IN N the most useful form This becomes a very good approximation as N becomes large Actual Stirling f i Rel N l N lnN l lnN Error Error i l v 2 i l 10 36291106 1510 1303 207 137 i 5 J 100 9333X10157 36374 Y 36052 1 322 089 9 l 2 1000 34024x102567 591213 590776 437 3007470 1 1 Z 3 g j 5000 3759114 I 3758597 517 g 0013 i 39 l 1 may a The Method of Lagrange Multipliers The Lagrange method of undetermined multipliers was introduced by the French mathematician Joseph Louis Lagrange 17361813 0 The problem To find extrema maxima or minima of a function fx1x2x3xn under constraints g1x1x2xn O gzx1x2xn 0 etc Solution One solves the set of n equations Q k1 lk2832 A3anm 0 8x1 8x1 8x1 8x1 Q1 kzagz u 0 8x2 8x2 8x2 8x2 etc where the Xi are the undetermined multipliers Example of the use of Lagrange Multipliers Problem To maximize the area A xy of a rectangle under the constraint that its perimeter P 2x 2y is fixed Thus f xy and the constraint is g 2x 2y P0 O 0 Solution The equations are QAXB O yX2O ky2 8x 8x AAag0 xyl220 xy By By Conclusion When a rectangle has a constant perimeter P the maximum area is achieved if the two sides are equal ie it s a square A square has the maximum area in this case SeCond example of Lagrange Multpliers Suppose we want to maximize I the area of a rectangle inscribed within a circle of radius 1 From the diagram the area is A 4xy and the constraint is gx2y21O quot With f lg 4xy sz y2 1 the equations are 4y 21x 0 k 2yx 4x 2ylx2y 0 So 4x 4ny 0 and x2 y2 0 Thus again the maximum area occurs for a square x y Also since the constraint is x2 y2 1 2x2 1 and x y 12 0707 Note that the area of the rectangle is 4xy 2 and the area of the circle is m2 quot 713 So the ratio of the areas is 27 0637 The Random Walk George Gamow has pictured this as a drunkard s walk This applies to a great number of physical phenomena Diffusion Brownian motion Dispersion of fruit flies Escape of photons from the interior of the sun mamas walk 0 Each step of length L is followed by a step in a random direction In two dimensions the distance R from the starting point is given by R2 III2 Y1 Y2 2 0 As a result the average distance travelled is given by db LN where N is the number of steps if the number of steps per unit time is constant the average distance is proportional to the square root of the time ltRgt AVt Random Walk Applications Dispersion of fruit flies With time fruit flies disperse from a central release point They can be followed by for example marking them with fluorescent dust Their average distances are seen to follow the random walk model Diffusion of light from the center of the sun Photons move only about one cm between collisions Since light moves at 3x101o cmls it would take only about 2 s to travel the 700000 km radius of the sun in a straight line In diffusing out it takes about 3x103911s for each step Thus there are 5x1021 steps and it takes m for light to get out 0 m distances in polymers One can consider a polymer as composed as a series of N monomer links of length L There are two major modifications that must be taken into account The polym strand cannot cross its own path This requires a selfavoiding walk SAW analysis The polymer Ilnks often are restricted in the angles that they can take with respect to each other One can test the SAW model on a computer The average distance scales as a fractal dimension d as N z ltRgtd Simulations give d z 133 Note that for a regular random walk d 2 exactly i 1 GI W39M Z Mmdrmq um x 92 Random Walks 0 steps 50 steps 30 5 10 0 10 20 30 4D 50 461 J1v y1 uvyILn m munx quotMVIVW L ugm Hm 71 Tubie 1 Mean distances traveled from release point and variance of the distribution of released ies about the release point from results obtained in 1945 using orange eyed Dramth mutants Day after Flies Mean distance re recaptured meters Variance m t 85 1 861 501 4096 1 309 2 m 622 7225 a 539 3 W 897 14161 2 892 4 351 882 15376 1 1144 395 311 1139 23409 1 1877 847 1215 28561 2 2168 scrum Duh presented in Dobzhanskyand Wright 1947 FRUr FCV D 39 1952570 y s 39224 5070 R 098 O I i 1 0 1 3 4 ml In day m L Mun distances traveled from mlenu pain Ind variance of the distribution I dud ies about the rel point ram nuns obhlned tn 1945 using orange ud WI mutants Muslin Flinn Maul M W neale macu Vwimce m39 2 SE 1 861 501 4096 x 309 I 2 7225 2 539 I 504 891 14161 a 4 861 38 15 376 2 1 144 I 311 113 23409 2 1877 I 3 7 1216 28561 2 2168 m M mud n Dabzhanskjmd Wright 1947 Personl Invo lug continued academicians have been formulth Wall Street39s money managers and investment adviser with 1 known as the random walk I centh thought is that price Ian do not follow patterns or trends Ind Hiera fore cnnnot be predicted ell of which strongly implies that conventional money management and investment advice aren39t worth much The concept was rst tested and found to be valid on commodity fu tures contracts Then a number of ac ademic researchers found that it ap plied to stocks More recently there terest rates are random too Now comes the nal insult the ran dom walk also applies to the prices of stock exchange seats the admission tickets to the Wall Street cartel That conclusion was reached in a study by G William Schwert an assistant pro fessor at the University of Roches ter s Graduate School of Manage ment Schwert studied the movements of New York and American Stock Ex change seat prices