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# Homework 14

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

## 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. $$kT=3E$$

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