PAPER No. 6: PHYSICAL CHEMISTRY-II (Statistical

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Randall Long
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1 Subject Paper No and Title Module No and Title Module Tag 6: PHYSICAL CHEMISTRY-II (Statistical 1: Introduction to Statistical CHE_P6_M1

2 TABLE OF CONTENTS 1. Learning Outcomes 2. Statistical Mechanics 3. Role of Statistical Mechanics in Chemistry 4. Fundamentals of Statistical Mechanics a. Maxwell Boltzmann Statistics b. Bose Einstein Statistics c. Fermi Dirac Statistics 5. Summary

3 1. Learning Outcomes After studying this module, you shall be able to Understand Statistical Mechanics Appreciate and learn the role of Statistical Mechanics in Chemistry Understand the relationship between statistical mechanics and quantum mechanics Learn about different types of statistics applied to types particles present in a system 2. Statistical Mechanics Statistical mechanics is a theoretical science of bulk matter (many-particle system). Or in simple terms, it gives the laws of motion of bulk matter. The term Statistical Mechanics was coined by the American mathematical physicist J. Willard Gibbs in Statistical Mechanics is made up of two words STATISTICAL + MECHANICS. The term Mechanics refers to the science of motion of the body. Mechanics word comes because we are dealing with motion of the bulk matter. The other term is Statistical which represents the use of probabilistic approach applied to study a bulk matter. Statistical mechanics is a branch of theoretical physics that studies the average behavior of a macroscopic system using probability theory. With this brief introduction to this new concept, the following questions might arise in your mind:

4 What is Statistical Mechanics? Role of Statistical Mechanics in Chemistry Fundamentals of Statistical Mechanics Difference between Statistical Mechanics, Quantum Mechanics and Classical Mechanics Applications of Statistical Mechanics This paper will deal with each of the above aspects in detail. In the next section, we will highlight the significance of Statistical mechanics in chemistry. Before proceeding further, let us understand the difference between bulk matter and macroscopic matter. It is important to understand the difference because throughout the discussion of Statistical thermodynamics, we will be using the term Bulk matter. Actually, bulk matter is macroscopic matter. And both the expressions Bulk matter and Macroscopic matter are used interchangeably and frequently convey the same meaning. Over here, Bulk matter refers to the agglomeration of many particles (atomic or subatomic) that define an entity as a system (which can be a macroscopic object). The behavior of the system is predicted by averaging the effect of atomic or sub-atomic motion. Macroscopic matter refers to the object that we can see and touch in real (table, bottle etc); It is considered as a single large particle (does not take into account the internal constitution of the macroscopic object).

5 2. Role of Statistical Mechanics in Chemistry Statistical mechanics is used to explain the thermodynamic behavior of many-particle system. One might think how thermodynamics comes into picture over here? In order to understand the emergence of consideration of statistical behavior to derive thermodynamic properties of a bulk system, one needs to learn and recapitulate the following concepts: We all know that Classical thermodynamics is an empirical or experimental science of macroscopic matter which predicts the properties of a system without considering the behavior of atomic or sub-atomic particles. Without due consideration to the constituent microscopic systems such as an atom or molecule or sub-atomic particle, that make up the macroscopic object, it becomes difficult for the learner to understand the evaluation of the thermodynamic properties. So, here comes the significance of theoretical evaluation of properties of a system considering the behavior of particles at atomic or sub-atomic levels. In order to calculate the properties of a macroscopic system considering the behavior of atomic or sub-atomic particles, one needs to know a) The behavior of a single constituent microscopic particle b) Number the microscopic particles present in the macroscopic system The properties (or behavior) of individual particles can be obtained using the methods of quantum mechanics. Quantum Mechanics is a theoretical science of microscopic matter (electrons and nuclei in atoms and molecules,). So, quantum mechanics gives the properties of single particle. The most important result of quantum mechanics is that the energy of a single atomic or sub-atomic particle is not continuous, but discrete (the significance of this statement will be seen in latter modules)

6 Quantum Mechanics is the science which takes both particle and wave character (dual behavior) of microscopic matter (considering the particles at atomic or sub-atomic level (atoms, molecules, electron, proton). At microscopic level, one needs to take into account both particle and wave character of matter as a consequence of de-broglie relation. λ = h mv To study the bulk properties of one mole of hydrogen gas (consisting of a large number of hydrogen molecules), the quantum mechanical results are taken as an input in statistical mechanics where some sort of statistical average over the entire system gives the bulk properties. Statistical Mechanics Quantum Mechanics Single particle Bulk behavior This is possible because the basic rule governing the motion of each molecule in the gas is same and follows quantum mechanical principles. Therefore, we use quantum mechanics as an input in Statistical Mechanics in order to study the statistical average of the bulk system.

