Class Note for CHEM 476 at UMass
Class Note for CHEM 476 at UMass
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Date Created: 02/06/15
Chem 476 Spring 2009 Statistical Thermodynamics and Kinetics Lecture Notes 1 M D Barnes Overview of Concepts and Constructs from nonstatistical Thermodynamics I Introduction Thermodynamics 7 the physics of ensembles of particles 7 is a strange and funny thing and our experience this spIing semester will be quite different in terms of concepts and mathematical procedures than our experience with quantum chemistry last fall On the surface many aspects of thermodynamics seem intuitive and simple A child understands that when you place a 39hot thing39 n quotcontactquot with a 39cold thing39 the cold thing becomes warmer and the hot thing gets cooler We intuitively grasp ideas of direction of spontaneous physical change For example we know that a gas sample doesn39t spontaneously compress and a rubber ball placed on a table won39t spontaneously start bouncing if we heat up the table Thermodynamics deals with macro7 scopic things we see perceive and experience every day but quantifying the behavior of ensemble systems 7 and understanding the connection to microscopic properties of the material that makes up the ensemble is not a trivial task at all The apparent disconnection between Thermodynamics and the quantum mechanics of atoms and molecules derives in part because of its historical development preceding the microscopic theory of electrons atoms and molecules As we will see much of the quotfine7grainedquot quantum behavior of atoms or molecules is invisible in the limit when a large number of quantum states are populated from our experience last fall this should strike you as very strange If the ensemble is made up from molecules with unique quantum properties distinct electronic translational vibrational and rotational states how come the changes in thermo7 dynamic properties don39t seem to care about that This is a very strange thing 7 and as we go through the course we will unders7 tand why this happens As a result the basic constructs of thermodynamics lnternal Ensemble Energy U Enthalpy H the Entropy S Helmholtz Free Energy A and the Gibbs Free Energy G require no reference to the identity or properties of the atomicmolecular species that make up the ensemble NonStatistical Thermodynamics expresses differential changes in these thermodynamic quot State Func7 tionsquot in tenns of any two quot State Variablesll of particles N Temperature T Volume V or Pressure P Here I very briefly summarize the basics of nonstatistical thermodynamics that will serve as a framework for our discussion of the statistical mechanics of ensembles As we will see the energy con guration of the ensemble that is finding the probability that an atom or molecule occupies a particular quantum state with energy E provides a complete description of the thermody7 namic state functions U H S A G not just the differential or integrated changes offered by nonstatistical thermodynamics Thus the c01respondence between the statistical and nonstatistical models is an important one thus I offer a brief and necessar7 ily somewhat superficial overview of the basic concepts and constructs of nonstatistical thermodynamics here 2 l Chem47575pr097NSthermorev 01 nb I The quotsystemquot state variables and equations of state I System and State Variables The state of our thermodynamic system closed collection of particles is defined by the following State Variables N gt particles in the system T gt Absolute temperature V gt Volume m3 P gt Pressure Forcearea kg m 1 5 2 I equatinns nf state and Ideal Gas Law The equation of state of our thermodynamic system relates one state variable in terms of the other 3 that is we can indepen7 dently specify 3 of the 4 state variables eg PfN T V or VfNT P etc If we know the equation of state deduced empirically for ideal gases then we know how one of the state variables changes as we change the other 3 The IDEAL GAS LAW derived empirically in the late 170039s provides a simple equation afstale to use as a basis for develop7 ing relations for thermodynamic state functions U H S A G It reads V V P N k T nR T where k is Boltzmann39s constant k 138 x 10 23 JK and the other perhaps more familiar form n is the number of moles and R is the Gas Constant R 8314 Jmol K 7 it39s just k multiplied by Avogadro39s number There are two basic assumptions of the ideal gas law 1 The particles in the ensemble DO NOT interact with each other 2 The particles are considered as point masses they have no volume There are lots of different equations of state that have been developed to account more accurately for particle interactions and finite molecular volume We39ll deal with some of these later in the homework One of the most interesting things that we will do in this semester in my view is to show how equations of state for any species 7 and at any level of detail in terms of molecular interactionsvolume 7 can be derived from statistical thermodynamics Chem476Spr09NSthermorev 01 nb I3 I Ensemble Internal Energy for an Ideal Gas The 1st Law of Thermodynamics I gas kinetic picture An interesting and somewhat subtle aspect of the ideal gas law is that PV has units of energy From a knowledge of the pressure you measure it the volume and temperature we know the number of particles Suppose we have a collection of gas particles in a cylinder with volume V at temperature T and piston head at the end of the cylinder with area A F ueEmquotisiunJa c l isiiun fsecla a ph mlli i n E m w The pressure has units of