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Chemistry LibreTexts

Homework 14 (Due 7/25/16 @ 10:00 a.m.)

  • Page ID
    52482
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    Section: _____________________________

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    Q1

    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.

    Q2

    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?

    Q3

    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).

    Q4

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

    Q5

    What is the molecular partition function at

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