🟣 This is Higher Level (HL) content.
The Convergence Limit
In any emission spectrum, as spectral lines move toward higher frequencies, the physical distance between them steadily decreases until they appear to merge. This phenomenon is the limit of convergence.
Why Do Lines Converge?
The main energy levels within the atom grow progressively closer together in terms of potential energy as the distance from the nucleus increases. The convergence limit corresponds to the point of ionization. The electron has absorbed enough energy to escape the atom entirely (\(n = \infty\)).
First Ionization Energy
📘 IB Definition
The first ionization energy (IE₁) is the minimum energy required to completely remove one mole of electrons from one mole of isolated gaseous atoms in their ground state, producing one mole of gaseous, singly charged positive ions.
\( X(g) \rightarrow X^+(g) + e^- \)
⚠️ The state symbols in this definition are non-negotiable in the mark scheme.
Worked Example: Calculating IE₁ from Spectral Data
Problem
The convergence limit of the hydrogen Lyman series occurs at a frequency of \(\nu = 3.28 \times 10^{15}\) Hz. Calculate the first ionization energy of hydrogen in kJ mol⁻¹.
Step 1: Calculate energy for one atom using Planck's equation
\(E = h\nu = (6.63 \times 10^{-34})(3.28 \times 10^{15})\)
\(E = 2.18 \times 10^{-18}\) J (per atom)
Step 2: Scale up to one mole (multiply by \(N_A\))
\(E = 2.18 \times 10^{-18} \times 6.02 \times 10^{23}\)
\(E = 1.31 \times 10^{6}\) J mol⁻¹
Step 3: Convert to kJ mol⁻¹
\(IE_1 = 1310\) kJ mol⁻¹
⚠️ Critical Examiner Trap
A prevalent and highly damaging error occurs when candidates calculate the energy for a single atom using \(E = h\nu\) but fail to multiply by Avogadro's constant. Ionization energy is strictly a molar measurement. Always check your final units match the requested kJ mol⁻¹.
Key Equations (from the Data Booklet)
Photon energy
\(E = h\nu\)
Wave equation
\(c = \nu\lambda\)