Convergence Limit & Ionization Energy
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 Spectral Lines Converge at Higher Frequency
The main energy levels within the atom grow progressively closer together with increasing distance from the nucleus. Consequently, transitions originating from these higher levels converge into a continuous band.
First Ionization Energy
First Ionization Energy (IE₁)
The first ionization energy (IE₁) is the minimum energy required to remove one mole of electrons from one mole of gaseous atoms under standard conditions:
X(g) → X⁺(g) + e⁻
Worked Example: Calculating IE₁ from Spectral Data
Calculating IE₁ from Convergence Frequency
Problem: The convergence limit of the hydrogen Lyman series occurs at \(v = 3.28 \times 10^{15}\text{ s}^{-1}\). Calculate the first ionization energy of hydrogen in \(\text{kJ}\cdot\text{mol}^{-1}\).
1. Single photon energy: \(E = h\nu = (6.63 \times 10^{-34})(3.28 \times 10^{15}) = 2.175 \times 10^{-18}\text{ J}\)
2. Multiply by Avogadro's constant: \(E_{\text{mol}} = (2.175 \times 10^{-18})(6.02 \times 10^{23}) = 1.309 \times 10^6\text{ J}\cdot\text{mol}^{-1}\)
3. Convert to \(\text{kJ}\cdot\text{mol}^{-1}\): 1310 kJ mol⁻¹ (3 sig figs)
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
State Symbols and Per-Mole Energy Conversion
A prevalent and highly damaging error occurs when candidates write equations for ionization energy omitting the (g) state symbols. Furthermore, forgetting to multiply the single photon energy by Avogadro's number (\(N_A\)) to express the answer in \(\text{kJ}\cdot\text{mol}^{-1}\) consistently costs marks.
Key Spectroscopic Equations
- Photon energy: \(E = h\nu\)
- Wave relation: \(c = \nu\lambda\)
- Combined formula: \(E = \frac{hc}{\lambda}\)
- Molar ionization energy: \(IE_1 = E \times L\) (where \(L = 6.02 \times 10^{23}\text{ mol}^{-1}\))
Wave equation
\(c = \nu\lambda\)
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