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Quantum Numbers - Specifying the electron state
Quantum Numbers for Electron Orbital
Now it turns out that if you consider an electron trapped inside a sphere (instead of in a one-dimensional box) it will have standing waves that are very similar to an electron bound to a positively charged nucleus by electrostatic attraction. Normally we operate in 4 dimensions (3 space and 1 time dimension), so in general we will need 4 quantum numbers to fully specify an electron state (i.e., standing wave).
Let's look at the quantum numbers needed to label the possible standing waves or states of an electron trapped by its electrostatic attraction to a positively charged nucleus. For an electron trapped by its electrostatic attraction to positively charged nucleus we use the three following quantum numbers to describe the electron state (orbital).
n : the principal quantum number
The principal quantum number has integral values of n = 1, 2, 3... As n increases, the electron orbital becomes larger and the electron spends more time farther from the nucleus.
ℓ : azimuthal quantum number
The azimuthal quantum number has integral values of ℓ = 0 to ℓ = n - 1 for each value of n. This quantum number defines the shape of the orbital. There are special letters assigned to each ℓ value (see table below).
| ℓ | symbol |
|---|---|
| 1 | p |
| 2 | d |
| 3 | f |
For an electron with n = 1 and ℓ = 0 we say that the electron is in the 1s state or 1s orbital
For an electron with n = 2 and ℓ = 0 we say ... 2 s orbital
For an electron with n = 2 and ℓ = 1 we say ... 2 p orbital
For an electron with n = 3 and ℓ = 2 we say ... 3 d orbital
mℓ : magnetic quantum number
The magnetic quantum number has integral values of mℓ = - ℓ to + ℓ including 0.
For example, if n = 3 and ℓ = 2 then the possible values of mℓ are -2, -1, 0, +1, +2
ms, : spin quantum number
The spin quantum number has only two possible values of +1/2 or -1/2. If a beam of hydrogen atoms in their ground state (n = 1, ℓ = 0, mℓ = 0) or 1s is sent through a region with a spatially varying magnetic field, then the beam splits into two beams.
Clearly the three quantum numbers, n, ℓ, mℓ, are not enough to completely describe the state of the H-atom, Another quantum number is required to describe whether it goes up or down in a spatially varying magnetic field. This property is called spin because if electrons were balls of charge spinning about their own axes, they would behave in this way in a magnetic field. Actually, this is not a correct picture. The spin quantum number shows up when the wave function of quantum mechanics is modified to include the effects of relativity.
The electron spin is very important in understanding the electronic structure of atoms containing many electrons (not just one like Hydrogen) because of the Pauli Exclusion Principle.
Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers n, ℓ, mℓ, ms.
Since there are only two values of ms, then any orbital can only hold two electrons with opposite vales of ms. Any more would violate Pauli's Principle.
In summary, you can think of electron orbitals as standing waves of the electron wave function when it is bound to a postively charged nucleus. Remember, however, that the electron itself is a particle, not a wave. That wave function squared gives the probability of finding the electron at any particular position.
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
This expression is for a single electron orbiting a single nucleus of charge +Z. If I had a mole of atoms like this, then I could multiply this expression by Avogadro's Number to get the total energy for all the atoms:
This equation is so popular that the number 1312 is named the Rydberg Constant and given the symbol RH = 1312 kJ/mole.
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= - RH/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.
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 → nf | Wavelength |
|---|---|
| 3 → 2 | λ=657 (red) |
| 4 → 2 | λ=487 (green) |
| 5 → 2 | λ=435 (blue) |
| ∞ → 2 | λ=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).
Homework from Chemisty, The Central Science, 10th
Ed.
6.33, 6.35, 6.37, 6.39