Ionisation, Excitation & Spectral Lines

PHYAtoms — Bohr Model & Spectra

Ionisation energy is the energy needed to remove the electron entirely from an atom (take it from its level to $n=\infty$ ), while excitation energy lifts the electron from a lower level to a higher bound level. Both are just energy gaps read off Bohr's $E_n = -13.6/n^2\,\text{eV}$ ladder, and when excited atoms fall back they emit a set of spectral lines counted by $n(n-1)/2$ .

ISC examines this because it ties together the Bohr energy levels, the idea of bound versus free states, and the bookkeeping of how many distinct wavelengths a de-exciting gas produces.

Bohr energy levels of hydrogen

E_n = -\dfrac{13.6}{n^{2}}\ \text{eV}

E_n

is the energy of level

n

, negative because the electron is bound. Ground state

E_1=-13.6\,\text{eV}

, then

E_2=-3.4\,\text{eV}

E_3=-1.51\,\text{eV}

E_4=-0.85\,\text{eV}

, approaching

0

n\to\infty

Ionisation energy from a level

E_{ion} = E_\infty - E_n = 0 - \left(-\dfrac{13.6}{n^{2}}\right) = \dfrac{13.6}{n^{2}}\ \text{eV}

E_{ion}

is the energy to free the electron from level

n

(take it to

n=\infty

). From the ground state (

n=1

) it is

13.6\,\text{eV}

— the ionisation energy of hydrogen.

Excitation energy (lower to higher bound level)

E_{exc} = E_{n_2} - E_{n_1} = 13.6\left(\dfrac{1}{n_1^{2}} - \dfrac{1}{n_2^{2}}\right)\ \text{eV}

n_1

is the starting (lower) level and

n_2

the target (higher) level. From

n=1

n=2

E_{exc} = 13.6(1 - 1/4) = 10.2\,\text{eV}

— the first excitation energy.

Number of spectral lines emitted

\text{lines} = \dfrac{n(n-1)}{2}

n

is the highest level the gas of atoms is excited to. For

n=4

, this gives

\dfrac{4\times 3}{2}=6

distinct lines, distributed among the Lyman (

\to 1

), Balmer (

\to 2

) and Paschen (

\to 3

) series.

Ionisation = excitation to $n=\infty$ : it costs $|E_n|$ , the magnitude of that level's energy. From the ground state of hydrogen this is exactly $13.6\,\text{eV}$ .
Excitation energy is the gap between two bound levels and is always less than the ionisation energy from the same starting level.
Read energies off the level ladder with their signs and subtract: $E_{exc}=E_{n_2}-E_{n_1}$ comes out positive for an upward jump.
For a gas excited to level $n$ , the number of emission lines is $n(n-1)/2$ — every possible downward transition between the $n$ levels.
Sort the emitted lines by their final level: those ending on $n=1$ are Lyman (UV), on $n=2$ Balmer (visible), on $n=3$ Paschen (IR). From $n=4$ you get $3$ Lyman, $2$ Balmer and $1$ Paschen line.
A single isolated atom makes only one jump at a time; the $n(n-1)/2$ count assumes a large collection of atoms taking all possible paths down.
Energies become less negative (closer to $0$ ) as $n$ rises, so the levels crowd together near the top — successive excitation energies get smaller.

Where the marks go

Confusing ionisation with excitation: ionisation goes to $n=\infty$ (electron freed), excitation goes to another bound level. From $n=1$ these are $13.6\,\text{eV}$ and $10.2\,\text{eV}$ respectively.
Sign errors when subtracting level energies — keep both energies negative and subtract carefully so the excitation energy turns out positive.
Using $n$ or $n^2$ for the number of spectral lines instead of $n(n-1)/2$ (for $n=4$ that is $6$ , not $4$ or $16$ ).
Misassigning series — forgetting that lines ending on $n=2$ (Balmer) are the visible ones, while $n=1$ (Lyman) lines are ultraviolet.

How the board asks it

Numericalionisation and excitation energy from a level
The ground state energy of hydrogen is $-13.6\,\text{eV}$ . Calculate the energy required to (i) ionise a hydrogen atom from the ground state and (ii) excite it from $n=1$ to $n=2$ .
Numericalnumber of spectral lines $n(n-1)/2$
A hydrogen atom is excited to the energy level $n=4$ . Calculate the number of spectral lines that can be emitted as the atoms return to the ground state.
Numericalexcitation energy as a gap between bound levels
Using $E_n=-\dfrac{13.6}{n^2}\,\text{eV}$ , calculate the energy of the photon emitted when a hydrogen atom de-excites from $n=3$ to $n=2$ , and state the series to which this line belongs.
Define / stateionisation vs excitation energy
Define the terms ionisation energy and excitation energy of an atom.
Give reasonsionisation = excitation to $n=\infty$
Give a reason why the ionisation energy of hydrogen from the ground state is greater than any of its excitation energies from the same state.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.