LectureT7

LectureT7 - Chapter 7 In this chapter we solve some of the interesting mysteries that have popped up along our path through thermodynamics We will

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5/28/2004 H133 Spring 2004 1 Chapter 7 In this chapter we solve some of the interesting mysteries that have popped up along our path through thermodynamics. ± We will make extensive use of the Boltzmann Factor that we derived in Chapter 6. ± We want to study the ² Maxwell-Boltzmann Distribution o Tells how molecular velocities are distributed in a gas with temperature T. ² Count Velocity States. ² Average Energy of a Quantum System ² Energy storage in a Gas molecule In our study of a small system in contact with a large reservoir we never stated that the two substances needed to be different. ± They don’t! In fact, we can think of a large container of gas and treat one of the molecules as the small system and the other molecules as the reservoir. ± We know from our study of an ideal gas that ± But we would like to know what the distribution of velocities looks like ² ² ²
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5/28/2004 H133 Spring 2004 2 Maxwell-Boltzmann Distribution So we are looking for something like this We know from our study in chapter 6 that Note: this is the probability that we are in a state with speed v but it is not the probability that it simply has speed v. ± If there are many states with speed v, the probability that the particle will have a speed v is proportional to the number of states with speed v. ± The number of states with speed between v-dv/2 and v+dv/2 is given by v 2 dv ² Rough Justification for this statement o o o o o T k mv T k E B B e e state / / 2 2 1 v) speed with Pr( =
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5/28/2004 H133 Spring 2004 3 Maxwell Boltzmann Distribution So now the probability that we are in any state within dv centered on v is We have a term which depends on mass (m), k B , and T, which has units of m 2 /s 2 . Let’s define that as a special parameter v p . Since v p is just a constant, we can put it in our relationship (it just changes the constant of proportionality) Notice that this version is unitless…like what we would expect for a probability. That means the constant of proportionality that is left to determine is also unitless. By demanding that the total probability is equal to 1 we can determine this remaining constant of proportionality. ± It is 4/ π 1/2 T k mv B dve v 2 / 2 2 v) around dv within Pr( 2 2 v) around dv within Pr( p v v e v dv v v p p 2 / 1 2 2 / 1 2 4 ) ( ) ( v) around dv within Pr( 2 = = =
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This note was uploaded on 06/03/2011 for the course H 133 taught by Professor Furnstahl during the Spring '11 term at Ohio State.

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LectureT7 - Chapter 7 In this chapter we solve some of the interesting mysteries that have popped up along our path through thermodynamics We will

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