# Orbital Energies

We saw earlier that the energy of the electron in a hydrogen atom depends only on the principal quantum number, n. The nucleus of a hydrogen atom has a charge of +1, however, if the electron is bound to a nucleus of arbitrary charge +Z, then the energy of the electron is

$\large E = - Ry \displaystyle \frac{Z^2}{n^2}$

where $Ry$ is the Rydberg unit of energy where 1 Ry = 2.179877125595425$\times 10^{-18}$J = 13.60572374378387 eV

This expression is for a single electron orbiting a single nucleus of charge +Z. Let's look carefully at this equation:

• As n increases (holding Z constant), then the energy increases (becomes less negative). In the limit that n goes to infinity then the energy goes to zero.
• As Z increases (holding n constant), then the energy decreases (becomes more negative). This makes sense, since a higher Z means a more positively charged nucleus, which holds the electron tighter.

## Hydrogen Atom Energy Levels

Let's look at the energy levels of the hydrogen atom. For the hydrogen atom Z=1 so En= - Ry/n2

Notice that the energy level spacing decreases as n increases, that the number of orbitals (i.e. l values) increase with n, and all orbitals with the same n have the same energy (degenerate). (H-atom only).

## Spectrum of Hydrogen

We can look at either absorption or emission spectra. Using our equation for the energy of the hydrogen levels we can write an equation for the change in energy of an electron that charges orbitals and emits or absorbs a photon.

$\Delta E = E_\text{final} - E_\text{initial} = h \nu = \displaystyle \frac{h c}{\lambda} = -Ry \, Z^2 \left(\frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$

With this equation we can calculate the frequency of light emitted or absorbed when an electron moves between orbitals of different principal quantum numbers. Using the equation above we can calculate the wavelengths for various transitions in the H-atom.

ni nfWavelength
3 2λ=657 (red)
4 2λ=487 (green)
5 2λ=435 (blue)
2λ=365 (purple)

The energy required to promote an electron to n = ∞ is called the ionization energy.

(Because this is the energy required to make an ion).

• Electronic Transitions for Hydrogen:
• Electronic Transitions for Hydrogen-Like Species:

#### Homework from Chemisty, The Central Science, 10th Ed.

6.33, 6.35, 6.37, 6.39