Question

Calculate the rotational partition function for CINO at \(500 .\) K where \(B_{A}=2.84 \mathrm{cm}^{-1}, B_{B}=0.187 \mathrm{cm}^{-1}\) and \(B_{C}=0.175 \mathrm{cm}^{-1}\)

Short answer

The rotational partition function for CINO at 500 K, given the rotational constants \(B_{A}=2.84 \mathrm{cm}^{-1}, B_{B}=0.187 \mathrm{cm}^{-1}\) and \(B_{C}=0.175 \mathrm{cm}^{-1}\), is approximately 39589.6.

Step-by-step solution

Identify the constants and given values

We are given the following values: - Temperature T = 500 K - Rotational constants B_A = 2.84 cm⁻¹, B_B = 0.187 cm⁻¹, B_C = 0.175 cm⁻¹ We will also use the following constants: - Planck's constant h = 6.626 × 10⁻³⁴ Js - Speed of light c = 2.998 × 10¹⁰ cm/s - Boltzmann constant k_B = 1.381 × 10⁻²³ J/K For CINO, the symmetry number σ is equal to 1 because it is a nonlinear molecule.

Convert the rotational constants to SI units

The rotational constants are given in wavenumber units (cm⁻¹), so we need to convert them to SI units (J) using the following formula: \[B = hc\tilde{B}\] where B is the rotational constant in J, and \(\tilde{B}\) is the rotational constant in wavenumber units (cm⁻¹). Using the given values of the rotational constants and the constants h and c, we get: - B_A = (6.626 × 10⁻³⁴ Js) × (2.998 × 10¹⁰ cm/s) × (2.84 cm⁻¹) = 5.36 × 10⁻²² J - B_B = (6.626 × 10⁻³⁴ Js) × (2.998 × 10¹⁰ cm/s) × (0.187 cm⁻¹) = 3.52 × 10⁻²³ J - B_C = (6.626 × 10⁻³⁴ Js) × (2.998 × 10¹⁰ cm/s) × (0.175 cm⁻¹) = 3.30 × 10⁻²³ J

Calculate the rotational partition function for CINO

Using the expression for the rotational partition function for a nonlinear molecule: \(q_{rot} = \frac{\pi^{1/2}}{1} \left(\frac{hc}{k_{B}T}\right)^{-3/2} (\frac{1}{B_{A}}\frac{1}{B_{B}}\frac{1}{B_{C}})^{1/2}\) we substitute the given and calculated values: \(q_{rot} = \frac{\pi^{1/2}}{1} \left(\frac{6.626 \times 10^{-34} \text{Js} \cdot 2.998 \times 10^{10} \text{cm/s}}{1.381 \times 10^{-23} \text{J/K} \cdot 500 \text{K}}\right)^{-3/2} (\frac{1}{5.36 \times 10^{-22} \text{J}}\frac{1}{3.52 \times 10^{-23} \text{J}}\frac{1}{3.30 \times 10^{-23} \text{J}})^{1/2}\) After evaluating the expression: \(q_{rot} \approx 39589.6\) #Phase2Solution# The rotational partition function for CINO at 500 K is approximately 39589.6.

Key concepts

Rotational Constants

Rotational constants are crucial in determining the rotational energy levels of molecules. These constants, denoted as \(B_A\), \(B_B\), and \(B_C\), characterize the rotational inertia of the molecule along different principal axes. They are measured in units of wavenumbers (cm⁻¹) and give insight into the geometry and internal structure of the molecule.
To convert these constants into the energy units of joules (J), you can use the formula \( B = hc\tilde{B} \), where \(h\) is Planck's constant and \(c\) is the speed of light. In our example, we converted the given rotational constants into joules to facilitate further thermodynamic calculations. Understanding rotational constants allows us to predict how a molecule will behave when exposed to thermal energy.

Nonlinear Molecules

Nonlinear molecules, such as the chloroiodomethane (CINO) molecule in our example, have a more complex rotational behavior compared to linear molecules. They possess three unique axes of rotation, leading to three distinct rotational constants \(B_A\), \(B_B\), and \(B_C\).
One unique property of nonlinear molecules is their symmetry number \(\sigma\). This number accounts for identical configurations through rotational symmetry. In CINO's case, the symmetry number is \(1\) because it has no axis aligning with symmetrically equivalent arrangements. Understanding the concept of nonlinear molecules is vital for accurately calculating the rotational partition function, as it differs from that of purely linear structures.

Boltzmann Constant

The Boltzmann constant \(k_B\) bridges small-scale physical energy scales with macroscopic temperature measurement. It is a fundamental constant in statistical mechanics, with a value of approximately \(1.381 \times 10^{-23} \, \text{J/K}\).
In the calculation of partition functions, including the rotational partition function, the Boltzmann constant is used in evaluating thermal energy availability per molecule at a specific temperature. It serves as a key parameter to express the relationship between the microscopic and macroscopic aspects of thermodynamic systems. In the solution, \(k_B\) helps convert quantum mechanical concepts into measurable temperature effects.

Partition Function

The partition function is a central concept in thermodynamics and statistical mechanics, representing a statistical sum over all possible energy states a system can occupy. In the context of rotations for molecules like CINO, the rotational partition function \(q_{rot}\) helps determine the distribution of energy states at a given temperature.
The formula for the rotational partition function for a nonlinear molecule integrates the rotational constants and temperature to provide the probability distribution among rotational states. For CINO at 500 K, \(q_{rot}\) equals approximately 39589.6, indicating a broad distribution over rotational energy states due to sufficient thermal energy. Partition functions ultimately allow us to predict molecular behavior in broader thermodynamic contexts, showing how molecules store and exchange energy.