Define molecular partition functions. Derive an expression for the translational partition function for motion along x- direction in a system.

Understanding Permutation and Configuration 1. Definition of Permutation Permutation refers to the arrangement of a subset of items where the order of arrangement is important. It calculates the number of ways to arrange a certain number of objects from a larger set, considering the sequence in whicRead more

## Understanding Permutation and Configuration

### 1. Definition of Permutation

**Permutation** refers to the arrangement of a subset of items where the order of arrangement is important. It calculates the number of ways to arrange a certain number of objects from a larger set, considering the sequence in which they appear. For instance, arranging the letters A, B, and C would yield different results such as ABC, ACB, BAC, BCA, CAB, and CBA β each arrangement being unique due to the order of the letters.

### 2. Definition of Configuration (Combination)

**Configuration**, often referred to as combination, involves selecting items from a larger set where the order of selection does not matter. Itβs used to calculate the number of ways to choose a subset where the sequence is irrelevant. For example, selecting 2 letters from A, B, and C would result in combinations like AB, AC, and BC, where AB is considered the same as BA.

## Differences Between Permutation and Configuration

**Order Importance:**In permutations, the order of items is important, while in configurations, the order is not considered.**Formulae:**The formula for permutations of \(r\) items from \(n\) is \(P(n, r) = \frac{n!}{(n-r)!}\), and the formula for configurations of \(r\) items from \(n\) is \(C(n, r) = \frac{n!}{r!(n-r)!}\).

## Calculation for Selecting Three Days Out of Seven

### 1. Permutation Calculation

To find the number of permutations of selecting 3 days out of 7 (considering the order in which they are selected matters), we use the permutation formula:

\[

P(7, 3) = \frac{7!}{(7-3)!} = \frac{7 \times 6 \times 5 \times 4!}{4!} = 7 \times 6 \times 5 = 210

\]

There are 210 different ways to select and arrange 3 days from a week.

### 2. Configuration Calculation

To find the number of configurations (combinations) of selecting 3 days out of 7 (where the order does not matter), we use the combination formula:

\[

C(7, 3) = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35

\]

There are 35 ways to select 3 days from a week without considering the order of the days.

## Conclusion

The choice between permutation and configuration depends on whether the order of items is important in a given context. Permutations are used when the sequence affects the outcome, while configurations are appropriate for scenarios where only the choice of items matters, not the sequence. In the example of selecting days from a week, the substantial difference in results between permutations and configurations (210 vs. 35) highlights the impact of considering order.

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Definition of Molecular Partition Functions Molecular partition function is a fundamental concept in statistical mechanics that quantifies the number of available quantum states for a molecule at a given temperature. It plays a crucial role in linking the microscopic quantum states of a system to itRead more

## Definition of Molecular Partition Functions

Molecular partition functionis a fundamental concept in statistical mechanics that quantifies the number of available quantum states for a molecule at a given temperature. It plays a crucial role in linking the microscopic quantum states of a system to its macroscopic thermodynamic properties. The partition function is a sum over all possible energy states of a system, weighted by the Boltzmann factor, \( e^{-\beta E} \), where \( \beta = \frac{1}{k_BT} \) (with \( k_B \) being the Boltzmann constant and \( T \) the temperature), and \( E \) represents the energy levels of the system.## Derivation of the Translational Partition Function for Motion Along the X-direction

## 1. Setting the Framework

The translational partition function quantifies the number of ways a particle can be distributed in space, considering its kinetic energy due to motion. For a single particle in a one-dimensional box (along the x-axis), the energy states can be described by the quantum mechanics of a particle in a box:

\[

E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}

\]

where:

## 2. Expression for the Partition Function

The translational partition function for a particle confined to move along the x-axis in a box of length \( L \) is given by:

\[

q_x = \sum_{n=1}^{\infty} e^{-\beta E_n}

\]

Substituting the expression for \( E_n \):

\[

q_x = \sum_{n=1}^{\infty} e^{-\beta \frac{n^2 \pi^2 \hbar^2}{2mL^2}}

\]

## 3. Approximating the Summation

For high temperatures or large boxes, the energy levels are closely spaced, allowing the summation to be approximated by an integral:

\[

q_x \approx \int_{0}^{\infty} e^{-\beta \frac{\pi^2 \hbar^2 x^2}{2mL^2}} dx

\]

To solve the integral, we perform a change of variables \( u = \frac{\pi \hbar x}{\sqrt{2mL^2 \beta}} \), which simplifies the integral:

\[

q_x \approx \frac{\sqrt{2mL^2 \beta}}{\pi \hbar} \int_{0}^{\infty} e^{-u^2} du

\]

The integral of \( e^{-u^2} \) from 0 to \( \infty \) is \( \frac{\sqrt{\pi}}{2} \), thus:

\[

q_x \approx \frac{\sqrt{2mL^2 \beta}}{\pi \hbar} \cdot \frac{\sqrt{\pi}}{2} = \frac{\sqrt{2\pi mk_BT}}{h}L

\]

This result shows that the translational partition function for motion along the x-direction is proportional to the length of the box and depends on the mass of the particle, the temperature, and the Boltzmann constant.

## Conclusion

The translational partition function for motion in one dimension provides insight into how quantum mechanical properties of particles contribute to macroscopic thermodynamic quantities. It illustrates the dependency of statistical properties on physical dimensions and conditions of the system, such as temperature and size. This concept is extensible to three dimensions and forms the foundation for understanding molecular behavior in gases and other phases.

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