UNIVERSITY OF RWANDA (UR) COLLEGE OF SCIENCE AND TECHNOLOGY (CST) SCHOOL OF SCIENCE CHEMISTRY DEPARTMENT ACADEMIC YEAR 2018-2019 ADVANCED PHYSICAL CHEMISTRY (15 Credits) Level: Year 2 Env. Chem and Org. Chem Instructor: Mr. Jean Bosco NKURANGA 1 COURSE OUTLINE 2 I. II. III. IV. V. VI. Introduction to statistical mechanics and statistical thermodynamics, Boltzmann distribution Partition function and molecular partition function (translation, rotation, vibration and electronic) and Entropy in statistical thermodynamics, calculation of thermodynamic variables by using partition functions (ideal gas properties). Heat capacities of solids Classical statistical mechanics (phase space, Ensembles and Ensembles average, equipartition, microcanonical, canonical and grand canonical ensembles Quantum Statistical Mechanics (B-E, F-D, phonons and photons) Colloidal Chemistry and its applications Advanced Physical Chemistry Year 2 Env. and Org. Chem 9/13/2019 Recommended readings 3 1. 2. 3. Alberty R.A and Silbey R.J. (2001) Physical Chemistry, 3rd Edition, John Willey and Sons, Inc. Puri, B.R., Sharama L.R. and Patthania M.S. (2008), Principles of Physical Chemistry, 43rd Edition, Washall publishing.co, Jalandhar-Delhi, India. Atkins P. and De Paula J. (2010) Physical chemistry , 9th Edition, Oxford University Press, Great Lakes U.K. ISBN-13: 978-1-429-21812-2 Advanced Physical Chemistry Year 2 Env. and Org. Chem 9/13/2019 NKURANGA Jean Bosco Address: Office: Room P215, Muhabura block Email:[email protected] • Assessment: 1. CW: 50% a. Test 1 (30 marks), Quiz (10 marks), Assignments (10 marks) b. Laboratory Practices:- 2. UE: 50% Prerequisites: Basic Mathematics, Quantum Chemistry, Classical thermodynamics and Probability and statistics 9/13/2019 Advanced Physical Chemistry Year 2 Env. and Org. Chem 4 Study Tips: • You start passing/failing this course on day one by attending/not attending all class lectures. • Practice makes perfect. If you are interested in understanding this course then confront as many readings as you possibly can. • Lecturers are here/there (office/lab) to help you NOW, not after you have done your exam and failed. 9/13/2019 Advanced Physical Chemistry Year 2 Env. and Org. Chem 5 I. Introduction to Statistical Thermodynamics 6 Statistical mechanics (SM) provides a link between quantum mechanics (or wave mechanics) and classical thermodynamics . Classical Thermodynamics (CT) deals with macroscopic properties of matter and describes the behaviour of large number of molecules in terms of state variables (P, V, n, T). Quantum mechanics (QM) deals almost exclusively with matter at the microscopic level, which is described by using a wave function, ψ . However, QM does not indicate which wave function of a molecule will represent the state of the system at a given time. It is noteworthy that neither CT nor QM is able to calculate the macroscopic properties of matter from microscopic structures of individual molecules. Since any observed equilibrium property of matter must be some kind of an average of large number of molecules, it is plain that statistical methods must be used to determine this property. Advanced Physical Chemistry Year 2 Env. and Org. Chem 9/13/2019 I. Introduction to Statistical Thermodynamics .. 