Question

Determine the total molecular partition function for \(\mathrm{I}_{2}\) confined to a volume of \(1000 . \mathrm{cm}^{3}\) at \(298 \mathrm{K}\). Other information you will find useful is that \(B=0.0374 \mathrm{cm}^{-1},\left(\widetilde{\nu}=208 \mathrm{cm}^{-1}\right)\) and the ground electronic state is nondegenerate.

Short answer

The total molecular partition function (q) for I₂ confined to a volume of 1000 cm³ at 298 K can be calculated by multiplying the translational, rotational, vibrational, and electronic partition functions together. After calculating each partition function component using the given information, we get: \[ q=\left(\frac{1.00e^{-3}m^{3}}{\Lambda^3}\right) \cdot \left(\frac{(1.381e-23 J K^{-1})(298 K)}{2(6.626e-34 Js)(3\times10^8 m/s)(0.0374 cm^{-1})}\right) \cdot \left(\frac{1}{1-e^{-\frac{(6.626e-34 Js)(3\times10^8 m/s)(208 cm^{-1})}{(1.381e-23 J K^{-1})(298 K)}}}\right) \cdot (1) \]

Step-by-step solution

Calculate the translational partition function

To calculate the translational partition function, we use the following formula: \[ q_{trans}=\frac{V}{\Lambda^3} \] where V is the volume and Λ is the thermal wavelength given by: \[\Lambda=\sqrt{\frac{h^2}{2\pi mk_{B}T}} \] Here, h is the Planck's constant, m is the molecular mass of I2, kB is the Boltzmann constant, and T is the temperature. We first need to find the molecular mass for I2 and convert it to kg/mol. The atomic mass of I is 126.9 amu; therefore, the molecular mass of I2 is 253.8 amu, which is equal to 253.8 * 1.66054e-27 kg. Now, we can find Λ: \[\Lambda=\sqrt{\frac{(6.626e-34 Js)^2}{2\pi(253.8 * 1.66054e-27 kg)(1.381e-23 J K^{-1})(298 K)}}\] And finally, the translational partition function: \[ q_{trans}=\frac{1.00e^{-3}m^{3}}{\Lambda^3} \]

Calculate the rotational partition function

The rotational partition function is given by the following formula: \[ q_{rot}=\frac{k_{B}T}{\sigma hcB} \] where σ is the symmetry number, h is the Planck's constant, c is the speed of light, B is the rotational constant, and T is the temperature. For I2, σ=2. Substituting the given values, we get: \[ q_{rot}=\frac{(1.381e-23 J K^{-1})(298 K)}{2(6.626e-34 Js)(3\times10^8 m/s)(0.0374 cm^{-1})} \]

Calculate the vibrational partition function

The vibrational partition function is given by: \[ q_{vib}=\frac{1}{1-e^{-\frac{hc\widetilde{\nu}}{k_{B}T}}} \] where h is the Planck's constant, c is the speed of light, ν̃ is the vibrational frequency, kB is the Boltzmann constant, and T is the temperature. Substituting the given values, we get: \[ q_{vib}=\frac{1}{1-e^{-\frac{(6.626e-34 Js)(3\times10^8 m/s)(208 cm^{-1})}{(1.381e-23 J K^{-1})(298 K)}}} \]

Calculate the electronic partition function

Since the ground electronic state is nondegenerate, the electronic partition function (q_elec) is 1.

Calculate the total partition function

Now, we can calculate the total molecular partition function (q) by multiplying the four partition functions together: \[ q=q_{trans} \cdot q_{rot} \cdot q_{vib} \cdot q_{elec} \] Substitute the values found in Step 1 through Step 4: \[ q=\left(\frac{1.00e^{-3}m^{3}}{\Lambda^3}\right) \cdot \left(\frac{(1.381e-23 J K^{-1})(298 K)}{2(6.626e-34 Js)(3\times10^8 m/s)(0.0374 cm^{-1})}\right) \cdot \left(\frac{1}{1-e^{-\frac{(6.626e-34 Js)(3\times10^8 m/s)(208 cm^{-1})}{(1.381e-23 J K^{-1})(298 K)}}}\right) \cdot (1) \] Now, we can calculate the total molecular partition function for I2 confined to a volume of 1000 cm³ at 298 K using the given information.

