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18.2: Most Atoms are in the Ground Electronic State

Writing the electronic energies as

\[E_1, E_2 ,E_3, ...\]

with corresponding degeneracies

\[g_1, g_2, g_3, ...\]

the electronic partition function is then given by the following summation

\[ q_{el}  = g_1 e^{E_1/k_BT} +  g_2 e^{E_2/k_BT}  + g_3 e^{E_3/k_BT} + ... \label{Q1}\]

Usually the electronic energy different is significantly greater than thermal energy \(k_BT\), that is

\[ k_B T \le E_1 \le E_2 <  E_3\]

If we treating \(E_1\) as the reference value of zero of energy, Equation \(Q1\) is then approximated as

\[q_{el} = g_1 \label{3.24}\]

which is the ground state degeneracy of the system.

Example

Find the electronic partition of \(H_2\) at 300 K.

Solution

The lowest electronic energy level of \(H_2\) is near \(- 32\; eV\) and the next level is about \(5\; eV\) higher. Taking -32 eV as the zero (or reference value of energy), then

\[q_{el} = e_0 + e^{-5 eV/ k_BT} + ...\]

At 300 K, T = 0.02\; eV and

\[q_{el} = 1 + e^{-200} +... \approx 1.0\]

Where all terms other than the first are essentially 0. This implies that \(q_{el} = 1\). The physical meaning of this is that only the ground electronic state is generally thermally accessible at room temperature

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