7 Quantum Mechanics Science of microscopic matter Gives laws of motion of atomic or subatomic particles Statistical Mechanics Science of bulk matter Uses quantum mechanical result of individual particles and averages them to give the laws of motion of bulk system Classical Mechanics Science of macroscopic matter Gives laws of motion of macroscopic object without taking into account of the internal structure of the system Now, for a bulk system consisting of large number of microscopic particles, the quantum mechanical approach can be utilized for writing the exact laws of motion for the system. The problem arises due to large number of microscopic particles that make up the system. For instance, if we consider one mole of a substance; it will contain Avogadro s number of atoms or molecules, i.e particles which is very large. To solve this system exactly, one would have to write down about equations of motion and integrate them to get the solution. Definitely, this becomes severely complex to solve. It is important to note over here that we are interested in calculating properties like volume, pressure, temperature, heat capacity etc., which do not depend on the motion (position and velocity) of individual particles, but, instead they depend on average motions of all the particles in a

8 macroscopic system. In all, the concerned properties depend on the statistical properties of the atomic or sub-atomic motion. Taking into account the quantum mechanical results of single particle and statistical behavior of the atomic or sub-atomic motion, one not only obtains theoretically the results of classical thermodynamics but also is also enabled to investigate and evaluate macroscopic properties of the system from knowledge of its microscopic constituents. This approach is referred to as Statistical. So, Statistical theoretically calculates the properties of the bulk system by taking some sort of statistical average of the microscopic behavior. The conceptual linking is shown in figure 1. One can say that Statistical plays a central role in linking microscopic and macroscopic worlds. Statistical acts as a bridge linking the quantized world (Quantum Mechanics) with the real macroscopic world (Classical Mechanics). Both Classical and Statistical are only valid for systems in equilibrium. Classical and Statistical are referred to as two different approaches of. At many places, it has been claimed that both these approaches are based on experimental data, but it is important to note that the manner in which the experimental data is used by two approaches is different. Classical uses data based on the properties of bulk matter such as density, heat capacity whereas Statistical uses properties of individual molecules such as bond length, symmetry etc. C(graphite) + O 2 CO 2 (g) Classical thermodynamics measures the entropy of formation of CO 2 (g) under standard conditions without taking into account the internal structure of the entities involved. Whereas, Statistical thermodynamics studies the behavior of atomic or sub-atomic particles which ultimately results in the Classical thermodynamics outcome.

9 QUANTUM MECHANICS Theoretical science Microscopic Matter Particle & Wave behavior Discrete Properties of individual particles STATISTICAL THERMODYNAMICS Theoretical science Bulk matter Uses Quantum Mechanical results Statistical Mechanics (Probability Theory) Properties of Bulk Matter CLASSICAL THERMODYNAMICS Empirical science Macroscopic matter Continous Properties of Bulk Matter Figure 1: Conceptual linking of quantum mechanics, statistical thermodynamics and classical thermodynamics Throughout this unit, we will dealing with various aspects of Statistical. 4. Fundamentals of Statistical Mechanics Statistical Mechanical treatment of a bulk system depends on the nature of microscopic particles constituting the system. Depending upon the nature of particles present in the bulk, there are three types of statistics that can be applied on the particles. These are: a. Maxwell Boltzmann Statistics, b. Bose Einstein Statistics, and c. Fermi Dirac Statistics,

10 named after their respective inventors. The three types of statistics are applied on a given system depending on nature of particles constituting the system and secondly on the rules governing the filling of various energy levels, i.e., the occupancy per level. Two sub categories of particle statistics are Classical statistics and quantum statistics. Classical statistics ignores the internal structure of the system Quantum statistics, on the other hand, takes account of the internal structure of the system. Maxwell-Boltzmann statistics is a class of classical statistics. While quantum statistics has been classified in two sub divisions- Fermi-Dirac statistics and Bose-Einstein statistics. Over here, we will briefly discuss the salient features of the above mentioned statistics and later on, we will take up each one in detail. a) Maxwell Boltzmann Statistics It is also referred to as classical statistics where 1. The internal structure of the system is ignored. 2. Particles in the system are considered distinguishable from each other. 3. There is no restriction on occupancy of energy levels.

b) Bose Einstein Statistics A sub class of quantum statistics where 1.

11 4. Particles which obey Maxwell-Boltzmann statistics are known as Boltzmannons or Maxwellons. Example: Maxwell Boltzmann statistics can be applied to a system comprising of large number of gas particles. b) Bose Einstein Statistics A sub class of quantum statistics where 1. The internal structure of the system is taken into account. 2. Particles in the system are considered indistinguishable from each other. 3. There is no restriction on occupancy of energy levels. 4. Particles have integer spin.

5. Particles which obey Bose - Einstein statistics are known as Bosons. Example: Photons c) Fermi Dirac Statistics It is a sub class of quantum statistics where 1.

12 5. Particles which obey Bose - Einstein statistics are known as Bosons. Example: Photons c) Fermi Dirac Statistics It is a sub class of quantum statistics where 1. The internal structure of the system is taken into account. 2. Particles in the system are considered indistinguishable from each other. 3. There is restriction on occupancy of energy levels. 4. Particles have half integral spin 5. Particle which obey Fermi Dirac Statistics are known as Fermions. Examples: Electrons Thus, all these types of statistics when applied to particles in bulk matter can be used to calculate the bulk properties.

Statistical thermodynamics applies statistical average on quantum mechanical results to calculate the properties of bulk

13 5. Summary 1. Statistical mechanics is a theoretical science of bulk matter. 2. Statistical thermodynamics is a link between Quantum Mechanics and Classical. 3. Statistical thermodynamics applies statistical average on quantum mechanical results to calculate the properties of bulk matter. 4. Depending upon the nature of the particles present in the bulk matter, 3 types of statistics can be applied on the system: Maxwell-Boltzmann statistics, Fermi-Dirac statistics and Bose-Einstein statistics.

Introduction. Chapter The Purpose of Statistical Mechanics

Introduction. Chapter The Purpose of Statistical Mechanics Chapter 1 Introduction 1.1 The Purpose of Statistical Mechanics Statistical Mechanics is the mechanics developed to treat a collection of a large number of atoms or particles. Such a collection is, for