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Homework 14

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    Assume that the probability of occupying a given energy state is given by the distribution

    \[P(E_i) =Ae^{−E_i/kT}\]

    where \(k\) is Boltzmann's constant.

    1. Consider a collection of three total states with the first state located at \(E_0=0\) and others at \(E_1=kT\) and \(E_2=2kT\), respectively, relative to this first state. What is the normalization constant for this distribution?
    2. How would your answer change if there are five states with \(E_0=kT\) in addition to the single states at 0 and 2kT (hint: this is a degeneracy problem)?
    3. Determine the probability (\(P_1\)) of occupying the energy level \(E_1=kT\) for the cases in which one and five states exist at this energy.


    The number of ways that \(A\) distinguishable particle can be divided into 2 groups containing \(a_1\) and \(a_2 =A -a_1\) objects is

    \[ W(a_1,a_2) = \dfrac{A!}{a_1!a_2!}\]

    this can be expanded to the number of ways that \(A\) distinguishable particle can be divided into groups containing \(a_1\), \(a_2\),… objects are

    \[ W(a_1,a_2, a_3, ...) = \dfrac{A!}{a_1!a_2!a_3! a_4!...} = \dfrac{\sum_k a_k}{\Pi a_k!}\]

    What are the weights and total energies for the following configurations of five particle systems (ordered in five state from 0 to 4 with energies of 0, E, 2E, 3E, and 4E, respectively):

    1. {5,0,0,0,0}
    2. {4,1,0,0,0}
    3. {3,2,0,0,0}
    4. {1,0,4,0,0}
    5. {0,0,0,0,5}
    6. {1,1,1,1,1}

    Which configuration has the greatest weight? How would you interpret this result?


    To find the dominating configuration (that with the largest weight), we must find the \(\{a_i\}\) with the maximum value of W. This is also occurs with the maximum of \(\ln W\) since that is a monotonic growlingly function. We can define entropy as

    \[S = k \ln W\]

    For each of the configurations in Q2, calculate the associated entropy. (Note: the "desire" for a system to move to the most probable configuration is based on statistics and is not a real driving force like force and energy).


    If the system in Q2 were in thermal equilibrium, identify the probability of finding the system in the five energy states assuming \(kT= 2E\).


    What is the molecular partition function at

    1. \(kT=E\)
    2. \(kT=2E\)
    3. \(kT=3E\)

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