For a given type of
in a given medium, the wavelength
l and the frequency n
can be related to the speed of propagation of the wave(wave velocity) as follows:
l n = c
E = h n
Where h== Planck's constant = 6.626 x 10-34 Js. When light strikes matter, in particular a molecule, the entire energy of the photon must be absorbed or emitted. Thus the color of the light that interacts with a particular piece of matter tells you about the change in energy that is possible in that matter,
If Light has properties of a particle, surely particles have properties of a wave.
DeBroglie showed that this was in fact the case, in the first equation that has both
wave and particle properties related:
p = h / l When one confines a wave to a particular region of space, the edges of the
containment place a constraint on the wavelength due to 'boundary conditions'.
is how you play different notes on the same guitar string by moving the position in
which you make the wave displacement zero, i.e. where your finger touches the fret.
Note that the 'boundary conditions' can be satisfied by many different waves (called harmonics) if each of those waves has a position of zero displacement at the right place. These positions where the value of the wave is zero are called nodes
(Sometimes we distinguish two types of waves, travelling waves and standing waves, by whether the nodes of the wave move or not. Our discussion of the atom will pretty much rely on the standing wave picture of the electron.)
If electrons are waves, then the wavelength of the electron must 'fit' into any orbit that it makes around the nucleus in an atom. This is the 'boundary condition' for a one electron atom. All orbits that do not have the electrons wavelength 'fit' are not possible, because wave interference will rapidly destroy the wave amplitude and the electron wouldn't exist anymore. This 'interference' effect leads to discrete (quantized) energy levels for the atom. Since light interacts with the atom by causing a transition between these levels, the color(spectrum) of the atom is observed to be a series of sharp lines. For the hydrogen atom:
This equation works for all one-electron atoms, not just hydrogen. (Here is a more complete derivation of the Bohr Atom's Properties. Danger: Advanced). This is precisely the pattern of energy levels that are observed to exist in the Hydrogen atom. Transitions between these levels give the pattern in the absorption or emission spectrum of the atom.
p = h / l
When one confines a wave to a particular region of space, the edges of the containment place a constraint on the wavelength due to 'boundary conditions'. This is how you play different notes on the same guitar string by moving the position in which you make the wave displacement zero, i.e. where your finger touches the fret.
A Note on Signs:
Obviously the change in energy of an atom can be either positive or negative depending on whether energy is absorbed by the atom from the light field or emitted by the atom to its surroundings. Yet frequencies (or wavelengths) of light MUST be positive numbers. Thus, in the equation above, n2 must be greater than n1 for the resulting frequency to be a positive number. Yet, if the initial principle quantum number of the atom (ninitial) is smaller than the final principle quantum number (nfinal), photon absorption has taken place and if nfinal is smaller than ninitial, photon emission has occurred.
To represent the observed spectra of one electron atoms using the above energy spacing,
it is useful to relate the energy of the photon to its wavelength through
E = hn
l n = c:
The energy patterns of atoms give the elements their characteristic 'flame' colors because the light they emit when heated has specific photon energy:
The number of nodes determines the energy. The Principle quantum number, n, is equal to the number of nodes plus 1, i.e. nodes = n-1. For a hydrogen atom, the energy it takes to make a radial node is equal to the the energy it takes to make an angular node.
For higher n, you can have a greater numbers of nodes. For n>=3,
you can have 2 angular nodes, and these are called d orbitals
Here are some more pictures of the atomic orbital shapes
The following shows the angular nodes for a 2p orbital:
Similarly for a 3d orbital (with two angular nodes):
In short, the energy of the atom is determined by the number of nodes which is related to the principal quantum number n by: nodes = n-1.
The number of angular nodes is labelled by a letter (s, p, d, f, g, h, i, ....)
Note: The number of radial nodes is the total number of nodes minus the number of angular
PJ Brucat || University of Florida