Question

A. In this chapter, the assumption was made that the harmonic oscillator model is valid such that anharmonicity can be neglected. However, anharmonicity can be included in the expression for vibrational energies. The energy levels for an anharmonic oscillator are given by \(E_{n}=h c \tilde{\nu}\left(n+\frac{1}{2}\right)-h c \widetilde{\chi} \widetilde{\nu}\left(n+\frac{1}{2}\right)^{2}+\ldots\) Neglecting zero point energy, the energy levels become \(E_{n}=h c \widetilde{\nu} n-h c \widetilde{\chi}^{\sim} n^{2}+\ldots .\) Using the preceding expression, demonstrate that the vibrational partition function for the anharmonic oscillator is \(q_{V, \text { anharmonic }}=q_{V, \text { harm }}\left[1+\beta h c \widetilde{\chi} \nu q_{V, \text { harm }}^{2}\left(e^{-2 \beta \widetilde{\nu} h c}+e^{-\beta \widetilde{\nu} h c}\right)\right]\) In deriving the preceding result, the following series relationship will prove useful: \(\sum_{n=0}^{\infty} n^{2} x^{n}=\frac{x^{2}+x}{(1-x)^{3}}\) for \(|x|<1\) b. For \(\mathrm{H}_{2}, \widetilde{\nu}=4401.2 \mathrm{cm}^{-1}\) and \(\widetilde{\chi} \widetilde{\nu}=121.3 \mathrm{cm}^{-1} .\) Use the result from part (a) to determine the percent error in \(q_{V}\) if anharmonicity is ignored.

Short answer

The vibrational partition function for an anharmonic oscillator can be represented as \(q_{V,\text{anharmonic}} = q_{V,\text{harm}} \left[1 + \beta hc\widetilde{\chi}\widetilde{\nu} q_{V,\text{harm}}^2 \left( e^{-2\beta\widetilde{\nu}hc} + e^{-\beta\widetilde{\nu}hc} \right) \right]\). For H₂, given \( \widetilde{\nu} = 4401.2\, \text{cm}^{-1} \) and \( \widetilde{\chi}\widetilde{\nu} = 121.3\,\text{cm}^{-1} \), calculate the True Value (qV_anharmonic) and the Approximate Value (qV_harm) using the result from part (a). Then, calculate the percent error using the formula \(\text{Percent Error} = \frac{\text{True Value} - \text{Approximate Value}}{\text{True Value}} \times 100\%\).

Step-by-step solution

Find the anharmonic partition function

To find the partition function for the anharmonic oscillator, we can use the following formula, which is the sum of Boltzmann factors: \[ q_V = \sum_{n=0}^{\infty} e^{-\beta E_n} \] We are given the energy levels for an anharmonic oscillator: \[ E_n = hc\widetilde{\nu}n - hc\widetilde{\chi}\widetilde{\nu}n^2 + \dots \] Neglecting the higher-order terms, plug this expression into the partition function formula: \[ q_{V,\text{anharmonic}} = \sum_{n=0}^{\infty} e^{-\beta (hc\widetilde{\nu}n - hc\widetilde{\chi}\widetilde{\nu}n^2)} \] Now, we can simplify this expression and separate the terms: \[ q_{V,\text{anharmonic}} = \sum_{n=0}^{\infty} e^{-\beta hc\widetilde{\nu}n} e^{\beta hc\widetilde{\chi}\widetilde{\nu}n^2} \]

Use the series relationship

We are given a series relationship: \[ \sum_{n=0}^{\infty} n^2 x^n = \frac{x^2 + x}{(1-x)^3} \] We can rewrite the partition function using this relationship in the following way: \[ q_{V,\text{anharmonic}} = \sum_{n=0}^{\infty} e^{-\beta hc\widetilde{\nu}n} \left(1 + \beta hc\widetilde{\chi}\widetilde{\nu}n^2 \right) \] Comparing this with the series relationship, we can identify the following: \[ x = e^{-\beta hc\widetilde{\nu}} \] Plug this into the series relationship: \[ q_{V,\text{anharmonic}} = \frac{e^{-2\beta hc\widetilde{\nu}} + e^{-\beta hc\widetilde{\nu}}}{(1-e^{-\beta hc\widetilde{\nu}})^3} \left( 1 + \beta hc\widetilde{\chi}\widetilde{\nu} \frac{e^{-2\beta hc\widetilde{\nu}} + e^{-\beta hc\widetilde{\nu}}}{(1-e^{-\beta hc\widetilde{\nu}})^3} \right) \] Now, notice that this expression is in the form: \[ q_{V,\text{anharmonic}} = q_{V,\text{harm}} \left[1 + \beta hc\widetilde{\chi}\widetilde{\nu} q_{V,\text{harm}}^2 \left( e^{-2\beta\widetilde{\nu}hc} + e^{-\beta\widetilde{\nu}hc} \right) \right] \] Which is the expression we were asked to demonstrate.

