Thermo_ISM_ch13

Thermo_ISM_ch13 - Chapter 13 The Boltzmann Distribution...

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13-1 Chapter 13: The Boltzmann Distribution Problem numbers in italics indicate that the solution is included in the Student’s Solutions Manual. Questions on Concepts Q13.1) What is the difference between a configuration and a microstate? A configuration is a general arrangement of total energy available to the system. A microstate is a specific arrangement of energy in which the energy content of specific oscillators is described. Q13.2) How does one calculate the number of microstates associated with a given configuration? The number of microstates associated with a given configuration is known as the weight of the configuration ( W ), and is given by: W = N ! a n ! n where N is the number of units or oscillators, and a n is the number of units or oscillators with a certain amount of energy (the occupation number). Q13.3) What is an occupation number? How is this number used to describe energy distributions? The occupation number represents the number of units occupying a given energy level. The distribution of energy over a collection of units can be specified as the number of units occupying a given energy level. Q13.4) Explain the significance of the Boltzmann distribution. What does this distribution describe? The Boltzmann distribution is the energy distribution associated with the dominant configuration of energy. It provides a quantitative description of the probability of a given unit occupying a certain energy level. The Boltzmann distribution also represents the energy distribution associated with a chemical system at equilibrium. Q13.5) What is degeneracy? Can you conceptually relate the expression for the partition function without degeneracy to that with degeneracy? Degeneracy is the case where more than one state exists at a certain energy level. The expression for the partition function ( q ) including degeneracy is:
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Chapter 13/The Boltzmann Distribution 13-2 q = g n e βε n n In this expression, g n is the degeneracy of a given energy level, ε n is the energy of the level, and the sum extends over all energy levels. Q13.6) How is β related to temperature? What are the units of kT ? β is the inversely proportional to temperature, and equal to ( kT ) –1 . The units of the Boltzmann constant are J K –1 ; therefore, the product kT has units of joules, or energy. Problems P13.1) a) What is the possible number of microstates associated with tossing a coin N times and having it come up H times heads and T times tails? b) For a series of 1000 tosses, what is the total number of microstates associated with 50% heads and 50% tails? c) How less probable is the outcome that the coin will land 40% heads and 60% tails? a) In this case, the number of coin tosses is equal to the number of units ( N ), and each unit can exist in one of two states: heads ( H ) or tails ( T ). Since the number of microstates is equal to the weight ( W ): W = N !
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