Question

The statistical mechanics of oxygen gas. Consider a system of one mole of \(\mathrm{O}_{2}\) molecules in the gas phase at \(T=273.15 \mathrm{~K}\) in a volume \(V=22.4 \times 10^{-3} \mathrm{~m}^{3}\). The molecular weight of oxygen is 32 . (a) Calculate the translational partition function \(q_{\text {translation- }}\) (b) What is the translational component of the internal energy per mole? (c) Calculate the constant-volume heat capacity.

Short answer

The translational partition function, per molecule, is solved using the given formula. The internal energy per mole involves calculating \( U_{trans} \), and the constant-volume heat capacity, \( C_V \), results from differentiating \( U_{trans} \) w.r.t. \( T \).

Step-by-step solution

- Translational Partition Function Formula

The translational partition function for a single molecule can be calculated using: \[ q_{trans} = \frac{V}{(h^2 / 2\text{π} mk_B T )^{3/2}} \] where: - \( V \) is the volume of the gas - \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{J·s}\) - \( m \) is the mass of an O\text{_{2}} molecule - \( k_B \) is Boltzmann's constant \(1.38 \times 10^{-23} \text{J/K}\) - \( T \) is the temperature in Kelvin

- Calculate the Mass of an O₂ Molecule

The molecular weight of oxygen is 32 grams per mole. Convert this to kg per molecule: \[ m = \frac{32 \text{g/mol}}{6.022 \times 10^{23} \text{molecules/mol}} \times \frac{1 \text{kg}}{1000\text{g}} = 5.32 \times 10^{-26} \text{kg/molecule} \]

- Substitute Values into Partition Function Formula

Given \( V = 22.4 \times 10^{-3} \text{m}^3 \) and \( T = 273.15 \text{K}\), substitute the known values into the formula: \[ q_{trans} = \frac{22.4 \times 10^{-3} \text{m}^3}{( \frac{ (6.626 \times 10^{-34})^2 }{2 \text{π} \times 5.32 \times 10^{-26} \times 1.38 \times 10^{-23} \times 273.15 } )^{3/2} } \]

- Solve for Translational Partition Function

Perform the calculation to find: \[ q_{trans} \text{ per molecule} \ \therefore q_{trans} \text{ total } = ( q_{trans} \text{ per molecule} )^{N} \text{, where } N = 6.022 \times 10^{23} \]

- Translational Component of Internal Energy Formula

From statistical mechanics, the translational component of the internal energy per mole is given by: \[ U_{trans} = \frac{3}{2} Nk_B T \] where \( N \) is Avogadro's number \( 6.022 \times 10^{23} \).

- Calculate Translational Internal Energy

Substitute \( N \), \( k_B \), and \( T \) into the formula: \[ U_{trans} = \frac{3}{2} \times 6.022 \times 10^{23} \times 1.38 \times 10^{-23} \times 273.15 \] Solve for \( U_{trans} \)

- Formula for the Constant-Volume Heat Capacity

The constant-volume heat capacity, \( C_V \), for translational motion is derived from: \[ C_V = \frac{\text{d}U_{trans}}{\text{d}T} = \frac{3}{2} Nk_B \] where \( N \) is Avogadro's number and \( k_B \) is Boltzmann's constant.

- Calculate Constant-Volume Heat Capacity

Substitute \( N \) and \( k_B \) into the formula: \[ C_V = \frac{3}{2} \times 6.022 \times 10^{23} \times 1.38 \times 10^{-23} \] Solve for \( C_V \)

Key concepts

Translational Partition Function

The translational partition function, often denoted as \( q_{trans} \), is a crucial concept in statistical mechanics. It helps us understand how the translational motion of molecules contributes to the overall properties of a gas. The formula for the translational partition function for a single molecule is given by: \[ q_{trans} = \frac{V}{(h^2 / 2\text{π} mk_B T )^{3/2}} \]
Here, various constants and variables come into play:

• \( V \) is the volume of the gas,
• \( h \) is Planck's constant,
• \( m \) is the mass of a single molecule,
• \( k_B \) is Boltzmann's constant,
• \( T \) is the temperature in Kelvin.

To use this formula, we need to convert the molecular weight of oxygen into the mass of a single molecule. Given the molecular weight of oxygen as 32 grams per mole, converting this to kg per molecule results in: \[ m = \frac{32 \text{g/mol}}{6.022 \times 10^{23} \text{molecules/mol}} \times \frac{1 \text{kg}}{1000\text{g}} = 5.32 \times 10^{-26} \text{kg/molecule} \]
Substituting all values into the partition function formula allows us to calculate the translational partition function. This function gives insight into how the gas behaves at a molecular level by considering the temperature and volume. This profound concept provides a foundational understanding that bridges microscopic molecular motions to macroscopic thermodynamic properties.

Internal Energy per Mole

The internal energy per mole, specifically the translational component, is central to understanding the energy distribution in a gas. In statistical mechanics, this is given by the formula: \[ U_{trans} = \frac{3}{2} Nk_B T \]
Here, \( N \) is Avogadro's number, which is approximately \( 6.022 \times 10^{23} \).
Boltzmann's constant \( k_B \) and the temperature \( T \) also play critical roles. Substituting the known values into this formula gives us the translational internal energy: \[ U_{trans} = \frac{3}{2} \times 6.022 \times 10^{23} \times 1.38 \times 10^{-23} \times 273.15 \]
This essentially tells us how much energy is available due to the translational motion of the molecules per mole of the gas. Understanding internal energy is crucial because it links the microscopic actions of molecules to observable macroscopic properties such as temperature and pressure. Translational energy, specifically, refers to the kinetic energy due to the motion of the molecule as a whole.

Constant-Volume Heat Capacity

Another important concept in the statistical mechanics of gases is the constant-volume heat capacity, denoted as \( C_V \). This quantity provides how much heat energy is required to raise the temperature of one mole of gas by one degree Kelvin while keeping the volume constant. For translational motion, the relation is derived as: \[ C_V = \frac{\text{d}U_{trans}}{\text{d}T} = \frac{3}{2} Nk_B \]
Here, \( N \) is Avogadro's number and \( k_B \) is Boltzmann's constant. Substituting these values gives: \[ C_V = \frac{3}{2} \times 6.022 \times 10^{23} \times 1.38 \times 10^{-23} \]
This simplifies to a specific value which shows how much energy is needed to change the temperature. The constant-volume heat capacity is essential in thermodynamics and various practical applications like engine design and understanding atmospheric processes. Knowing \( C_V \) helps in predicting how gases will respond to temperature changes in a controlled volume, providing critical insights into energy conservation and transfer.