force in X directionarea which can be expressed as A px 11 collisions 1 P Fx Area collision unit time A the change in momentum A pX from a single collision is Apx 2mvx where vX is the X component of the velocity Since the measured pressure results from many different collisions we don39t know vX for every collision we can replace vX with an ensemble average ltvXgt The factor of 2 comes from the fact that the collision with the piston head is elastic momentum is conserved so the momentum imparted TO the piston is 2 m ltvXgt m is the mass of the particle Now the number of collisions that occur per unit time on the piston head can be computed by noting that the number of particles N39 that collide with the piston head is given by ltVXgt unit time A So 11 collisions V 39 1 N unit time V unit time the term here represents the volume fraction containing particles that will encounter the surface of the piston head in unit time V39 lt vx gt A unit time V V lt vx gt A unit time 1 V unit time SO 4 l Chem47575pr097NSthermorev 01 nb ncollisions lt v gt A unit time 1 unit time V unit time ncollisions lt vX gt A 1 n unit time 2 V because only 12 of the particles with lt vX gt are going in x direction So we can express the pressure in terms of the average speed of the particles in the x directions as 1 lt vX gt A 1 II P 2mltvlgt n mltvgt2 2 V A V and if we wanted to be really picky about it we would note that the quantum mechanical expectation value lt vX gt is zero so better to replace lt vX gt2 with lt vx2 gt Comment on elaxsical vs quantum mechanical uncertainties It39s important to bear in mind an important distinction between classical and quantum mechanical uncertainties and expectation values In this problem we are talking about an ensemble averaged lt vX gt whose uncertainty spread in V values comes from the fact that we have a broad range of different classical states each with a different trajectory The uncertainty spread in the corresponding quantum observable is there even if we prepare every particle in the precise same quantum state Thus ensemble averages of quantum observables are often given using the notation such as ltlt A gtgt where the inner braeket denotes the quantum expectation value of the observable A averaged over all coordinate space of whatever wavefunction we use and the outer brackets denote an average over all possible different states I Kinetic Energy nf an Ideal Gas In the previous section we related the pressure of a system to one component of the meanesquare velocity Combining this result with the equation of state for an ideal gas gives us the ensemble kinetic energy for the system For an Ideal gas I P k T V and from the result in the last section II P m lt v12 gt V implying m lt v2 gt k 391 divide by 2 m lt v2 gt k39I 2 k T T we make the quite reasonable assumption that the yquot and quot2quot components are equal to each other energy is partitioned equally amongst the three different translational degrees of freedom then the total kinetic energy of the ensemble is we see that the ensembleeaveraged xecomponent of kinetic energy is for an ideal gas where T is the temperature Now if quotxquot u mltv2gt IIIltV12gt 2 2 k39I39 k39I39 RT 3 Ukinetic N T T T ENkT If there are no other degrees of freedom eg electronic vibrational rotational available to the particles then Ukinetic U the total ensemble energy of the system I The First Law of Thermodynamics The last result tells us that for a mono atomic ideal gas ignoring electronic energy the ensemble energy U gt f N T The energy scales linearly with number of particles and linearly with the temperature Thus if we keep N fixed and we keep the volume fixed we can obviously increase or decrease the ensemble energy by transfer ring energy in the form of HEAT to or from the system We would express this change in energy in a differential form as dU constant N V aq where 3 q represents a differential amount of heat transferred tofrom the system The distinction in notation between 39the bar39 and 39not the bar39 is important Differential quantities expressed without the bar are referred to as exact differentials meaning that the integral of such an exact diffrerential between an initital and final value is independent of the path taken to get from the initial value to the final value A familiar example is potential energy in a 1D gravitational problem The gravitational potential energy of an object in such a problem is Vh m g h m is the mass g is the acceleration due to gravity and h is ther vertical distance The above picture shows a fellow trying to get to the top of a tower He can get there by a variety of different ways eg a ladder or very long spiral staircase His change in potential energy AV 2 mg h 1 h 0 is the same whether he takes the ladder or the stairs BUT the amount of physical effort work that he has to expend to get there clearly depends on the path Exact differentials have the useful property among others U2 I d1UU2U1AU U1 However inexact differentials as indicated with the bar like E q must be integrated along a specific path and the integrated quantities are therefore pathdependent Jiaqq path where q is the sum total integral over path of the heat transferred Of course we can change the ensemble energy of the system by using part of energy stored in the gas sample to do work for us internal combustion engines do this all the time So we could imagine thermally isolating our system so no heat can be transferred in our out of the system such a thermodynamic system is called adiabatic and we would write the differential change in U as dU