How Do You Calculate Ionization Energy From Spectral Data? (HL)

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The Reasoning

In an atomic emission spectrum, lines become closer together as the electron's energy approaches the ionization threshold. At the convergence limit, the electron reaches $n=\infty$, where it is no longer bound to the atom.

For the first ionization energy, the transition must begin from the ground state. For hydrogen, this means using the convergence limit of the Lyman series, where transitions end at $n=1$. A series ending at a higher level gives the energy required to ionize an electron from that excited level, not from the ground state.

Be careful with wavenumber units. Spectral tables often report $\tilde{\nu}$ in $\text{cm}^{-1}$, but $c$ in the data booklet is in $\text{m s}^{-1}$. Convert $\text{cm}^{-1}$ to $\text{m}^{-1}$ by multiplying by $100$, or use a value of $c$ expressed in $\text{cm s}^{-1}$. Mixing these units produces an answer smaller by a factor of $100$.

Also distinguish the ionization threshold from the highest visible line: the limit is usually obtained by extrapolating where the lines converge, even if no line is drawn exactly at that point.

For example, hydrogen's ground-state convergence wavelength is approximately $91.2\text{ nm}=9.12\times10^{-8}\text{ m}$:

Convert this to molar ionization energy:

The common misconception is that any line in the spectrum gives the ionization energy. Individual lines represent transitions between bound energy levels; only the correct convergence limit represents ionization.

Exam Technique

This is Structure 1.3.6 AHL content. In a calculation, identify the convergence limit, convert units before substitution, use constants from the data booklet, and report the final answer in $\text{kJ mol}^{-1}$ with appropriate significant figures.