Question

Consider rotation about the C-C bond in ethane. A crude model for torsion about this bond is the "free rotor" model where rotation is considered unhindered. In this model the energy levels along the torsional degree of freedom are given by \\[ E_{j}=\frac{\hbar^{2} j^{2}}{2 I} \text { for } j=0,\pm 1,\pm 2, \ldots \\] In this expression \(I\) is the moment of inertia. Using these energies, the summation expression for the corresponding partition function is \\[ Q=\frac{1}{\sigma} \sum_{j=-\infty}^{\infty} e^{-E_{l} / k T} \\] where \(\sigma\) is the symmetry number. a. Assuming that the torsional degree of freedom is in the hightemperature limit, evaluate the previous expression for \(Q\) b. Determine the contribution of the torsional degree of freedom to the molar constant-volume heat capacity. c. The experimentally determined \(C_{v}\) for the torsional degree of freedom is approximately equal to \(\mathrm{R}\) at \(340 .\) K. Can you rationalize why the experimental value is greater that than predicted using the free rotor model?

Short answer

In the high-temperature limit, the partition function for the torsional degree of freedom in ethane becomes infinite. The molar constant-volume heat capacity is approximated using the expectation value of the energy and by differentiating it with respect to temperature. The experimental value of the heat capacity is greater than the prediction from the free rotor model because the model neglects repulsive interactions between hydrogen atoms, affecting the heat capacity calculation.

Step-by-step solution

a. Evaluating the partition function in the high-temperature limit

To evaluate the partition function in the high-temperature limit, we first take the limit of the partition function as the temperature increases. In this limit, the Boltzmann factor becomes: \[ e^{-E_{l} / k T} \approx 1 - \frac{E_l}{kT} + \frac{(E_l)^2}{2(kT)^2} \] In this limit, the partition function becomes: \[ Q = \frac{1}{\sigma} \sum_{j=-\infty}^{\infty} \left( 1 - \frac{\hbar^2 j^2}{2IkT} + \frac{(\hbar^2 j^2)^2}{8(IkT)^2} \right) \] The summation can be broken down into three separate summations: \[ Q = \frac{1}{\sigma} \left( \sum_{j=-\infty}^{\infty} 1 - \sum_{j=-\infty}^{\infty} \frac{\hbar^2 j^2}{2IkT} + \sum_{j=-\infty}^{\infty} \frac{(\hbar^2 j^2)^2}{8(IkT)^2} \right) \] The first summation is equal to \( +\infty \), and the second and third summations converge as t -> infinity. Thus, the partition function in the high-temperature limit is: \[ Q = \frac{+ \infty}{\sigma} \]

b. Torsional degree of freedom contribution to molar constant-volume heat capacity

The molar constant-volume heat capacity is defined as: \[ C_v = \frac{d \langle E \rangle}{dT} \] where \( \langle E \rangle \) is the expectation value of the energy. The expectation value of the energy can be calculated as: \[ \langle E \rangle = \frac{1}{Q} \sum_{j=-\infty}^{\infty} E_l e^{-E_{l} / k T} \] In the high-temperature limit, we use the approximation for the Boltzmann factor from before: \[ \langle E \rangle \approx \frac{1}{Q} \sum_{j=-\infty}^{\infty} E_l \left( 1 - \frac{E_l}{kT} + \frac{(E_l)^2}{2(kT)^2} \right) \] Now we differentiate \( \langle E \rangle \) with respect to T: \[ \frac{d \langle E \rangle}{dT} = \frac{1}{Q} \sum_{j=-\infty}^{\infty} \frac{\partial}{\partial T} \left( E_l \left( 1 - \frac{E_l}{kT} + \frac{(E_l)^2}{2(kT)^2} \right) \right) \] We substitute this to our expression for \( C_v \): \[ C_v \approx \frac{1}{Q} \sum_{j=-\infty}^{\infty} \frac{\partial}{\partial T} \left( E_l \left( 1 - \frac{E_l}{kT} + \frac{(E_l)^2}{2(kT)^2} \right) \right) \]

c. Compare experimental value and free rotor model prediction for heat capacity

The experimentally determined molar constant-volume heat capacity for the torsional degree of freedom is approximately equal to \( \mathrm{R} \) at \( 340 \) K. In the free rotor model, we have derived an expression for the heat capacity in the high-temperature limit. However, in reality, the torsional motion in ethane is not completely unhindered (as assumed in the free rotor model). The torsional potential in ethane has an energy barrier due to the repulsive interaction between hydrogen atoms attached to adjacent carbon atoms. This interaction is ignored in the free rotor model, which leads to the heat capacity being smaller than the experimental value. In conclusion, the experimental value is greater than the prediction using the free rotor model because the model fails to account for the repulsive interaction between hydrogen atoms when preparing the expression for heat capacity.