constant N aq gt 0 aw where d w is the inexact differential corresponding to the mechanical work done byon the system If our system does work by expansion under adiabatic conditions no heat transferred the ensemble energy must decrease we lost energy in order to perform the work Thus the sign convention for expansioncompression work otherwise known as quotPVquot work is as follows dU constant N aq gt 0 pdl V Combining the two eXpressions for dU in terms of d w and E q we arrive at the First Law of Thermodynamics in differential form dU constant N aq pdl V Thus the first law tells us how to compute changes in ensemble energy by addingsubtracting heat thermal energy andor expansioncompression work I Entropy The 2nd Law of Thermodynamics Direction of Spontaneous Change I nonstatistical mechanical definition of entropy and direction of spontaneous change An interesting issue arises directly from considering the functional form of the ensemble energy we derived for a monatomic ideal gas three modes of translation only and the first law Consider the two systems shown in the picture below El lhll 39IL FA Fin H T V5 Fig The two cylinders contain the same number of particles at the same temperature T but with different volumes VA and VB thus the pressures are different PA and PB If we assume the ideal gas law as the equation of state the two systems have the same energy recall U is only a function of N and T for an ideal gas But clearly these two systems are di erent thus ensemble energy U is apparently not a unique way of defining the thermody namic state of the system there are an infinite number of different ideal gas systems we can imagine that have the same energy The question is then how do we describequantify this difference and why is it important Since the two systems have the same energy there is no energetic driving force to intericonvert from PA 0 P3 lntuitively we know that there is a direction of spontaneous change Gas samples don39t spontaneously ie without applying an external force thereby doing work compress 7 they expand So if we consider the two systems in the figure the system on the left VA PA can spontaneously expand to PB at constant T 7 provided that it is in contact with a thermal heat reservoir that provides a quantity of heat q necessary to compensate for the work done by the system in expanding from PAVA gt P3 V3 I Miernsenpie reversibility But that amount of heatwork involved in the transformation depends on the path We consider two possible paths for the transformation PAVA gt P3 V3 at constant N and T reversible and irreversible The distinction between the two words is subtle a reversible change implies an infinitesimal change in one or more state variables so that the system could be restored exactly to it39s original state not just the original values of the state variables First we consider an example of an irreversible transformation Imagine that the system defined by the state variables N T PAVA is abruptly expanded against and external pressure P3 The work that is done by the system is trivial to calculate dwirrev Pext W a wirrev Pa f le Pa Va VA A If VE VA gt O the quantity of work is negative lntemal energy decreases because the system DID work on surroundings thus the quantity of heat that must be absorbed by the system to maintain constant temperature is qirrev wirrev Pa Va VA or if we wish k39I Va VA qirrev Va Now consider a reversible transformation Microscopic reversibility implies that the transformation from N T PA VA gtN T P3 V3 occurs via an infinite number of infinitesimally small external pressure steps where the external pressure is equal to the internal pressure kT Pext V 1le V3 wrev k391 kTLog i v vA Clearly the amount of work done and heat transferred is different for the reversible and irreversible transformations Prove to yourself The definition of Entropy S from nonstatistical thermodynamics is in differential form Ham 1115 391 which interestingly is an exact differential French Engineer dei Carnal invented this construct in the early 180039s For proof that the entropy is an exact differential 7 and therefore a state function 7 see section in text on the Carnot Cycle As you learned in general chemistry the direction of spontaneous change is associated with a POSITIVE change in entropy Ham AS Ids I gt 0 for spontaneous process 391 In other words we can determine whether any physical or chemical transformation is spontaneous in the absence of any addi7 Ham T tional energetic driving force by computing the somewhat obscure integral for over a reversible path If that integral is positive 7 meaning that heat energy was added to the system during the transformation at temperature T 7 then the transformation is spontaneous I The first law in terms nf entrnpy and vnlume differentials Since U is a state function the differential dU is exact 7 and therefore independent of the path Thus we can use the definition of the entropy 5 am dis 391 and express the first law in differential form as dlU les ple This lets us express again in differential form all of the 3 remaining thermodynamic state functions in terms of combinations of 2 quotnaturalquot variables entropy somewhat obscure and volume 7 although we could if we wished express the ensemble energy differential in terms of whatever state variables we wish I entrnpy as a quotstate attributequot When we recast the 1st Law of thermodynamics as d U T d S 7 p d V the obvious implication is that internal energy is influenced both by changing entropy and volume Thus entropy plays a role here similar to a state variable 7 but