Can anyone define magnetic quantum number?

  • Thread starter land_of_ice
  • Start date
  • #1
land_of_ice
137
0
all it says in the chemistry book is that a magnetic quantum number is the spatial orientation of the orbital ?
That doesn't really tell you much. What exactly do they mean by that?
also on the web the definitions don't really say much, they seem to have pages and pages of stuff that the magnetic quantum number relates to but they don't really go into detail about what it is.
 

Answers and Replies

  • #3
morrobay
Gold Member
858
691
Magnetic quantum number:
The #2 levell contains a maximum of 8 electrons
2s shelll contains 2 electrons of opposite spin
2p shell contains 6 electrons
For the 6 electrons to be different in the 2p shell the 2p shell is subdivided into 3 sub orbitals :
2px , 2py ,2pz containing two electrons each.
The 3d level is also sub divided .
 
Last edited:
  • #4
morrobay
Gold Member
858
691
Magnetic quantum number:
The #2 levell contains a maximum of 8 electrons
2s shelll contains 2 electrons of opposite spin
2p shell contains 6 electrons
For the 6 electrons to be different in the 2p shell the 2p shell is subdivided into 3 sub orbitals :
2px , 2py ,2pz containing two electrons each.
The 3d level is also sub divided .

Let me complete this answer, obviously I did not refresh/refer enough.
There are 4 quantum numbers that identify the electron :
1.n = Principal quantum number (energy level ) 1,2,3,4
2. l = Subsidiary quantum number ( corresponding to s,p,d,f shells,
derived from the spectral terms: sharp, principal, diffuse,
fundamental
3. m = magnetic quantum number. Although the three 2p sub orbitals
above ( on the x,y,z, axis ) are equal in energy, under the influence
of a magnetic field there is a division in spectral lines.
This necessitates the magnetic quantum numbers: when n =2 , for the p shell the magnetic quantum numbers have values -1, 0,+1. When n = 3 for the d shell electron orbitals , m has values -2,-1 ,0, +1,-2
4. Electron spin number with values +1/2 and - 1/2
 
Last edited:
  • #5
Borek
Mentor
29,212
3,902
3. Magnetic quantum number. Although the three 2p sub orbitals
above ( on the x,y,z, axis ) are equal in energy, under the influence
of a magnetic field there is a division in spectral lines.
This necessitates the magnetic quantum numbers, which are
designated: 0,-1.+1,-2,+2

You may want to reword this part of your explanation. It seems to suggest 5 possible values of magnetic quantum number for three 2p orbitals. That's not true, there are three 2p orbitals (2px, 2py, 2pz) because there are three possible values of m for 2p orbital (-1, 0, +1).
 
  • #6
morrobay
Gold Member
858
691
You may want to reword this part of your explanation. It seems to suggest 5 possible values of magnetic quantum number for three 2p orbitals. That's not true, there are three 2p orbitals (2px, 2py, 2pz) because there are three possible values of m for 2p orbital (-1, 0, +1).

Reworded/corrected , thanks
 
  • #7
alxm
Science Advisor
1,848
9
The magnetic quantum number is indeed the spatial orientation of the orbital. To attempt a description: You only have particular allowed values for the angular momentum, which is the velocity at which the electron is moving perpendicular to the electron-nucleus line. This easy to visualize, because they're not classical particles, and they don't move in particular 'orbits'. The electrons can only have particular values for their total momentum (the principal quantum number), and that momentum can be divided into the linear momentum (which would be in the direction in and out from the nucleus) and the angular momentum, which is perpendicular to that. Combine that with the orientation of the angular momentum, and these are the three numbers which dictate the shape of an orbital.

An 's'-type orbital has no angular momentum; in some sense the electron is entirely moving in the inward-outward direction. It's spherically symmetric because of this. Correspondingly it doesn't have an orientation in space, the electron is moving the same amount in every direction, no matter how you rotate it. So it only has an m of zero. If it's a p,d or f orbital, it has angular momentum and is no longer spherically symmetric. So there are several independent ways in which the angular momentum can be oriented in space. There's a finite number because the components of the angular momentum are quantized as well. The details of what the shape looks like and which orientations it can take have to do with the mathematics of http://en.wikipedia.org/wiki/Spherical_harmonics" [Broken], which is a bit tricky. But the result is that you have 2*l+1 possible orientations, i.e. if l=2 then the magnetic quantum number can be -2,-1,0,1,2.

