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Unformatted text preview: A Guide to Orbitals Guide
Chem 21L Sarah Elizabeth Crider and Warren S. Warren (based on lecture, the textbook, and sources online) Review of some of the more general Review concepts about orbitals.
► ► An orbital is a distribution in space that would be a stable state for an electron to occupy. It has a well-defined energy. Shape of the orbital depends on quantum numbers associated with the orbital “Chemist’s” orbitals can be positive or negative at different positions. Technically, an orbital with a single value of ml≠0 is complex (it has a real and imaginary part) but you can combine + ml and –ml to make two real orbitals, which we and will always do in this class. So just remember that whenever you see ml≠0 it is always a combination of + ml and –ml and The square of the orbital is the probability distribution, but the different phases are important as well because they determine how orbitals combine (for example, to make bonds).
n = principal quantum #, l = orbital quantum #, ml = angular momentum quantum #. ► ► Here are the common shapes of orbitals. Keep in mind that as you get into higher energy levels the angular shape stays the same, but you get more sign changes as r increases. ► l = 2 (d-orbital) ► l = 0 (s-orbital) Common Shapes of Orbitals Common ►l = 1 (p-orbital) ► l = 3 (f-orbital) For historical (but stupid) reasons, the names given to the different values of l are s, p, d, f, then alphabetical order (l=4 is g, l=5 is h, l=6 is i, …)
http://www.orbitals.com/orb/ Energy Levels, Orbitals and Nodes Energy
► ► ► To visualize many of the possible shapes of orbitals, including those of higher energy levels, here is a link to a table of the orbitals through the n=10 orbitals. http://orbitals.com/orb/orbtable.htm http://orbitals.com/orb/orbtable.htm Notice that as you go up in n the shape of the orbital changes, the in that nodes appear within the orbital. For example as you increase energy levels from n=1 to n=5 for the s-orbitals, the number of nodes increases from 0 to 4 respectively. Therefore the higher you get in energy, the greater the number of nodes that appear in a specific orbital. n=1 n=2 n=3 n=4 n=5 http://www.orbitals.com/orb/ Types of Nodes Types
► For any orbital, there are a total of (n-1) nodes. (n-l-1) (n of these nodes are radial nodes, l are angular nodes. of ► Recall from the Orbitron tutorial, there are various types of nodes within other orbitals:
In many cases, the radial wave function passes through In zero. These regions are described as radial nodes, or radial spherical radial nodes since this describes their shape. spherical In other cases, the angular wave function passes through In zero. These regions are described as angular nodes, or angular nodal planes in those cases where they are planar. Not all nodal angular nodes are planar: some are conical, for instance.
http://winter.group.shef.ac.uk/orbitron/glossary.html The nodes of the s-orbitals. The
► Here are the s-orbitals with their nodes labeled a little more clearly. They have only spherical radial nodes, and the number of nodes is equal to n-1. 1s 2s 3s http://wps.prenhall.com/wps/media/objects/602/616516/Media_Assets/Chapter05/Text_Images/FG05_10a-c.JPG The nodes of the s-orbitals. The
1s, 2s, 3s orbitals (sliced open to see structure) ► Radial Distribution Plot ( 4πr2 |ψ|2 vs. r) The plot of 4πr2|ψ|2 gives the relative probability of finding electrons at different values of r The 1s orbital is nonzero The everywhere (although r2|ψ|2 is of course zero when r=0) The 2s orbital (in pink) passes The through zero once, therefore the 2s orbital has one spherical node. The ns orbital has (n-1) values of The r where it vanishes, hence (n-1) radial nodes s orbitals look the same from orbitals every direction, so they have no angular nodes. The np orbital series has one angular node, so it has (n-2) np radial nodes. Note in the drawings below that there is a specific distance from the center where the 3p orbital vanishes, and two distances where the 4p orbital vanishes.
For the 2p orbital n-l-1 =2-1-1=0, so there are no radial nodes. For the 3p orbital n-l-1 =3-1-1=1, so there is 1 radial node. For the 4 p orbital n-l-1 =4-1-1=2, so there are 2 radial nodes.
