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