Question

NO is a well-known example of a molecular system in which excited electronic energy levels are readily accessible at room temperature. Both the ground and excited electronic states are doubly degenerate and are separated by \(121.1 \mathrm{cm}^{-1}\) a. Evaluate the electronic partition function for this molecule at \(298 \mathrm{K}\) b. Determine the temperature at which \(q_{E}=3\)

Short answer

The electronic partition function for the given NO molecule at 298 K is approximately 2.067. The temperature at which the electronic partition function equals 3 is approximately 240.9 K.

Step-by-step solution

Understanding the electronic partition function

The electronic partition function, denoted as \(q_E\), is a measure of the number of accessible electronic states for a molecule at a given temperature. It is mathematically defined as the sum of the Boltzmann factors for all energy levels, taking into account the degeneracy of each level: \[q_E = \sum_{i} g_i e^{-\beta \varepsilon_i}\] where \(g_i\) is the degeneracy of the \(i\)-th energy level, \(\varepsilon_i\) is the energy of the \(i\)-th level, and \(\beta = \frac{1}{k_B T}\) with \(k_B\) being the Boltzmann constant and \(T\) the temperature in Kelvin.

Evaluate the electronic partition function for the given molecule at 298 K

For the given molecule, we know that both the ground and excited electronic states are doubly degenerate (i.e., \(g_1 = g_2 = 2\)) and separated by 121.1 cm\(^{-1}\). To compute the partition function at 298 K, we first need to convert the energy separation from wavenumbers (cm\(^{-1}\)) to Joules. We can use the conversion factor 1 cm\(^{-1}\) = 0.00012399 eV and then convert eV to Joules using 1 eV = 1.60218 × 10\(^{-19}\) J: \[\varepsilon_2-\varepsilon_1= 121.1 \,\mathrm{cm}^{-1} \times 0.00012399\, \frac{\mathrm{eV}}{\mathrm{cm}^{-1}} \times 1.60218\times 10^{-19}\, \frac{\mathrm{J}}{\mathrm{eV}} \approx 2.04 \times 10^{-20}\, \mathrm{J}\] Now we can calculate the Boltzmann factor for the excited state at 298 K: \[\beta = \frac{1}{k_B T} = \frac{1}{1.38065 \times 10^{-23} \,\mathrm{J\, K^{-1}} \times 298\, \mathrm{K}} \approx 2.415 \times 10^{22}\, \mathrm{J^{-1}}\] \[e^{-\beta (\varepsilon_2 - \varepsilon_1)} = e^{-2.415 \times 10^{22}\, \mathrm{J^{-1}} \times 2.04 \times 10^{-20}\, \mathrm{J}} \approx 0.0334\] Now we can plug the results into the partition function formula: \[q_E = g_1 + g_2 e^{-\beta (\varepsilon_2 - \varepsilon_1)} = 2 + 2 \times 0.0334 \approx 2.067\] So, the electronic partition function for this molecule at 298 K is approximately 2.067.

Determine the temperature at which the electronic partition function equals 3

We are given that we need to find the temperature for which the electronic partition function equals 3: \[3 = q_E = g_1 + g_2 e^{-\beta (\varepsilon_2 - \varepsilon_1)}\] We can solve for \(\beta\): \[\beta = \frac{\ln\left(\frac{q_E - g_1}{g_2}\right)}{-(\varepsilon_2 - \varepsilon_1)} = \frac{\ln\left(\frac{3 - 2}{2}\right)}{-(2.04 \times 10^{-20}\, \mathrm{J})} \approx 3.0 \times 10^{21}\, \mathrm{J^{-1}}\] Now we can find the temperature: \[T = \frac{1}{k_B \beta} = \frac{1}{1.38065 \times 10^{-23} \,\mathrm{J\, K^{-1}} \times 3.0 \times 10^{21}\, \mathrm{J^{-1}}} \approx 240.9\, \mathrm{K}\] So, the temperature at which the electronic partition function equals 3 is approximately 240.9 K.

Key concepts

Boltzmann Constant

The Boltzmann constant, denoted as k_B, is a fundamental physical constant that plays a key role in the field of statistical mechanics. It connects macroscopic variables such as temperature to microscopic behavior, particularly the energy states of particles.

When you see the Boltzmann constant in equations, you're looking at a bridge between the thermal energy of particles and temperature. This constant is crucial when calculating the Boltzmann factor, which is expressed as e^-(E/k_BT), where E is the energy level, T is the temperature in Kelvin, and e is the base of natural logarithms. The Boltzmann factor gives the probability that a system will be in a state with energy E at temperature T.

In practical scenarios such as the one seen in the exercise with the NO molecule, the Boltzmann constant is used to translate the energy difference between electronic states from Joules to an exponential factor that illustrates how population distribution among energy levels changes with temperature.

Energy Levels Degeneracy

In quantum mechanics, degeneracy refers to the phenomenon where multiple quantum states share the same energy. Degeneracy is denoted by g_i, which represents the number of states that have the same energy level ε_i.

When calculating the electronic partition function, the degeneracy is critical because it effectively multiplies the contribution of that energy level to the total number of accessible states. For instance, as illustrated in the exercise, the NO molecule has a doubly degenerate ground state and a doubly degenerate excited state (g₁ = g₂ = 2). This means that the ground state and the first excited state each have two quantum states with the same energy, respectively. When computing the partition function, the degeneracy translates into a higher weight for those energy levels in the sum, directly influencing the statistical properties of the molecule, such as population distribution and chemical reactivity, among many others.

Understanding degeneracy is essential to accurately determine the partition function, which is in turn necessary for predicting the outcome of chemical reactions and understanding the material properties at the atomic level.

Temperature Dependence of Partition Function

The partition function is not a static value but is highly dependent on temperature, a concept that is at the heart of understanding molecular behavior in thermodynamics. As temperature increases, the thermal energy available to a molecular system increases as well, making higher energy states more accessible and thereby increasing the partition function.

This temperature dependence can be observed through the Boltzmann factor within the partition function, which includes an exponential term that changes significantly with temperature. For example, as seen in the solution for the NO molecule, a higher temperature would result in a larger numerical value for e^{-β(ε₂ - ε₁)}, indicating a greater population of molecules in the excited state.

The exercise also demonstrates how to determine the specific temperature at which the partition function equals a particular value. This demonstrates the sensitivity of the partition function to temperature changes and emphasizes the predictive power of statistical mechanics in understanding how systems respond to thermal variations. Such insights are invaluable in fields such as materials science, chemistry, and even astrophysics, where temperature affects everything from reaction rates to stellar spectra.