7 Statistical Thermodynamics/mechanics is the discipline, which deals with the computation of the macroscopic properties of matter from the data on the microscopic properties of individual atoms (or molecules). Statistical thermodynamics helps to understand the significance of thermodynamic variables (T, S), which do not possess any microscopic or mechanical interpretation. Contributors to S T: J.C. Maxwell, L. Botzmann, Willard Gibss, Maxwell Planck, A. Einstein, S.N. Bose, E. Fermi, P.M.A Dirac, Tolman, R.H Fowler, Guggenhein, Born, Debye, L. Onsager, L.D. Landau, N. Bogoliubov, J.G. Kirkwood, I. Prigogine, N. Winner, J.E. Mayer, K.G. Wilson and R. Kubo and Chandrasekhar (1910-1995), who applied quantum statistics to stellar dynamics (white dwarfs). I. Introduction to Statistical Thermodynamics .. Scope of Statistical Mechanics/Thermodynamics 8 Statistical mechanics can be applied to simple ideal systems such as monoatomic and diatomic gases. For application to interacting systems such as such as liquids, where strong intermolecular forces exist), the details of the intermolecular potential energy, which is not always known accurately, have also to be taken into account. That is why statistical mechanics of liquids is a difficult but fascinating subject. Gases under high pressures, too, are difficult to treat statistically since they deviate strongly from ideality. Recently, statistical methods have been applied to successfully to simple liquids and dense gases. Progress in this area has been made possible by application of both the advanced mathematical methods and high speed computers, which otherwise highly intractable differential and integro-differential equations involved in advanced theoretical treatments. The third law of Thermodynamics is well discussed in Statistical Thermodynamics based on the partition function. The goal of ST is to describe macroscopic, thermodynamic thermodynamic properties Physical Chemistry Year 2 Env. andproperties. Org. Chem 9/13/2019 In termsAdvanced of microscopic atomic and molecular I. Introduction to Statistical Thermodynamics .. 9 1.2 Types of Statistics Different physical situations encountered in nature are described by three types of statistics: the Maxwell-Boltzmann (M-B) or Classical Statistics the Bose-Einstein (B-E) statistics, Fermi-Dirac (F-D) Statistics The M-B statistics is also referred to classical statistics while B-E and F-D statistics are called Quantum statistics. These three types of statistics are described for microcanonical ensemble. Advanced Physical Chemistry Year 2 Env. and Org. Chem 9/13/2019 I. Introduction to Statistical Thermodynamics .. 10 Maxwell-Boltzmann Probability (ni gi 1)! wBE i 1 ni !( gi 1)! ni n gi wB N ! i 1 ni ! n n gi ! wFD i 1 ni !( gi ni )! ni n gi wMB i 1 ni ! wB and wMB yield the same distribution. In M-B statistics, the particles are assumed to be distinguishable and any number of particles May occupy the same energy level. Particles obeying M-B statistics are boltzons Advanced Physical Chemistry Year 2 Env. and Org. Chem 9/13/2019 I. Introduction to Statistical Thermodynamics .. 11 ni 1 fi i gi e ni 1 fi i gi e 1 ni 1 fi i gi e 1 Boltzmann Bose-Einstein Fermi-Dirac In B-E statistics, the particles are indistinguishable And any number of particles may occupy a given Energy level, Ei. This statistics is obeyed by particles having integral spin such as H2 , N2, D2 and Helium-4 and photons. • Particles obeying B-E statistics are bosons ( Symmetric wave function) In F-D statistics, the particles are indistinguishable, but one particle may occupy a given Energy level, Ei. This statistics is obeyed by particles having half-integral spin (protones, electrons, He3, NO, etc. • Particles obeying F-D statistics are fermions (antisymetric wave function) ni gi ni gi ni gi 1 i e 1 kT 1 i e Boltzmann kT i e fi kT fi Bose-Einstein fi Fermi-Dirac 1 1 12 I. Introduction to Statistical Thermodynamics .. Maxwell-Boltzmann Statistics Consider a system of N distinguishable particles occupying energy levels, 0 , 1, 2 ,..........., n The total number of arrangements for placing no particles in the ground state energy level 0 , n1 particles in the first excited energy level 1 , n2 particles in the second energy level 2 , and so on, is known as the thermodynamic probability, W, of a given macrostate. In