entropy is clearly not something we can time at least not in a transparent way like volume temperature number of particles etc Entropy IS a state7function but does not by itself define the state of the thermodynamic system If we recall the example in the beginning of the section on entropy we considered two gas samples with the same N T but different volumes and pressures PAVA and P3 V3 Since both systems have the same energy assuming ideal gas behav7 ior the state function U clearly does not uniquely define the state of the system The two systems DO have different values of entropy therefore knowledge of U and S give us a unique description of the system All other thermodynamic functions are derived from this I Thermodynamic State Functions U H A G We39ve already seen how we can use the definition of entropy to write the 1st law as a combination of exact differentials dlU les ple Now we39ll use this to define the 3 other thermodynamic state functions and cast them in a similar differential form 7 but of different variables I enthalpy Enthalpy H is defined as HEUpV For 1 mole of an ideal gas 3 U R39I 2 p v R 391 so the enthalpy gt 5 HUpv RT 2 In differential form it becomes le les lep because 12PV p a V V alp I Helmhnltz FreeEnergy The Helmholtz FreeiEnergy A is defined as A E U T 5 That is A represents the amount of energy present in the system that is quotfreequot to do work such as expansion against an external pressure Thus in an absolute sense if we know the ensemble energy U and the entropy S the quantity of energy T S represents the quotrandomquot thermal energy of the system that we cannot convert into work Thus subtracting TS from U gives us the quotfree energyquot In differential form it becomes dl A 5 dl 391 9 dl v becausealT S SaZT TaZS I Helmhnltz FreeEnergy The Gibbs FreeiEnergy G is defined as G E H T S G is analogous to A in that it represents the amount of energy present in the system that is quotfreequot to do work such as expansion against an external pressure from the enthalpy Thus in an absolute sense if we know the ensemble Enthalpy H and the entropy S the quantity of energy T S represents the quotrandomquot thermal energy of the system that we cannot convert into work 10 l Chem47575pr097NSthermorev 01 nb In differential form it becomes le Sdl391 lep I Maxwell39 s relatinns and partial derivatives The four differential expressions for U H A and G allow us to calculate changes in any thermodynamic quantity as we change T V p S Constant N In summary they are again dlU les ple le les lep 1111 Sdl391 ple 11 Sdl391 lep Each of the above differential expressions allows to relate partial derivatives of a particular variable dlU 6U 6U les deV d15 d 65v avs The latter part is just a general expression for the differential of a function of 2 variables in this case S and T Thus 6U OH H 9 as v 6V 5 likewise from the differential expression for a H 6H 6H 391 and V 65 9 Op 5 likewise from the differential expression for a A 6A 6A S and 9 61 v 6V 39139 and finally from the differential expression for a G 6G 6G S and V 6T 9 DP 1 Because each of the 4 differentials a U a H a A and a G are exact the following crossipartial derivatives are equal as a property of exact differentials Consider a generic function f that depends on two variables x y The differential for f xy is 6 f 6 f dlfxy dlx dly 3 x Y 6 Y x if a f is exact then the following equivalence holds Chem47575pr09NSthermoirev 01 nb l 11 6 6f 6 6f Ei iyj Ei iu From the 4 differential expressions for U H A and G we get a set of 4 equivalent partial derivatives 6U 6U dlUleS pdlv and dlS div 65v avs 6U 391 and 5 v 6U 9 av 5 therefore 61 Op av 5 as v this constitutes one of 4 different Maxwell relations that allow us to express one potentially obscure partial derivative in terms of another hopefully less obscure 7 partial derivative Here they are you should be able to derive these on your own From dH le les lep 33 31 FromdA dlA Sdl391 ple 65 Op 515 andfrode as Sdl391 lep 6 5 6V Op T 61 p the latter two are the most useful since they relate to partial derivatives of state variables in terms of another via the choice of equation of state I Chemical Potential and Chemical Equilibrium The final thing we need to consider here is how the various thermodynamic state functions vary with changing number of particles holding all other state variables constant This of course is important in many chemical processes such as mixing reaction etc and this quantity is called the chemical potential 14 quotXII The chemical potential of species is defined as 12 l Chem47575pr097NSthermorev 01 nb 6 any state function U H A G M a v 1 1 all otherparticles ny nx That is the chemical potential of species quotxquot can be expressed as the derivative of any state function 7 U H A or G 7 with respect to the number of particles of quotxquot holding all other state variables constant The condition for chemical equilibrium 7 a critically important construct in chemistry 7 between species quotxquot and quotyquot 7 is that their chemical patent39mls are equal ie M My The idea is that chemical potential plays a role analogous to electric potential thermal potential gravitational potential etc in that the potential di erence defines the direction of charge heat mass ow If we place two systems in contact with each other so that particles of x and y can move between the two systems the chemical potential difference between the two systems tells us the direction that the particles will move in response to the chemical potential gradient
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