I'll try to explain. If this goes over your head, don't worry too much: It's more than most chemists and physicists know about the topic. Basically, what you're looking at here is the number of linearly independent ways the electron can move while having those particular values of n and l. So in the case of p orbitals, the simplest case where l is non-zero, you have three different orientations, along each spatial axis. These are linearly independent: There's no way you can add a py orbital and a pz orbital together to get a px orbital. Any state that has n=1, l=1 can be written as a linear combination of these three orbitals, which corresponds to an electron partially occupying several orbitals. (a 'superposition state' in quantum mechanics)

This isn't explicitly quantum-mechanical behavior though, to make a direct two-dimensional analogy, it's like the http://physics.usask.ca/~hirose/ep225/animation/drum/anim-drum.htm" [Broken]. See the animations there? The fundamental vibration is cylindrically symmetric. It also has no nodes (areas where the inside of the drum are level with the edge), i.e. it's completely convex/concave. That's a two-dimensional version of what a 1s-orbital looks like. The second vibration is also cylindrically symmetric, but it has a circular node - between the center which is pointing one way, and the outer part which is pointing in the other way. That's analogous to a 2s orbital (except it has a spherical nodes rather than a circle).

The third vibration has a node-line running down the center, and the two bumps are symmetric along that axis, but have opposite signs (point in opposite directions). This is the two-dimensional equivalent of a 2p orbital. But if you look at it closely, you can tell that it's not the only one: If you rotate the thing 180 degrees, it's the same, but if you rotate it 90 degrees, you get another, equivalent, vibrational mode. Since you only have two independent directions on a plane, you only have two such 'orbitals' on a drum. But p-orbitals, being three-dimensional, have three independent directions. And the node is not a line on the surface but a plane in space.

For higher values of the angular momentum, you get the http://x3f.xanga.com/721c904073133207432985/q161436131.gif" which have five possible orientations. Just as with the p orbitals, three of these are just rotated versions of each other, and they have two nodal planes. But then you have an additional pair (dz2 and dx2-y2) which look a bit different (weird, even). There's a hint in the names though. A 1s orbital has the mathematical function e-r, where r is the radius. Obviously it's spherically symmetric. A 2px orbital has the mathematical function xe-r, and x is the x coordinate. (py and pz are the same for different coordinates) So in the plane where x=0 (the yz plane), the px orbital is zero. And if you plot the shape where this function has some value (ignoring the sign), you get the well-known double-balloon shape. So the 2p orbitals are basically the 1s orbitals, multiplied by x, y or z.

The 3d-orbitals then, are the 2p orbitals, multiplied by x, y and z again. So you might expect there to be nine d-type orbitals and not five: dx2,dxy, dxz, dyx dy2, dyz, dzx, dzy, dz2.

But obviously, xy = yx, so those two are the same. So you're left with six then: x2, y2, z2, xy, xz, yz. So why are there five d-orbitals and not six? The answer to that is because of the restriction that that the total angular momentum is constant, which (and this is a bit hand-waving) amounts to demanding that a linear combination of them can form a sphere. For a sphere, x2 + y2 + z2 = constant. So the three orbitals that correspond to the squared terms aren't independent, and one of them can be eliminated, e.g. by having the z2 and x2 - y2 orbitals. By changing variables obviously you can have other combinations too, but that just amounts to rotating it all. There are only five independent orbitals, because there are only five linearly independent ways of describing a particle with l=2. (Although the 'cartesian' form of the orbitals, with all six, is sometimes used in quantum chemistry)

So for a given value of l, there are 2l+1 values of m, corresponding to the unique ways an electron can move and still have that l value. The m value corresponds to the component of the angular momentum along some axis. (Again, no matter how you rotate it, you still have the same number of m values)

Finally, the reason why it's called the magnetic quantum number is because electrons are charged particles. A particle that has angular momentum is moving in a 'loop', and a charged particle moving in a loop gives rise to a magnetic field. Again I should emphasize that electrons don't move in a classical path, but their combination of charge and angular momentum still works the same way it does with classical physics. So, if an atom is an external magnetic field, the different orbitals will have different energies, because they correspond to different orientations relative the field. And so they have different energies. Ultimately, that means you end up with the Zeeman effect, where the spectroscopic lines split due to the changes in the energy levels.

You also have the fact that electrons have their own intrinsic magnetic moment due to spin. The interaction of the spin-magnetic-moment with the magnetic field of the orbital gives rise to what's called spin-orbit coupling. This also causes a splitting of the levels, although it's much smaller.
 
Last edited by a moderator:

Suggested for: Can anyone define magnetic quantum number?

Replies
2
Views
828
Replies
2
Views
816
Replies
9
Views
850
Replies
15
Views
2K
Replies
9
Views
1K
Replies
3
Views
2K
Replies
4
Views
1K
Top