Ch 0 Introduction - University of Waterlooscienide2.uwaterloo.ca/.../Chem350_statmech/Ch_0_Introduction.pdf- Michel!Le!Bellac,!Fabrice!Mortessagne,!and!G.!George!Batrouni: ... M Quantum!Mechanics!

Winter 2014 Chem 350: Statistical Mechanics and Chemical Kinetics

Introduction 2

Preface

These lecture notes are based on the Chem350 class as it was taught in the winter of 2012. The lecture notes follow the class more or less verbatim and reflect what might be written on the black board, or course notes taken by students in the class. The notes are concise, focusing on mathematical formulations and pictorial representations of the material. The mathematical derivations tend to be complete. The textbooks by Reid and Engel, and by Metiu, will be complementary as they have more verbal explanations of the underlying ideas. These lecture notes contain all the material that will be covered and should be a good source for study.


Introduction 3

Statistical Mechanics and Chemical Kinetics: Syllabus Syllabus Chemistry 350: Introduction to statistical mechanics and chemical kinetics. Professor Marcel Nooijen ESC 330B e-‐mail [email protected], tel: 37708 Class: MWF 10.30-‐11.20, RCH 110 Tutorial: M, 11.30-‐12.20, RCH 110 Computer lab (selected Mondays), M 10.30-‐12.20, Friesen lab, C2-‐160, to be announced in class. Contents: In this course will first discuss the principles and applications of statistical mechanics. Statistical mechanics provides the link between quantum mechanics and thermodynamics, and these three branches of physical chemistry, together with chemical kinetics determine much of the theoretical underpinnings of chemistry. In this class we will discuss the basic results derived from statistical mechanics as they are used in many chemical applications, in particular gas phase chemistry, computational chemistry and transition state theory. In the second part of the class we will discuss chemical kinetics, which has a strong connection to experiment. We will discuss how chemical kinetics, i.e. the determination of the concentration profiles of various molecular species as a function of time, can be used to determine reaction mechanisms. If time permits we will also consider transport phenomena and some aspects of nonequilibrium statistical mechanics. Computations using computers will form an integral part of the class, and some of the class will take place in a computer lab. I strongly advocate the use of Mathematical programs such as Matlab or Mathematica, such that “real life” problems can be treated in the exercises. Additional problem sets will also require the use of Matlab. During the course I will provide a basic introduction to Matlab. Text book: Thermodynamics, Statistical Thermodynamics and Kinetics, third edition, by Thomas Engel and Philip Reid, chapters 12-‐19. ISBN-‐13:978-‐0-‐321-‐76618-‐2. There is also a student solutions manual under the same name. ISBN-‐13:978-‐0-‐13-‐260650-‐9. Besides the textbook, lecture notes are posted on the class web site. Grading: There will be one midterm test (33%) and a comprehensive final exam (33%). Practicing P-‐chem is essential to master the material and 34% of the grade is earned through triweekly problem sets (including computer problems). During all of the tests you can bring your own summary of the material. It will serve you well if you make a short summary of each chapter/topic covered and the most pertinent equations, while reading it. This summary can then be used during the exams. Problem solving is the most effective way to learn P-‐Chem, and working together is a good way to solve problems. I encourage you to do so after you have thought about the question and tried to work the problem. As tutors (and teachers) know, explaining the solution to someone else makes it even clearer. After consultation, though, please be sure to write out the entire solution on your own.


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Further reading:

-‐ Horia Metiu, Physical Chemistry, Statistical Mechanics. This book puts a strong emphasis on the main threads of the topic and conveys the applications of statistical mechanics. It does not cover the foundations of the theory in great depth, but it is very good in explaining the underlying principles.

-‐ H. Callen: Thermodynamics and Introduction to Thermostatistics. This is a beautiful text on the foundations of thermodynamics, starting from entropy as the fundamental function.

- Myron Tribus: Thermostatics and Thermodynamics: Has a very complete discussion on the use of different variables and statistical foundations.

- David Chandler: Introduction to Modern Statistical Mechanics (Paperback) - Terryl L. Hill: An Introduction to Statistical Thermodynamics (a classic Dover. More advanced

theory) - Michel Le Bellac, Fabrice Mortessagne, and G. George Batrouni: Equilibrium and non-‐

equilibrium statistical Thermodynamics (a fairly advanced recent physics text). - Donald A. McQuarrie: Statistical Mechanics (a standard text, hard cover. Graduate level) - Chris Cramer: Essentials of Computational Chemistry (theories and models. A good

introductory text). Topics to be covered.

