Given a gaseous system of \(N_{\mathrm{A}}\) indistinguishable, weakly interacting diatomic molecules: (a) Each molecule may vibrate with the same frequency \(v\), but with an energy \(\epsilon_{i}\), given by $$ \epsilon_{i}=\left(\frac{1}{2}+i\right) h v \quad(i=0,1,2, \cdots) $$ Show that the vibrational partition function \(Z_{y}\) is $$ Z_{v}=\frac{e^{-h v / 2 k T}}{1-e^{-h v / k T}} $$ (b) Each molecule may rotate, and the rotational partition function \(Z\), has the same form as that for translation, except that the volume \(V\) is replaced by the total solid angle \(4 \pi\), the mass is replaced by the moment of inertia 1, and the exponent \(\frac{3}{2}\) (referring to three translational degrees of freedom) is replaced by \(\frac{2}{2}\) since there are only two rotational degrees of freedom. Write the rotational partition function. (c) Taking into account translation, vibration, and rotation, calculate the Helmholtz function. (d) Calculate the pressure. (e) Calculate the internal energy. (f) Calculate the molar heat capacity at constant volume.

Step by step solution

01

Define the vibrational energy levels

Given a diatomic molecule with vibrational frequencies, the vibrational energy levels are given by \[\[\begin{equation}\begin{aligned}\text{Energy level:} \text{\( \epsilon_{i} = \left(\frac{1}{2} + i\right) h v\, (i = 0,1,2, \cdots) \)}\end{aligned}\end{equation}\]\]

02

Write the vibrational partition function

The vibrational partition function \(Z_{v}\) can be written as the infinite sum of Boltzmann factors for each energy level: \[Z_{v} = \sum_{i=0}^{\infty} e^{-\epsilon_{i}/kT}\]

03

Substitute the expression for \(\epsilon_{i}\)

Substitute \( \epsilon_{i} = \left( \frac{1}{2} + i\right) hv\) into the partition function: \[Z_{v} = \sum_{i=0}^{\infty} e^{- \left( \frac{1}{2} hv + ihv \right) / kT}\]

04

Separate exponential terms

The sum of exponential terms can be separated: \[Z_{v} = e^{-hv / 2kT} \sum_{i=0}^{\infty} \left( e^{-hv / kT} \right)^i\]

05

Use geometric series

Recognize the geometric series on the right-hand side: \[\sum_{i=0}^{\infty} \left( e^{-hv / kT} \right)^i = \frac{1}{1 - e^{-hv / kT}}\]. Applying this results with the factor, the total partition function is \[Z_{v} = \frac{e^{-hv / 2kT}}{1 - e^{-hv / kT}}\]

06

Rotational partition function

For the rotational partition function with two degrees of freedom and moment of inertia \(I\), the partition function is given by \[Z_{rot} = \frac{8\pi^2I kT}{h^2}\]

07

Translational partition function

For the translational partition function with the volume \(V\) and mass \(m\), it can be written as \[Z_{trans} = \left( \frac{2\pi m k T}{h^2} \right)^{3/2} V\]

08

Total partition function

The total partition function considering translation, vibration, and rotation is: \ \[Z_{total} = Z_{trans} \cdot Z_{rot} \cdot Z_{v}\]

09

Helmholtz function

The Helmholtz function \(A\) is given by \[A = -kT \, \ln(Z_{total})\].

10

Pressure calculation

Using the Helmholtz function and partition function, the pressure \(P\) is derived by \[P = kT \, \left( \frac{\partial \, \ln(Z_{total})}{\partial V} \right)_{T}\]

11

Internal energy

The internal energy \(U\) can be calculated as \[U = A + TS = kT^2 \left( \frac{\partial \, \ln(Z_{total})}{\partial T} \right)_V\]

12

Heat capacity

The molar heat capacity at constant volume \(C_V\) is calculated from the internal energy: \[C_V = \left( \frac{\partial U}{\partial T} \right)_V\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Diatomic Molecules

Diatomic molecules consist of two atoms bonded together. Examples include oxygen (O2) and nitrogen (N2). These molecules can undergo different types of motion: translational, rotational, and vibrational. Understanding these motions is crucial, as each type contributes to the energy and partition functions of the molecule. Translational motion refers to the movement of the molecule as a whole, while rotational motion involves spinning around an axis. Vibrational motion occurs when the distance between the two atoms changes periodically.

For diatomic molecules, their ability to vibrate and rotate adds complexity to their thermodynamic properties. By considering these motions, we can derive important quantities in a system of diatomic molecules, such as energy levels and partition functions.

Boltzmann Factors

Boltzmann factors are crucial in statistical mechanics as they help calculate probabilities of states. For a state with energy \(\epsilon_i\), the Boltzmann factor is given by \( e^{-\epsilon_i / kT}\). This factor quantifies the likelihood of a system being in a particular energy state at temperature \( T \).

For the vibrational energy levels of a diatomic molecule, we use the expression \( \epsilon_i = \left( \frac{1}{2} + i \right) hv \). The Boltzmann factor for each state thus becomes \( e^{-\left( \frac{1}{2} + i \right) hv / kT} \), and these factors are used to sum over all possible states in the partition function.

Helmholtz Function

The Helmholtz function (or free energy) is a thermodynamic potential that measures the work obtainable from a closed system at constant temperature and volume. It is denoted as \( A = -kT \, ln(Z) \). Here, \( Z \) represents the total partition function of the system, which includes contributions from translational, rotational, and vibrational motions.

By combining these contributions, we can calculate the total Helmholtz function for the system. This function is important as it helps determine the equilibrium state of the system and provides insights into other properties like pressure and internal energy.

Rotational Partition Function

The rotational partition function accounts for the molecule's rotational energy levels. For a diatomic molecule with a moment of inertia \( I \), and two degrees of rotational freedom, the rotational partition function is given by \( Z_{rot} = \frac{8\pi^2I kT}{h^2} \). Here, \( k \) is the Boltzmann constant and \( T \) is the temperature.

This formula is derived by replacing the mass with the moment of inertia and considering the rotational degrees of freedom, which influences the rotational states a molecule can access. These accessible states are integral to calculating the thermodynamic properties of diatomic gases.

Translational Partition Function

The translational partition function considers the kinetic energy states due to the molecule's movement in space. For a system containing volume \( V \) and molecules of mass \( m \), the translational partition function is given by \( Z_{trans} = \left( \frac{2\pi m k T}{h^2} \right)^{3/2} V \).

This partition function arises from integrating over all possible translational states of the molecule. The resulting value helps determine the contribution of translational motion to the overall partition function and, consequently, to the thermodynamic properties, such as pressure and energy, of the gas.