Question

Calculate the rotational partition function for the interhalogen compound \(\mathrm{F}^{35} \mathrm{Cl}\left(B=0.516 \mathrm{cm}^{-1}\right)\) at \(298 \mathrm{K}\).

Short answer

The rotational partition function for the interhalogen compound F35Cl with the rotational constant \(B = 0.516\ \text{cm}^{-1}\) at \(298\ \text{K}\) can be calculated using the rigid rotor model equation: \(q_\text{rot} = \frac{kT}{hcB}\) By substituting the given values and constants, we find that the rotational partition function \(q_\text{rot} \approx 444.26\).

Step-by-step solution

The constant term in the formula is \(\frac{kT}{hcB}\), where k is the Boltzmann constant, T is the temperature, h is the Planck constant, and B is the rotational constant. Using the given values, calculate the constant term: \(\frac{kT}{hcB} = \frac{(1.3806 \times 10^{-16}\ \text{erg/K})(298\ \text{K})}{(6.6261 \times 10^{-27}\ \text{erg s})(0.516\ \text{cm}^{-1})}\) #tag_Step 2# Calculate the rotational partition function

Now use the formula for the rotational partition function: \(q_\text{rot} = \frac{kT}{hcB}\) Substitute the constant term that we calculated in Step 1: \(q_\text{rot} = \frac{(1.3806 \times 10^{-16}\ \text{erg/K})(298\ \text{K})}{(6.6261 \times 10^{-27}\ \text{erg s})(0.516\ \text{cm}^{-1})}\) Now, solve for q_rot: \(q_\text{rot} \approx 444.26\) So, the rotational partition function for the interhalogen compound F35Cl at 298K is approximately 444.26.

Key concepts

Boltzmann Constant

The Boltzmann constant (\( k \)) is a fundamental physical constant that plays a pivotal role in the realm of thermodynamics and statistical mechanics. It serves as a bridge between macroscopic and microscopic physics by relating the temperature of a gas with the average kinetic energy of its molecules.

In the context of the rotational partition function, the Boltzmann constant is utilized to calculate the distribution of energy levels that molecules can occupy at a given temperature. In our exercise, the Boltzmann constant's value is used (\( 1.3806 \times 10^{-16}\text{ erg/K} \)), providing insight into how the rotational energy levels are populated by the interhalogen compound molecules at 298 K.

An important application of the Boltzmann constant is in the formula \( q_{\text{rot}} = \frac{kT}{hcB} \), which neatly encapsulates the relationship between the thermal energy at temperature T and the quantized rotational energy levels defined by the rotational constant B.

Planck Constant

The Planck constant (\( h \) is another fundamental physical constant crucial to quantum mechanics. Named after Max Planck, it defines the quantization of energy in the universe. Its role is fundamental in calculating the energy of a photon from its frequency, with the simple relationship \( E = hu \), where \( u \) is the frequency.

In our exercise, the Planck constant appears in the denominator of the rotational partition function formula. The value of the Planck constant used here is \( 6.6261 \times 10^{-27}\text{ erg s} \) which, in combination with the Boltzmann constant and rotational constant, helps determine the correct scale for the energy levels that are quantized by the rules of quantum mechanics.

This understanding is essential for students as it directly influences the total number of accessible rotational states at a given temperature, affecting the calculated partition function for the given interhalogen compound.

Rotational Constant

The rotational constant (\( B \) is a specific value that characterizes the rigidity of a rotating diatomic or linear polyatomic molecule. It is directly related to the moment of inertia (\( I \) of the molecule, and inversely proportional to it as given by \( B = \frac{h}{8\pi^{2}Ic} \), where \( c \) is the speed of light in vacuum.

In simpler terms, the rotational constant gives us an idea of how 'spread out' a molecule's mass is concerning its axis of rotation. The higher the value of B, the more tightly packed or lower the moment of inertia of the molecule. In the step-by-step solution, the rotational constant for the interhalogen compound F\textsuperscript{35}Cl is given as 0.516 cm\textsuperscript{-1}.

Grasping the concept of the rotational constant is crucial for students, as it influences the spacing between the rotational energy levels. Molecules with larger moments of inertia will have smaller rotational constants and therefore more closely spaced energy levels.

Interhalogen Compound

Interhalogen compounds are a class of molecules composed of two different halogens. Common examples include chlorine trifluoride (\( \text{ClF}_3 \) and bromine monofluoride (\( \text{BrF} \). These compounds typically have one halogen acting as a central atom bonded to one or more different halogen atoms.

In our exercise, the interhalogen compound of interest is F\textsuperscript{35}Cl, which is composed of fluorine (\( \text{F} \) and chlorine (\( \text{Cl} \). These molecules have distinct rotational spectra because of their diatomic nature and the different masses of the atoms involved.

The study of the rotational partition function of interhalogen compounds like F\textsuperscript{35}Cl is especially important in the understanding of their thermodynamic properties and behavior under various temperatures. Knowing the rotational partition function helps predict how the compound will interact with light, which is essential in fields such as spectroscopy and atmospheric chemistry.