Key concepts

Translational Partition Function

When we talk about the translational partition function, we're looking at how molecules move around in space. This concept is crucial because it helps us understand how many different states a molecule can be in, purely based on its movement.
This partition function is calculated using the formula: \[ q_{trans}=\frac{V}{\Lambda^3} \] where \(V\) stands for volume, and \(\Lambda\) represents the thermal wavelength.

• The thermal wavelength, \(\Lambda\), is computed as: \[\Lambda=\sqrt{\frac{h^2}{2\pi mk_{B}T}} \]
• Here, \(h\) is the Planck's constant, \(m\) is the mass of the molecule, \(k_{B}\) is the Boltzmann constant, and \(T\) is the temperature.

By knowing the molecular mass of iodine (\(\mathrm{I}_2\)), and applying the known constants, we can find \(\Lambda\) and subsequently \(q_{trans}\). This function essentially tells us how the molecule behaves in space, like in a gas confined to a specific volume.

Rotational Partition Function

The rotational partition function gives insight into how molecules rotate. For diatomic molecules like \(\mathrm{I}_2\), this rotation is around an axis perpendicular to the bond connecting the two atoms. This function is given by: \[ q_{rot}=\frac{k_{B}T}{\sigma hcB} \] where \(\sigma\) is the symmetry number, \(h\) is Planck's constant, \(c\) is the speed of light, \(B\) is the rotational constant, and \(T\) is the temperature.

• The symmetry number, \(\sigma\), represents how many indistinguishable rotations a molecule can have. For \(\mathrm{I}_2\), \(\sigma=2\).
• This partition function provides a count of how many quantum states are available to a molecule when it only considers rotational movement.

Understanding the rotational partition function is key to exploring molecular behavior, especially in conditions where rotation impacts energy levels, like in specific temperature ranges.

Vibrational Partition Function

The vibrational partition function deals with how molecules vibrate at various temperatures. This is particularly interesting for diatomic molecules since they can be thought of as two masses connected by a spring. The function is given by: \[ q_{vib}=\frac{1}{1-e^{-\frac{hc\widetilde{u}}{k_{B}T}}} \] Here, \(\widetilde{u}\) is the vibrational frequency, \(h\) is Planck's constant, \(c\) is the speed of light, \(k_{B}\) is Boltzmann's constant, and \(T\) is temperature.

• This function describes the number of available vibrational states for a molecule.
• Vibration pertains to the stretching and compressing of bonds within a molecule.

The vibrational partition function is particularly useful in understanding molecules at different temperatures, as higher temperatures enable access to higher vibrational states. It's vital for studying molecular interactions and spectral properties.

Thermal Wavelength

The thermal wavelength \(\Lambda\) is a fascinating concept because it gives us a scale of how 'spread out' or 'wavy' a molecule appears due to its kinetic energy. It's calculated as: \[\Lambda=\sqrt{\frac{h^2}{2\pi mk_{B}T}} \] where all variables are as previously defined.

• The thermal wavelength is important in the context of the translational partition function, where it modifies the volume's role to assess available states.
• A smaller thermal wavelength indicates higher momentum and vice versa.

Understanding \(\Lambda\) helps in discerning molecular behavior at different temperatures, as it essentially measures how the thermal energy translates into motion, bridging the gap between quantum mechanics and macroscopic observations.

Ground Electronic State

The concept of the ground electronic state is crucial when considering the energy levels of molecules. It's essentially the state with the lowest energy.

• For \(\mathrm{I}_2\) in this context, the ground electronic state is nondegenerate, meaning it does not have any other electronic state with the same energy.
• When calculating the electronic partition function \(q_{elec}\), if the ground state is nondegenerate, \(q_{elec} = 1\).

The ground electronic state's significance lies in its role as the baseline for assessing energy changes due to molecular interactions at varied temperatures. It simplifies the calculation of the electronic partition function, making it a central point in energy state analysis.