Question

The effect of symmetry on the rotational partition function for \(\mathrm{H}_{2}\) was evaluated by recognizing that each hydrogen is a spin \(1 / 2\) particle and is, therefore, a fermion. However, this development is not limited to fermions, but is also applicable to bosons. Consider \(\mathrm{CO}_{2}\) in which rotation by \(180^{\circ}\) results in the interchange of two spin 0 particles. a. Because the overall wave function describing the interchange of two bosons must be symmetric with respect to exchange, to what \(J\) levels is the summation limited in evaluating \(q_{R}\) for \(\mathrm{CO}_{2} ?\) b. The rotational constant for \(\mathrm{CO}_{2}\) is \(0.390 \mathrm{cm}^{-1}\). Calculate \(q_{R}\) at \(298 \mathrm{K} .\) Do you have to evaluate \(q_{R}\) by summation of the allowed rotational energy levels? Why or why not?

Short answer

In summary, the allowed J levels for CO2 in the rotational partition function are even values (J=0, 2, 4, 6, ...) to maintain the symmetric wave function's characteristics. The rotational partition function at 298 K can be calculated using the given rotational constant (B = 0.390 cm⁻¹) and the provided temperature (T = 298 K). Due to the converging nature of the partition function, manual summation of all allowed energy levels is not necessary, as various techniques (such as numerical methods or approximations) can be utilized for more efficient calculations.

Step-by-step solution

Identify symmetry requirements for CO2 and allowed J levels

Given that the CO2 molecule consists of two bosons, their wave function must be symmetric with respect to exchange. This means that the allowed J levels for CO2 in the rotational partition function will be even values (J=0, 2, 4, 6, ...). These even values will maintain the symmetric characteristics of the wave function describing the interchange of two bosons.

Calculate the rotational partition function qR for CO2 at 298 K

We are given the rotational constant B for CO2 (B = 0.390 cm⁻¹) and the temperature T (T = 298 K). Since the allowed J levels are all even values, we can calculate the rotational partition function qR as follows: \(q_{R}(T) = \sum_{J=0, 2, 4, 6, ...} g_J e^{-\frac{E_J}{k_\mathrm{B}T}}\) Where \(g_J\) is the degeneracy of the Jth level (given by \(g_J = 2J + 1\)), \(E_J\) is the energy of the Jth level (given by \(E_J = BJ(J+1)\)) and \(k_\mathrm{B}\) is the Boltzmann constant (\(k_\mathrm{B}=1.38064852 × 10^{-23}\) J/K). To obtain \(q_R(T)\), we need to sum over all allowed J levels. However, we don't need to perform the summation manually, as it is a convergence series, and we can use statistical mechanics principles to calculate the partition function.

Evaluate whether summation of energy levels is necessary for qR

As previously mentioned, the partition function is a converging function, meaning that beyond a certain point, adding more terms to the sum will not change the overall value by much. In other words, the partition function converges quickly and the contribution of higher energy levels approaches zero. Since we know that the sum converges, we don't need to calculate the partition function by manually summing all allowed energy levels. Instead, we can use various techniques (such as numerical methods or approximations) to calculate the partition function more efficiently. These techniques will often yield a result within an acceptable margin of error, without having to sum over all allowed energy levels. In summary, the allowed J levels for CO2 in the rotational partition function are even values, maintaining the symmetric wave function's characteristics. The rotational partition function at 298 K can be calculated using the provided rotational constant, and we do not need to evaluate the partition function by manually summing all allowed energy levels due to convergence and the ability to utilize other techniques for more efficient calculations.

Key concepts

Symmetry in Molecular Systems

Symmetry plays a critical role in understanding molecular systems, especially when evaluating rotational partition functions. In molecules like \( ext{CO}_2\), symmetry affects the allowable rotational energy levels. For \( ext{CO}_2\), rotating by \({180}^\circ\) results in an equivalent structure due to its symmetric linear shape. Symmetry determines which rotational quantum numbers \(J\) are permitted, impacting how we calculate the rotational partition function. This means only even \(J\) values will be allowed for \( ext{CO}_2\) because it maintains the symmetric characteristics of the molecule. By limiting the \(J\) levels to even numbers, it reflects the fundamentally symmetric nature of the physical molecule, which must be consistent with its quantum mechanical description.
Understanding symmetry helps in simplifying calculations and in ensuring that the physical and theoretical models align.

Bosons and Fermions

Bosons and fermions are fundamental types of particles distinguished by their spin characteristics. Bosons, like the particles in \( ext{CO}_2\), have integer spins (spin 0 in this case), and their wave functions are symmetric under particle exchange.
This symmetry leads to specific rules about permissible states and energy levels when constructing a partition function.

• Bosons: Obey Bose-Einstein statistics; their wave functions are symmetric. For \( ext{CO}_2\), this symmetry requirement means only even \(J\) levels are considered.
• Fermions: Have half-integer spins (e.g., 1/2); obey Fermi-Dirac statistics. Their wave functions are antisymmetric, requiring different symmetry considerations.

Considering whether particles are bosons or fermions is essential when analyzing molecular rotational states, as it influences the partition function structure and thus the thermodynamic properties of the system.

Rotational Energy Levels

Rotational energy levels are quantized states that a molecule can occupy as it rotates. For a linear molecule like \( ext{CO}_2\), the energy for each rotational level \(E_J\) is calculated using: \[ E_J = BJ(J+1) \] where \(B\) is the rotational constant, and \(J\) is the rotational quantum number. Each level's degeneracy \(g_J\) is given by \(2J + 1\).
Given \( ext{CO}_2\)'s symmetrical nature, only even \(J\) values are allowed, which greatly affects the calculation of the rotational partition function.

• Each allowed rotational energy state contributes to the molecule's overall energy distribution at a given temperature.
• The significance of contributions from higher \(J\) levels diminishes as energy increases and the partition function converges.

Recognizing these levels and their degeneracy allows for the insight needed to calculate molecular thermodynamics using statistical mechanics principles.

Statistical Mechanics

Statistical mechanics provides the framework for understanding and predicting molecular behavior, including symmetry and symmetry-induced effects. In the context of rotational energy, it enables the calculation of partition functions. The rotational partition function is: \[ q_{R}(T) = \sum_{J=0, 2, 4, \ldots} g_J e^{-\frac{E_J}{k_BT}} \]
Here, \(g_J\) denotes the degeneracy, \(E_J\) the energy of the \(J\)th level, and \(k_B\) the Boltzmann constant. Statistical mechanics implies that at equilibrium, systems explore available energy states proportionally to their Boltzmann factors \(e^{-E_J/k_BT}\).

• The convergence of the partition function simplifies calculations by focusing on significant contributions.
• Approximations in statistical mechanics allow efficient calculation without summing all states, thanks to the rapid decay of higher-energy state contributions.

This approach enables the determination of important thermodynamic variables, underpinned by symmetry and quantum mechanical principles.