. In analogy to the characteristic vibrational temperature,...

. In analogy to the characteristic vibrational temperature, we can define a characteristic electronic temperature by $$ \Theta_{\text {elec, } j}=\frac{\varepsilon_{e j}}{k_{\mathrm{B}}} $$ whefe $\varepsilon_{e j}$ is the energy of the $j$ th excited electronic state relative to the ground state. Show that if we define the ground state to be the zero of energy, then $$ q_{\text {elec }}=g_{0}+g_{1} e^{-\Theta_{e b c .1} / T}+g_{2} e^{-\theta_{\text {elkc }-2} / T}+\cdots $$ The first and second excited electronic states of $\mathrm{O}(\mathrm{g})$ lie $158.2 \mathrm{~cm}^{-1}$ and $226.5 \mathrm{~cm}^{-1}$ above the ground electronic state. Given $g_{0}=5, g_{1}=3$, and $g_{2}=1$, calculate the values of $\Theta_{\text {elec, } 1}$, $\Theta_{\text {elec, } 2}$, and $q_{\text {elec }}$ (ignoring any higher states) for $\mathrm{O}(\mathrm{g})$ at $5000 \mathrm{~K}$.

. In analogy to the characteristic vibrational temperature, we can define a characteristic electronic temperature by
$$
\Theta_{\text {elec, } j}=\frac{\varepsilon_{e j}}{k_{\mathrm{B}}}
$$
whefe $\varepsilon_{e j}$ is the energy of the $j$ th excited electronic state relative to the ground state. Show that if we define the ground state to be the zero of energy, then
$$
q_{\text {elec }}=g_{0}+g_{1} e^{-\Theta_{e b c .1} / T}+g_{2} e^{-\theta_{\text {elkc }-2} / T}+\cdots
$$
The first and second excited electronic states of $\mathrm{O}(\mathrm{g})$ lie $158.2 \mathrm{~cm}^{-1}$ and $226.5 \mathrm{~cm}^{-1}$ above the ground electronic state. Given $g_{0}=5, g_{1}=3$, and $g_{2}=1$, calculate the values of $\Theta_{\text {elec, } 1}$, $\Theta_{\text {elec, } 2}$, and $q_{\text {elec }}$ (ignoring any higher states) for $\mathrm{O}(\mathrm{g})$ at $5000 \mathrm{~K}$.

Key Concepts

Characteristic Temperature

This is a temperature scale that relates an energy level to thermal energy by dividing the energy by the Boltzmann constant. It provides a means of comparing energy differences between quantum states to the thermal energy available at a given temperature, thereby indicating how thermally accessible different states are.

Electronic Partition Function

The electronic partition function is a sum over the electronic energy states of a system, incorporating the degeneracy of each state and the Boltzmann factor. It plays a crucial role in statistical mechanics by quantifying how the population of electronic states is distributed at thermal equilibrium, influencing thermodynamic properties such as internal energy and entropy.

Degeneracy

Degeneracy refers to the number of distinct quantum states that share the same energy level. In the context of the partition function, each electronic state is weighted by its degeneracy, indicating that states with higher degeneracy contribute more significantly to the overall statistical distribution of the system's energy.

Boltzmann Factor

The Boltzmann factor, expressed as exp(-E/(kB T)), represents the probability weighting for a state with energy E at a given temperature. It determines how likely a state is to be occupied relative to the ground state, with higher energy states becoming increasingly less populated as the energy difference grows relative to the thermal energy available.

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