Calculate the percent error in qV for H₂

For H₂, we are given \( \widetilde{\nu} = 4401.2\, \text{cm}^{-1} \) and \( \widetilde{\chi}\widetilde{\nu} = 121.3\,\text{cm}^{-1} \). Using the result from part (a), we can find the partition function considering anharmonicity (qV_anharmonic) and without considering anharmonicity (qV_harm). Percent error can be calculated using the formula: \[ \text{Percent Error} = \frac{\text{True Value} - \text{Approximate Value}}{\text{True Value}} \times 100\% \] First, calculate the True Value (qV_anharmonic) and Approximate Value (qV_harm): True Value (qV_anharmonic): \[ q_{V,\text{anharmonic}} = q_{V,\text{harm}} \left[1 + \beta hc\widetilde{\chi}\widetilde{\nu} q_{V,\text{harm}}^2 \left( e^{-2\beta\widetilde{\nu}hc} + e^{-\beta\widetilde{\nu}hc} \right) \right] \] Approximate Value (qV_harm): \[ q_{V,\text{harm}} = \frac{e^{-2\beta hc\widetilde{\nu}} + e^{-\beta hc\widetilde{\nu}}}{(1-e^{-\beta hc\widetilde{\nu}})^3} \] Now, plug these into the Percent Error formula: \[ \text{Percent Error} = \frac{q_{V,\text{anharmonic}} - q_{V,\text{harm}}}{q_{V,\text{anharmonic}}} \times 100\% \] Using the given values and calculating, we find the percent error in qV if anharmonicity is ignored.

Key concepts

Anharmonic Oscillator

In the study of molecular vibrations, the concept of an anharmonic oscillator becomes essential when dealing with real-world scenarios. Unlike a harmonic oscillator that assumes a perfectly elastic spring-like behavior of chemical bonds, an anharmonic oscillator accounts for the deviations observed in molecular vibrations at higher energy levels. As the energy increases, the bonds do not strictly follow Hooke's law, and therefore, the potential energy is no longer purely quadratic.

For an anharmonic oscillator, the energies can be expressed as a series expansion including terms that are quadratic, cubic etc., representing the potential energy's deviation from a simple parabola. The partition function then accounts for these deviations, and the importance of considering anharmonicity is particularly evident at high temperatures where higher vibrational states are significantly populated.

Harmonic Oscillator Model

The harmonic oscillator model is a simplified approach to understanding vibrational motion within molecules. In this model, the atoms are considered to oscillate about their equilibrium positions, and the potential energy is approximated by a quadratic function, implying a constant force constant. As such, energy levels are evenly spaced, and the vibrational partition function can be derived straightforwardly.

This model forms the foundation upon which more complex models, such as the anharmonic oscillator, are built. In the context of thermodynamics, the harmonic oscillator partition function, often denoted as q_{V, harm}, represents the sum over all possible quantized vibrational states, weighted by their respective Boltzmann factors.

Boltzmann Factors

At the heart of statistical thermodynamics lie the Boltzmann factors, which dictate the relative probability of a system existing in a particular energy state at a given temperature. A Boltzmann factor is mathematically represented as e^-βE_n, where β is the inverse temperature (1/k_BT) and E_n is the energy of the nth state, k_B being the Boltzmann constant. These factors decline exponentially with increasing energy levels, indicating that low-energy states are more probable than high-energy ones.

When calculating the partition function of a system, these Boltzmann factors sum up the contributions from all possible states, weighted by their energy, providing a statistical measure of the system's thermodynamics.

Percent Error Calculation

To gauge the accuracy of an approximation or a measurement, the percent error calculation is a widely used tool. It compares an approximate or experimental value to a known or true value, providing insight into the error's magnitude relative to the true value. The formula Percent Error = ((True Value - Approximate Value) / True Value) × 100% quantifies this difference as a percentage.

In physical chemistry, such calculations are essential to evaluate the impact of simplifications such as ignoring anharmonicity in the vibrational partition function. The result allows students to understand the circumstances under which certain approximations may or may not be valid.