statistical thermodynamics, the problem is to determine how many microstates correspond to a given macrostate. W N! N! n n1 !n2 !n3 !..... n j ni i 1 n N ni i 1 13 I. Introduction to Statistical Thermodynamics .. Maxwell-Boltzmann Statistics … In B-M distribution, it may happen that a given energy level occur in more than one quantum state with the same energy, so there is degeneracy or multiplicity, of the energy, i . ni n gi W N ! constant i 1 ni ! The entropy, S and the probability, W of a given state of a system are related by the Boltzmann Equation, which is the most famous Equation in Statistical mechanics. S k B ln W How can determine the maximum probability, at equilibrium? S k ln Wmax 14 I. Introduction to Statistical Thermodynamics .. Maxwell-Boltzmann Statistics … In order to find a distribution that will make W a maximum , it is more convenient, however, to optimize the linearized W, so logarithm of W. ln W ln W d ln W dn 1 n 1 n2 n ln W d ln W dni n i i 1 ln W dn 2 n3 ln W dn .............. 4 ni dni For a closed system of independent particles: n n (i ) N ni constant i 1 n ln W dn 3 n4 (ii ) U ni i constant (i ) dN dni 0 Differentiate i 1 Advanced Physical Chemistry i 1 n (ii ) U i dni 0 i 1 Year 2 Env. and Org. Chem. n n i 1 i 1 ln W ln N ! ni ln gi ln ni ! ct 15 I. Introduction to Statistical Thermodynamics .. Maxwell-Boltzmann Statistics … The derivation of the B-M uses the striling’s approximation (SA): ln x ! x ln x x n n i 1 i 1 ln W ln N ! ni ln gi ln ni ! ct Stirling Approx. gi ln n i dni 0 i 1 i n i 1 i 1 ln W N ln N ni ln gi ni ln ni ct Remember the considered system is closed, so ni and our N is constant, the values of dni are not independent of one another and the energy of the system is constant! The introduction of the Lagrange’s Undetermined multipliers α and β for ∑dni and ∑ɛidni, respectively. n Differentiating w.r.t ni, N and gi are constants ! ? n i 1 n i 1 d ln W ln gi dni ln ni dni g d ln W ln i dni 0 ni i 1 n ni gi e n i ni 1 i gi e 16 I. Introduction to Statistical Thermodynamics .. M-B Statistics and Partition Function ni gi e i gi i e This equation is one of the form of the Boltzmann distribution law, which is for the most Probable distribution for a macrostatate, i.e., it gives the occupation numbers of the molecular energy levels for the most probable distribution in terms of the energies, ɛi , The degeneracy, gi and the undetermined Lagrange’s multipliers, α and β. The quantity Z is known i i i n n ge ge ge Pi i n i n i i Z gi e i as the partition function, a function that enables N Z i i i 1 gi e gi e the calculation of all i 1 i 1 thermodynamic variables! Advanced Physical Chemistry Year 2 Env. and Org. Chem 9/13/2019 17 I. Introduction to Statistical Thermodynamics .. Bose-Einstein Statistics 0 , 1, 2 ,..........., n Consider a system of N indistinguishable particles such that ni particles are in the ith energy level with degeneracy, gi . The ni particles have to be distributed among gi states. For the sake of simplicity, let us consider that the i th energy level has (gi - 1) partitions, which are sufficient to separate energy level into gi intervals. In B-E statistical thermodynamics, the possible number of distributions of ni particles among the gi states may be determined by permuting the array of partitions and particles. n ( g 1) C( n i g 1) i i ni gi 1! ni ! gi 1! N ni i 1 n U i ni i 1 18 I. Introduction to Statistical Thermodynamics .. Bose-Einstein Statistics In B-E statistics, the thermodynamic probability, W for the system of N particles, i.e., the number of ways of distributing N particles among the various energy levels is given by: ni gi 1! W constant i 1 ni ! gi 1 ! n n ln W ln ni gi 1! ln ni ! ln gi 1! constant i 1 1) ni+ gi -1 ≈ ni+ gi since ni is large 2) gi -1 ≈ gi since gi is large 3) Stirling’s Approximation: lnx! =xlnx-x Approxi mations: n n n i 1 i 1 i 1 n gi d ln W ln i i 1 ni n Differentiate w.r.t ni ln W (ni gi ) ln ni gi ni ln ni gi ln gi constant B-E distribution of N particles among the various energy levels dni 0 the Lagrange’s Undetermined multipliers α ∑dni and β ∑ɛidni, gi ni i 1 e 19 I. Introduction to Statistical Thermodynamics .. 