I. General principles of statistical mechanics a. Statistical mechanics in a nutshell b. Discussion on statistics c. Boltzman distribution d. Interpretation Statistical mechanics e. Advanced topics (briefly)

i. Various types of ensembles ii. Connection to Thermodynamics

II. Statistical mechanics of molecules in the gas phase. a. Partition function for independent particles b. Translational, vibrational, rotational, electronic, nuclear partition functions c. Poly-‐atomics and calculations using Gaussian (Lab). d. Chemical equilibrium e. Advanced topic: Quantum statistical mechanics for ortho and para Hydrogen

III. Transition state theory: rates of elementary reactions a. Statistical mechanics treatment of transition states and reaction rates b. Tunneling, and kinetic isotope effect c. Calculations of reaction rates using Gaussian (Lab) and Matlab.

IV. Chemical kinetics a. Rate laws and reaction mechanisms b. Temperature dependence c. Complex reaction mechanisms and catalysis d. Radical-‐chain reactions and polymerization e. Using Matlab software to simulate reaction profiles (Lab).

V. Miscellaneous topics (if time permits, which is not so likely).


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a. Kinetic theory of gases b. Transport phenomena c. Nonequilibrium statistical mechanics


Introduction 6

Introduction Physical chemistry is divided into:

Microscopic laws of physics

-‐ Electromagnetism -‐ Quantum Mechanics -‐ (quantum field theory)

For Quantum Mechanics: Schrodinger’s equation : HΨ = Eψ i! ∂∂tψ = Hψ

The wave function of every particle, where ( )231 2 3 10, , ....x x x xψ and H is the Hamiltionian

operator 2

ˆ2 4

i ji

i i ij o

q qPHm r πε

= +∑ ∑

QM is our most detailed description of the world (for a chemist). Everything else would/should follow First Principle theory. Inputs : e , Zα1

, 1m , 2m , h Imagine “Ultimate Google” from QM. worldψ describes everything that can (physically) exist. Use a Google search machine to zoom in on the events that interest you. This is fantasy, but it also captures out scientific view of the world.

Macroscopic Laws of Physics

i) Thermodynamics (Thermostatics, although it’s no longer heat related.) → describes systems in equilibrium or shifting from one equilibrium to another

o very few variables to describe system ( ), , ,T P V µ which are all measurable experimentally (useful)

o In thermodynamics lots of experimental data is used such as 298o

f HΔ

o Something else, special about thermodynamics, is that it can only calculate changes in thermodynamic quantities , ,U S HΔ Δ Δ etc..

ii) Chemical Kinetics: addresses the question of how fast processes or chemical reactions proceed

o By monitoring concentrations over time (concentration profiles), plus kinetic models (theory + simulation) we can understand reaction mechanisms

o Catalysis, enzymatic catalysis o Long time limit → chemical equilibrium → thermodynamics o Use experimental data: elementary rates of reaction, functions of T

Statistical Mechanics : provides the bridge to connect the microscopic (QM) world to the macroscopic description


Introduction 7

It uses microscopic (often quantum) input to provide absolute values for thermodynamic quantities , , , , , , ,V PS A G C Cµ β κ . Similar principles can be used to describe rates of elementary reactions which

forms the basic inputs for kinetics.

Underlying principle: Use statistical averages that involve huge numbers ( 2310 ), of more or less independent or weakly interacting entities. With such large numbers statistics becomes “certainty” since the fluctuations become very small. -‐ non equilibrium statistical mechanics: A difficult topic that describes behavior of macroscopic systems as they change in time.

Eg. Reactions in a flow reactor, reactions where energy is fed in. Transport properties are the simplest examples.

Founding fathers of Stat-‐Mech: Maxwell: kinetic theory of gases Boltzmann: microscopic definition of entropy Gibbs: ensembles and classical mechanics Bose, Einstein, Fermi, Dirac: particle statistics Ehrenfest: concepts in stat-‐mech Just some remarks

1) Systems are almost never in a state of fully stable equilibrium. If there were a fully stable equilibrium, it would be very “boring”; only low energy molecules would exist ( 2 2 3, ,CO H O NH ).

Most of the equilibriums we talk about are metastable states (relatively stable but degrade after a while), a local but not global equilibrium. The barriers (and kinetic humps) are too high to allow reaching of a global equilibrium. This is a fundamental feature of all systems.

-‐ In statistical mechanics, we speak of “accessible” states


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2) Quantum Mechanics in practice is: -‐ good for molecules in gas phase (especially non-‐interacting) -‐ good for solid state periodic crystals

-‐ Quantum gets more complicated for: liquids, large floppy molecules (like proteins,

conformations etc..) for these there isn’t quite a Quantum Mechanical method yet, classical mechanics still predominates.

Microscopic Physics ← Chemical Physics Quantum Mechanics Statistical Mechanics bridges the two Thermodynamics (equilibrium) Macroscopic Physics ← Physical Chemistry Kinetics (how fast? How does it change?) Transport properties (diffusion, heat)

Documents

Ch 0 Introduction - University of Waterlooscienide2.uwaterloo.ca/.../Chem350_statmech/Ch_0_Introduction.pdf- Michel!Le!Bellac,!Fabrice!Mortessagne,!and!G.!George!Batrouni: ... M Quantum!Mechanics!