Key concepts

Free Rotor Model

The Free Rotor Model is crucial for understanding molecular rotations around bonds, specifically in molecules like ethane. Here, rotation about the C-C bond is viewed as unhindered, allowing us to simplify calculations significantly. This model assumes that the energy levels associated with this rotation, or the torsional degree of freedom, can be expressed as \( E_{j} = \frac{\hbar^{2} j^{2}}{2 I} \). In this equation, \( j \) represents an integer quantum number, \( \hbar \) is the reduced Planck's constant, and \( I \) denotes the moment of inertia, which is vital as it reflects how mass is distributed relative to the axis of rotation.
This model allows the assumption that molecules behave like free rotors without energy barriers impeding rotation, which simplifies the description of rotational dynamics and transitions. However, real molecules, such as ethane, experience barriers to rotation, often ignored in this model. Therefore, while the Free Rotor Model serves as a starting point for understanding molecular dynamics, it doesn't fully capture all the complexities of molecular interactions.

Partition Function

In statistical mechanics, the Partition Function \( Q \) provides key insights into the statistical properties of a system in thermodynamic equilibrium. It sums over all possible states of a system, weighted by the Boltzmann factor \( e^{-E/kT} \), where \( E \) is the energy of a state, \( k \) is the Boltzmann constant, and \( T \) is the temperature. For the free rotor model, the partition function for torsional motion in ethane can be written as:
\[ Q = \frac{1}{\sigma} \sum_{j=-\infty}^{\infty} e^{-E_{j}/kT} \]
here, \( \sigma \) is the symmetry number, which corrects for the symmetrically equivalent states.

• Calculating \( Q \) involves summing over energy states, which represents the distribution of molecules across these states at a given temperature.
• In the high-temperature limit, the partition function helps predict thermodynamic quantities like internal energy and heat capacity.

Understanding \( Q \) is essential because it links the microscopic states of a system with macroscopic observable phenomena, bridging the gap between quantum mechanics and thermodynamics.

Molar Constant-Volume Heat Capacity

The molar constant-volume heat capacity \( C_v \) describes how much heat energy is required to change a mole of a substance by a degree in temperature, holding volume constant. This property is critical for evaluating energy exchanges in systems where volume doesn't change. It is defined as:
\[ C_v = \frac{d \langle E \rangle}{dT} \]
where \( \langle E \rangle \) is the expectation value of the energy. For the free rotor model, an approximation is used which, unfortunately, underestimates the real molecular behavior because it assumes an unhindered rotation.

• The model does not account for interactions between molecules, such as the repulsive forces in molecules like ethane.
• Consequently, while the model suggests lower heat capacity values, real systems show higher heat capacities due to these overlooked factors.

Understanding this discrepancy highlights why real-world experiments often observe heat capacities that deviate from basic theoretical predictions.

High-Temperature Limit

The High-Temperature Limit is a simplification used in thermodynamics where it is assumed that thermal energy greatly exceeds the energy spacing between quantum states. In this context, each energy level becomes almost equally populated due to the high thermal motion. This approximation simplifies various calculations, including those of the partition function.
At high temperatures, the Boltzmann factor is approximated by:
\[ e^{-E/kT} \approx 1 - \frac{E}{kT} + \frac{(E)^2}{2(kT)^2} \]
This means the expression for the partition function becomes dominated by higher-energy states, simplifying to a calculation where the contributions from lower energy levels diminish, allowing us to focus on the leading terms.

• This approximation is particularly useful for calculating properties like the molar heat capacity at constant volume because it allows us to predict behavior without complex integrals.
• It also serves as an essential tool in approximating behavior when exact solutions are difficult to obtain.

However, care must be taken since this limit doesn't precisely capture behaviors at lower temperatures where state populations differ significantly.

Boltzmann Factor

The Boltzmann Factor is a central element in statistical mechanics that expresses the probability of a system being in a certain state as a function of its energy. It is expressed as:
\[ e^{-E/kT} \]
where \( E \) is the energy of the state, \( k \) is Boltzmann's constant, and \( T \) is temperature.

• This exponential term shows that states with lower energies are exponentially more likely to be occupied than those with higher energies, explaining the natural tendency of systems to settle into minimum energy configurations.
• In the context of the free rotor model, the Boltzmann factor helps compute the partition function by determining the weight of each energy state in the sum over states, essentially linking microscopic quantum mechanical descriptions to macroscopic thermodynamic phenomena.

Overall, the Boltzmann factor is fundamental in determining how energy is distributed across states in a system, illustrating the relationship between energy, temperature, and molecular behavior.