0 , 1, 2 ,..........., n Fermi-Dirac (F-D) Statistics Consider N indistinguishable particles such that ni particles are in the ith energy level with degeneracy, gi and ni < gi . The first particle have to be placed in any one of the gi states and for each one of these choices, the second particle may be placed in any one of the remaining, (gi - 1) states, and so on, up to (gi - ni) states . gi ! # Arrangements Thus, the number of arrangement is given by gi ni ! In F-D statistical thermodynamics, the particles are indistinguishabe, so the above expression has to be divided by the possible number of permutations of ni particles. g ni Cg i i gi ! ni ! gi ni ! n W i 1 n gi ! constant ni ! gi ni ! N ni i 1 n U i ni i 1 20 I. Introduction to Statistical Thermodynamics .. Fermi-Dirac Statistics In F-d statistics, the thermodynamic probability, W for the system of N indistinguishable particles, i.e., the number of ways of distributing N particles among the various energy levels is n given by: gi ! W i 1 ni ! gi ni ! constant n ln W ln gi ! ln ni ! ln gi ni ! constant i 1 1) ni and gi are large 2) gi > ni since gi is large 3) Stirling’s Approximation: lnx! =xlnx-x Approxi mations: n n n i 1 i 1 i 1 g n d ln W ln i i i 1 ni n Differentiate w.r.t ni ln W (ni gi ) ln gi ni ni ln ni gi ln gi constant F-D distribution of N particles among the various energy levels dni 0 the Lagrange’s Undetermined multipliers α ∑dni and β ∑ɛidni, gi ni i 1 e 21 I. Introduction to Statistical Thermodynamics .. Fermi-Dirac Statistics Fermi-Dirac distribution of plot for an electron in a metal Advanced Physical Chemistry Year 2 Env. and Org. Chem 9/13/2019 I. Introduction to Statistical Thermodynamics .. 22 Evaluation of the Lagrange’s Undetermined Multiplayers n n N ni gi e N From B-M statistics, i ni gi e i i 1 e i 1 n i g e N e i i 1 Substitution i gi e ni N Z n gi e i N Z Z gi e i i 1 • The partition function, Z is a quantity of immense importance in statistical thermodynamics as it enables the calculation of the Value of any thermodynamic function for the system. • However, one need to determine the value of β prior to Z evaluation. ? Advanced Physical Chemistry i 1 n with N Year 2 Env. and Org. Chem Determine the value of β I. Introduction to Statistical Thermodynamics .. Evaluation of the Lagrange’s Undetermined Multiplayers 23 From B-M distribution statistics, the thermodynamic probability in terms of ln W is given by: n n n n i 1 i 1 i 1 i 1 ln W ln N ! ni ln gi ln ni ! N ln N ni ln gi ni ln ni gi e i ni N Z ln ni ln N ln gi i ln Z n n i 1 n i 1 ln W N ln N ni ln gi ni ln N ln Z ln gi i n n i 1 i 1 ln W N ln N ni ln gi N ln N N ln Z ni ln gi ni i i 1 ln W U N ln Z In ST (Boltzmann) : S k B ln W S Nk B ln Z k B U I. Introduction to Statistical Thermodynamics .. 24 Evaluation of the Lagrange’s Undetermined Multiplayers n Z gi e i S NkB ln Z kB U i 1 From the combined statement of the first and the second law of Classical Thermodynamics, the Internal energy is an exact differential function of S and V, so U=f (S,V) dU TdS pdV U U dU ds dV S V V S S NkB ln Z kB U S kB U V 1 S U V T Differentiate w.r.t U At constant volume, dV=0 Nk B Z S k k U B B Z U V U V U V Nk B Z S Z V U U V dU TdS 1 S U V T 1 k BT k k U B B U V Z U Z N V I. Introduction to Statistical Thermodynamics .. 25 Evaluation of the Lagrange’s Undetermined Multiplayers Advanced Physical Chemistry Year 2 Env. and Org. Chem 9/13/2019