Structure 1.3.6—In an emission spectrum, the limit of convergence at higher frequency corresponds to ionization. AHL

**What You’ll Learn:**

AHL Only

- Explain the trends and discontinuities in first ionization energy (IE) across a period and down a group.
- Calculate the value of the first IE from spectral data that gives the wavelength or frequency of the convergence limit.
- The value of the Planck constant h and the equations E = h f and c = λ f are given in the data booklet.

**Keywords**

Emission spectra, excited states, lower energy levels, electromagnetic spectrum, wavelength, frequency, energy, continuous spectrum, line spectrum, hydrogen emission spectrum, energy transitions, Balmer series, Lyman series, Paschen series, Rydberg formula, principal quantum number (n), maximum number of electrons, Aufbau principle, main energy level, sublevels (s, p, d, and f), atomic orbitals, s orbital, p orbitals (p_{x}, p_{y}, and p_{z}),

d orbitals (d_{xy}, d_{xz}, d_{yz}, d_{x²-y²}, and d_{z²}), orbital shapes and orientations, nodal planes, electron configuration, chemical environment, opposite spin, Aufbau principle, Hund’s rule, Pauli exclusion principle, full electron configurations, condensed electron configurations, noble gas core, orbital diagrams (arrow-in-box diagrams)

.

**Syllabus Links**

Structure 3.1—How does the trend in IE values across a period and down a group explain the trends in properties of metals and non-metals?

Nature of science, Tool 3, Reactivity 3.1—Why are log scales useful when discussing [H+] and IEs?

## In an emission spectrum, the limit of convergence at higher frequency corresponds to ionization:

As the energy of emitted photons increases, the frequency of the electromagnetic radiation also increases. When the energy of the emitted photons is sufficient to remove an electron from the atom, ionization occurs. In an emission spectrum, the limit of convergence at higher frequencies corresponds to this ionization energy.

## Trends and discontinuities in first ionization energy (IE) across a period and down a group:

Across a period (left to right in the periodic table), the first ionization energy generally increases. This trend is due to the increasing effective nuclear charge experienced by the valence electrons as more protons are added to the nucleus, causing a stronger attraction between the nucleus and electrons, which requires more energy to remove an electron.

However, there are discontinuities in this trend, such as between the second and third elements (e.g., between beryllium and boron) and between the fifth and sixth elements (e.g., between nitrogen and oxygen) in a period. These discontinuities can be attributed to electron-electron repulsion in the p orbitals, where the electron being removed is from a doubly occupied orbital, or from a new, higher energy sublevel.

Down a group (top to bottom in the periodic table), the first ionization energy generally decreases. This decrease is due to the increasing atomic radius and the addition of electron shells, which result in increased shielding of the valence electrons from the nucleus. Consequently, it requires less energy to remove an electron from the atom.

## Calculate the value of the first IE from spectral data that gives the wavelength or frequency of the convergence limit:

To calculate the first ionization energy (IE) from spectral data, you can use the Planck constant (h) and the equations E = hf and c = λf, which are provided in the data booklet.

First, determine the frequency (f) of the convergence limit using either the wavelength (λ) or frequency provided in the spectral data:

- If the wavelength is given, use the equation c = λf, where c is the speed of light, to solve for the frequency (f).
- If the frequency is given, you can directly use it in the next step.

Next, calculate the energy (E) associated with the convergence limit using the equation E = hf, where h is the Planck constant and f is the frequency you determined in the previous step.

The energy (E) you calculate will be in joules. To convert it to electron volts (eV), divide the energy (E) by the elementary charge (e = 1.602 x 10^{-19} C). The resulting value represents the first ionization energy (IE) of the element.

## Worked Example

Let’s work through an example of calculating the first ionization energy (IE) from spectral data. Suppose you are given the following information:

Wavelength of the convergence limit (λ) = 9.112 x 10^{-8} m

- First, determine the frequency (f) using the wavelength (λ) and the equation c = λf, where c is the speed of light (c = 3.00 x 10
^{8}m/s).

Rearrange the equation to solve for the frequency (f): f = c / λ

f = (3.00 x 10^{8} m/s) / (9.112 x 10^{-8} m)

f ≈ 3.29 x 10^{15} Hz

- Next, calculate the energy (E) associated with the convergence limit using the equation E = hf, where h is the Planck constant (h = 6.63 x 10
^{-34}Js) and f is the frequency you determined in the previous step.

E = (6.63 x 10^{-34} Js) x (3.29 x 10^{15} Hz)

E ≈ 2.18 x 10^{-18} J

- Convert the energy (E) to electron volts (eV) by dividing the energy (E) by the elementary charge (e = 1.602 x 10
^{-19}C).

First ionization energy (IE) = E / e

IE = (2.18 x 10^{-18} J) / (1.602 x 10^{-19} C)

IE ≈ 13.6 eV

Thus, the first ionization energy of the element is approximately 13.6 electron volts (eV).

Questions

- What is the significance of the limit of convergence at higher frequencies in an emission spectrum?
- How does the first ionization energy generally change across a period in the periodic table?
- How does the first ionization energy generally change down a group in the periodic table?
- Why are there discontinuities in the first ionization energy trend across a period?
- What factors contribute to the decrease in ionization energy as you move down a group in the periodic table?
- What is the relationship between the energy of emitted photons and the frequency of electromagnetic radiation?
- Given the wavelength of the convergence limit in an emission spectrum, describe the steps to calculate the first ionization energy of an element.
- If the frequency of the convergence limit in an emission spectrum is 4.0 x 10^15 Hz, calculate the first ionization energy of the element in electron volts.
- A given element has a convergence limit wavelength of 1.5 x 10^(-7) m in its emission spectrum. Calculate the first ionization energy of the element in electron volts.
- How do electron-electron repulsion and the addition of a new, higher energy sublevel contribute to the discontinuities observed in the first ionization energy trend across a period?