3.012 Fundamentals of Materials Science Fall 2005 Lecture 21: 11.22.05 Two Postulates of Statistical Mechanics and the Microscopic Definition of Entropy Today: LAST TIME .........................................................................................................................................................................................2�A simple model: the Einstein solid..............................................................................................................................................2�MICROSTATES IN ISOLATED SYSTEMS.............................................................................................................................................3�The first postulate of statistical mechanics ................................................................................................................................4�The second fundamental postulate of statistical mechanics7 ....................................................................................................5�THE MICROSCOPIC DEFINITION OF ENTROPY ...................................................................................................................................7�Testing the microscopic definition of entropy............................................................................................................................8�The first postulate satisfies the second law ..............................................................................................................................10�THE BOLTZMANN FACTOR AND PARTITION FUNCTION.................................................................................................................11�REFERENCES ...................................................................................................................................................................................12�Reading: Engel and Reid 31.1-31.4 Supplementary Reading: Lecture 22 – Connectiing molecular events t u3.012 Fundamentals of Materials Science�Fall 2005 Last time A simple model: the Einstein solid •� Our introduction to the connection of thermodynamic quantities to microscopic behavior begins by considering a simple model of a monoatomic crystalline solid- models of materials are the starting point for statistical mechanics calculations. •� Suppose for definiteness we have a crystalline solid: As a model for how this material behaves in response to temperature, we propose that the most important degree of freedom available to the atoms to respond to thermal energy is vibration of the atoms about their at-rest positions. o� The bonding between atoms creates a potential energy well in which the atoms are centered at their at rest position. Oscillations of the atoms about the at rest position can be induced by thermal energy in the material. The potential is called a harmonic potential because of its shape-similar to the potential of a spring in classical mechanics. Because the oscillation of the atoms is constrained by bonding to center about their at-rest positions in the crystal lattice, the energy of vibration for each atom is quantized: Model for a 3-atom solid as 1D harmonic oscillators n 7 6 5Energy 4 �E = (9/2)hv - (7/2) hv = hv 3 2 1 0 Microstate Figure by MIT OCW. 2 •� …where h is Planck’s constant (h = 6.62x10-27 gm cm sec-2) and ν is the frequency of the atomic vibrations. The total energy of the solid is the sum of the individual energies of each oscillator: Lecture 22 – Connectiing molecular events t u3.012 Fundamentals of Materials Science�Fall 2005 Microstates in Isolated Systems o� The total number of allowed microstates is a parameter we will refer to again and again; we give it the symbol Ω. For the system above, Ω = 6. The collection of all Ω microstates for a given system is called its ensemble. (For the case of a system with fixed (E,V,N) it is referred to as the microcanonical ensemble). Distinguishable vs. indistinguishable atoms/particles •� Two cases arise in modeling real systems: one where we can identify each atom uniquely, and the case of atoms being identical and indistinguishable. •� If we had indistinguishable atoms, then we would only be able to observe the unique microstates, whose number is given the symbol W: Lecture 22 – Connectiing molecular events t u3.012 Fundamentals of Materials Science�Fall 2005 o� We have collected the individual microstates now into 2 groups characterized by the number of atoms in each energy level (n0, n1, and n2). We will call the set of microstates that has a given set of occupation numbers a state (as opposed to microstate). The total number of distinguishable states is W. •� For our overly-simplified 3-atom model with a low total energy of Etotal = (7/2)ε, the number of distinct arrangements is small. However, for a material containing a mole of atoms at room temperature, the number of possible ways to occupy the available energy levels is enormous. Thus, rather than writing diagrams of all the possible microstates, we become concerned with the probability of finding a certain set of microstates j in the ensemble that have a given distribution of the atoms among the energy levels. The first postulate of statistical mechanics •� The first postulate of statistical mechanics tells us the frequency of each of the possible states or�microstates occurring in the ensemble:�� This postulate is often called the principle of equal a priori probabilities. It says that if the microstates have the same energy, volume, and number of particles, then they occur with equal frequency in the ensemble. o This postulate tells us what the pj’s for the two states in our 3-atom Etotal=7/2εsystem are: � We have a total of W = 2 states, (3 microstates in each of the two unique states j=1, j=2). Thus the probability for each state is: Each state has a frequency of 50% in the ensemble. Lecture 22 – Connectiing molecular events t u3.012 Fundamentals of Materials Science�Fall 2005 The second fundamental postulate of statistical mechanics7 The second postulate connects ensemble averages to measured thermodynamic quantities •� Statistical mechanics makes one postulate that connects the energy-level model of a material to its macroscopic thermodynamic properties: •� What it means: o� The postulate tells us that we can calculate the thermodynamic
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