Statistical Mechanics (Unit 2).docx

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Massey University
123 .201
John Harrison

123201 Notes Statistical Mechanics  relating the thermodynamic properties to behaviour at an atomic level The Quantum Structure of Matter There are a variety of ways that a molecule can store energy:  Translational  Rotational  Vibrational  Electrical The Quantum States are the certain values that are allowed for each type of energy (as all the types of energy above are quantised) εi= the energy of quantum state i ∑ ni= the number of molecules in quantum state i ∑ ∑ Differential: The total energy can be altered by either changing the population in each level (di) or changing the energy of the levels themselves (dε) – often it is hard to change the energy i levels, so the population of each energy level is changed The entropy is a measure of the spread of the occupied levels Probability and Entropy Basic Probability Concepts n A represent the number in the category A N = total number of possible outcomes Independent Events occur when the outcomes in each event does not depend on the outcome of any other events. Mutually Exclusive Events is when one outcome prevents that value from being obtained as an outcome of any of the other events. Multiplicity of Events is a weighting for each type of event. There can be multiple ways of the same outcome occurring. Correlated Events is when the outcome of one event influences the outcome of another 1 123201 Notes Relationship between Probability and Entropy S = Entropy k = Boltzmann’s Constant W = probability of finding the molecules in a particular configuration or microstates R = ideal gas constant and A = Avogadro’s Number  A maximum for S is equivalent to the most probable state (and largest W) There can be a significant different between the distribution of energy in the most probable state and the next probable. In this example there is more than 500 ways of distributing the energy in the most probable state. Lattice Models  take advantage of the quantum mechanical nature of matter, where not all values for all quantities are possible A Lattice Model for Gas Pressure Arranging N objects into M sites: ( ) ( ) The probability that the molecules are concentrated in one area is very unlikely, with the most probable arrangement that sites are filled over the whole space. 2 123201 Notes A Lattice Model for Diffusion W = number of permutation possibilities  The most even distribution of particles corresponds to the largest number of possible permuations  The system will evolve towards the state of maximum multiplicity (most even distribution of particles) relates to the Second Law of Thermodynamics A Lattice Model for a Polymer Chain This is used for predicting the polymer behaviour when it is fixed at one end to a surface In this example the red polymer unit remains fixed to the surface. The possible configurations (R:B:Y) are: (1:2:3), (1:2:2), (1:1:2), (1:1:1) However not all the configurations have the same weight, multiple ways of arranging the units can arise in certain configurations The average distance from the surface polymer is equal to the weighted average of the furthest distance: Average Length = (1x3 +2x2+2x2+2x1)/7 = 1.86 This proves that the most likely state for a polymer is the partially contracted state (this is a statistical interpretation of elasticity). 3 123201 Notes A Lattice Model for Heat Flow  The tendency is for the energy to be distributed between two evenly sized systems  When the systems are not evenly sized, the most likely case is the one with the highest multiplicity. This corresponds to the situation where the fraction of molecules in the excited states are the same for the two systems (both left and right) The ratio between systems is related to the temperature of the system – hence this is a statistical interpretation of the zeroth law of thermodynamics. (This does not imply that both systems have the same internal energy as they may have different heat capacities). Entropy and Constraints When all possible states are equally likely the principle maximum entropy takes the form of a uniform distribution over all these possible states. p = probabilities ∑ n = different outcomes
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