from 1926 through has been evidence that changes in in 39 A lm In Mklxchnnlo Seats I For some twentyodd years now Mend found that Brenton unced Vestment hen stocks Wall Street s amend randomly Despite the connotation of sense lsss meanderinz a random m1de n t involve prices being set randomly The concept requires only Host price changes be random The undedyilll theory holds that changes are millions ic unpredictable because they come in response to new information It39s an efficient market Schwert found that all this applies to exchange seats as well A one might expect seat prices move in step with the levels of stock prices and trading volume Intuitiverl it would seem that someone trading in the thin market for exchange seats could pro t by acting quickly on the news about these matters Schwert s nd ings show that any such pro ts would be hard to come by The market for seats is an e icient one and seat prices themselves adjust quickly to the news 39 39 Schwert also found that seat prices are more volatile than stock prices and thus seats are a much riskier in standard measure of risk is the 3 called beta which is essentially39 a measure of volatility A stock wit 13 between 1926 and 1972 Being broker on the Amex is even riskie the beta of its seats was 18 Some of the sharp movementsi sent prices originate in Washington rather than Wall Street Back iii March of 1934 for instance the pricefi f of an NYSE seat dropped fromy 190000 to 83000 after the Seedquot 39 ties and Exchange Act of 1934 was i 7L troduced in Congress In June of this year an NYSE seat sold for a mere 50000 d0wn from more than 500 000 in 1968 Once again the news from Washington had a lot to do with the decline The current depressed level re ects the advent of negotiate commissions two years ago and5fear that Congress will eventually oree theoreation of a centralmarketifgr stock trading If those fears prove correct seat values are aptrtof walk down to a lower levelthaneve same fteen years was 13 percent for taxexempt investors versus the 22 percent they could have had from stocks Investors in the higher tax bracket got tor s bane And all the minus signs in the tables showing returns on govern ment securities con rm another econom ic postulate one that was laid out by ing to Wall Street s yield cult ll 39 that began emerging inf1974 l iid been picking up adherents fever slnee l particularly among 39pension39 Potential Energy Surfaces j For a nonlinear molecule with N atoms there are 3 degrees of translational freedom for the whole molecule 3 degrees of rotational freedom 3N6 internal degrees of freedom remaining The most stable geometry is a minimum with respect to all of the 3N6 internal de grees of freedom internal coordinate A reaction proceeds along one coordinate for which the energy is a maximum In a multidimensional space the potential energy surface reaction coordinate is a hypersurface and the transition state is a maxi mum with respect to the reaction coordinate but a minimum with respect to all the other coordinates saddle pointquot Eotential Energy Surfaces Cont Thus we picture the transition state as a saddle point on a multidimensional otential ener surface sort of a high spot on a pass between the mountains We imagine that reactants must have enough energy an activation energy to get up and over the high spot Tunneling can also occur and for certain types of reactions eg electron transfers hydrogen abstractions this passage through the barrier may be more important than the classical process of going over the barrier One reaction that has been very much studied both experimentally and theoretly is the reaction between a hydrogen atom and a hydrogen molecule A hydrogen abstraction this is the simplest reaction possible in which one bond is broken and another is formed For the linear encounter the activation energy is 42 kJmol and the linear transition state has both HH distances at 93 Pm the bond distance in molecular hydrogen is 741 pm Potential Energy Surfaces HAHB HC HAHBHc l Tab 3964 l l 3quot O REC10 5cm P on 10 12510a cm Figure 193 Potential energy surface for the reaction HA HBHC HAHB Hg for a linear approach and departure HA HBHC gt HAHB Hg The potential energy surface for this reaction for 6 180 is described by means of a contour diagram in Fig 193 This surface has been calculated using ab initio methods with con guration interaction as discussed in Section 114 and the error at any point on the surface is behaved to be less than 003 eV 29 kJ m011 As HA approaches HBHC along the minimum energy path the potential energy of the system increases until the saddle point is reached at At this point RAB REC 93 pm and the potential energy of the system is 043 eV 42 kJ Incl 1 the highest along the dashed line Since the saddle po39mt is 043 eV higher than the potential of HA and HBC at an in nite dis tance this energy must be supplied from relative kinetic energy or vibrational energy in order for the reaction to occur In the upper rioht hand corner of Fig 193 there is a high plateau with energy of 432 kJ mol l l l l a Ra bl Figure 194 a Trajectory for a nonreactive inelastic collision of a hydrogen atom and a hydrogen molecule 7 Trajectory for a reactive collision From P Siegbalm and B Liu J Chem Phys 682457 1978 and C J Horowitz Chem Plzyx 682466 3978 l

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