http://www.sccj.net/CSSJ/jcs/v8n1/a12/fig8.gif What about p-orbital radial nodes? What The Nodal Planes of p-orbitals. The
► ► The first set of orbitals to contain angular nodes are the p-orbitals. The number of angular nodes in any orbital is equal to l. For p orbitals, the single angular node is always a plane. http://wps.prenhall.com/wps/media/objects/602/616516/Media_Assets/Chapter05/Text_Images/FG05_12.JPG The d -orbitals. The
Here are the 5 different shapes of the 3d-orbitals. Determine the number of angular nodes for each orbital. Answer is on the next slide. http://www.geo.arizona.edu/xtal/geos306/d-orbitals.gif The d -orbitals. The
Pictured here are the 5 different shapes of the 3dorbitals. Determine the number of angular nodes for each orbital. Answer is on the next slide. d –orbitals correspond to l = 2 so there are 2 angular nodes. (Remember that for the dz2 orbital the two nodes are in the shape of cones, not flat planes as in the other 3d orbitals.) http://www.geo.arizona.edu/xtal/geos306/d-orbitals.gif The f-orbitals. The
Now let’s look at the f-orbitals. How many planes (and cones) are in each orbital? Answer on the next slide… http://www.geo.arizona.edu/xtal/geos306/f-orbitals.gif The f-orbitals. The
Now let’s look at the f-orbitals. How many planes (and cones) are in each orbital? Answer on the next slide… 3 planes 3 planes 3 planes 3 planes 3 planes 3 planes 1 plane plus 2 cones http://www.geo.arizona.edu/xtal/geos306/f-orbitals.gif Identifying/Naming Orbitals Identifying/Naming
► The 4 quantum numbers provide information
► about the energy, size, shape and directionality.
Therefore, if you have the shape of an orbital with respect to a set of coordinates, you can assign 3 of the 4 quantum numbers to the orbital. ms is the only number that you’d be unable to assign. ► http://wps.prenhall.com/wps/media/objects/602/616516/Media_Assets/Chapter05/Text_Images/FG05_TB02.JPG How do I determine the ml value for a How given orbital?
► ► To find the ml quantum number, look down the zaxis, and look at how the wavefunction depends on φ. φ. Orbitals that look the same in all directions, looking down the z-axis, must have ml=0 (like the 3d orbital to the right, or the pz orbital). States with m=±1 go from plus, to minus States and back, looking down the z-axis. ► Here are the two 3d orbitals made from orbitals with |ml|=1. Looking down the z-axis, as φ goes from 0 to 2π, goes they look like cos (φ) or sin (φ). So you see one nodal plane, if you look down the z-axis. The px and py orbitals have the same dependence. States with |m|=2 go plus-minusStates plus-minus as you go around z. ► ► Here are the two 3d orbitals with |ml|=2. Looking down the z-axis, as φ goes from 0 to 2π, they look like cos (2φ) or sin (2 goes φ). In other words, you see two nodal planes. In general, states with fixed |ml| look like cos (ml φ) or sin (ml φ). So you see |ml| nodal planes. The probability distribution (|ψ|2) for two different orbitals is shown below. Identify the orbitals and give the quantum numbers for each orbital . Answers on the next slide. Practice Problem #1 Practice http://wps.prenhall.com/wps/media/objects/602/616516/Media_Assets/Chapter05/Text_Images/FG05_20-08UN.JPG The probability distribution (|ψ|2) for two different orbitals is shown below. Identify the orbitals and give the quantum numbers for each orbital. Practice Problem #1 Practice 3py (3, 1, +/-1, +/-1/2) 4dz2 (4, 2, 0, +/-1/2) http://wps.prenhall.com/wps/media/objects/602/616516/Media_Assets/Chapter05/Text_Images/FG05_20-08UN.JPG Practice Problem #2 Practice
► This problem is a lot tougher. ► Identify this type of orbital, and assign its quantum #’s. ► (next slide for answer)
z y http://csi.chemie.tudarmstadt.de/ak/immel/script/redirect.cgi?filename=http://csi.chemie.tudarmstadt.de/ak/immel/tutorials/orbitals/index.html Practice Problem #2 Practice
► z y http://csi.chemie.tudarmstadt.de/ak/immel/script/redirect.cgi?filename=http://csi.chemie.tudarmstadt.de/ak/immel/tutorials/orbitals/index.html Answer: This is a 5f orbital (quantum #’s n = 5, l = 3, ml = 0). ► Why? First find l. There are 3 angular nodes (2 cones and one plane) therefore l = 3. If l = 3 this must be an f-orbital. By using the equation for nodal surfaces you can find n. ► # nodal surfaces = n – l – 1 ► There is one radial node located at the white arrow. ► So . . . 1= n – 3- 1 ► Then . . . 5 = n ml = 0 because there is symmetry about the z axis. Scared yet?? Scared
► Don’t be. Nobody expects you to instantly identify a 17h orbital! Warren wouldn’t even give you the last problem on an exam without more pictures. ► Exams will always have enough pictures of an orbital so that you can clearly determine the number of nodal cones, nodal planes and nodal spherical shells-or we will tell you those numbers. Practice Problem #3 Practice
x y z B
x z y ► ► ► Name the orbital type: How many nodal planes are present in this orbital? What are the first two quantum #s of this orbtial?
(n = 4, l = 3) 4, 3)
http://csi.chemie.tu-darmstadt.de/ak/immel/script/redirect.cgi?filename=http://csi.chemie.tudarmstadt.de/ak/immel/tutorials/orbitals/index.html 4fyz2 4f 3 (see image B) (see Still want more practice??? Still
► Go to this website and download “Orbital Viewer” http://www.orbitals.com/orb/ov.htm http://
► In this program you can build your own orbitals by entering n, l and ml values. ...
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- Fall '08