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Jan 9, 2024

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Problem Set 8 – Quantum Numbers and Orbitals Chem 105 1. (a) State which quantum number is associated with the notions of shell, subshell, and specific orbital of an electron. The principle quantum number ( n ) = shell The angular momentum quantum number ( ℓ ) = subshell The magnetic quantum number ( m l ) = specific orbital (b) What values of l correspond to the letters s, p, d, and f? Why do we use letters to denote the values of the angular momentum quantum number, l ? ( ℓ ) = 0  s ( ℓ ) = 1  p ( ℓ ) = 2  d ( ℓ ) = 3  f Letters are used to denote the values of ( ℓ ) so that all three quantum numbers that uniquely identify an orbital can be written next to one another without confusion (for example, we write 2p x instead of of 211). (c) What type of orbital is designated by n = 4, l = 2, m l = 0, m s = ½ ? 4d 2. For the n=5 shell, (a) how many subshells are there? For n=5, ℓ=0,1,2,3, or 4 so there are 5 sub-shells. (b) How many orbitals are in each subshell? There are 2ℓ+1 orbitals (or values of m l ) in each sub-shell, so  ℓ=0 has 1 orbital  ℓ=1 has 3 orbitals  ℓ=2 has 5 orbitals  ℓ=3 has 7 orbitals  ℓ=4 has 9 orbitals (c) How many total orbitals are there? Adding the numbers of orbitals from the previous problems gives us a total of 25 orbitals in energy shell n=5. Alternatively, the total number of orbitals is equal to n 2 , which would also give us 25 orbitals. 3. Which subshell corresponds to each of the following sets of quantum numbers? a. n = 2, l = 1 2p b. n = 5, l = 3 5f c. n = 3, l = 2 3d d. n = 4, l = 3 4f

4. Draw diagrams showing the relative sizes, shapes and orientations of the 1s, 2p, and 3d orbitals of the hydrogen atom. 5. (a) What is a node in an electron orbital? A node is a region of space in an atom where there is zero probability of finding an electron of a particular energy. (b) How many total nodes will any given orbital have? It depends on the principle quantum number of the orbital. Total nodes is equal to n-1 (c) What is the difference between a planar and a radial node? Radial nodes are spherical, centered on the nucleus. Planar nodes are planes that cut through the nucleus (d) How can you tell how many planar and radial nodes an orbital will have? An orbital with the quantum number ℓ will have ℓ planar nodes. The remainder of the nodes will be radial. Since a particular orbital has n-1 total nodes, it will have n – ℓ – 1 radial nodes. (e) For the n=3 shell, how many total nodes will each orbital have? How many nodes are radial and how many are planar? Each orbital in this shell must have 2 total nodes (n-1 = 3-1 = 2). There are 3 sub-shells: s (ℓ=0), p (ℓ=1), and d (ℓ=2) with 1, 3, and 5 orbitals (values of ml), respectively. For the s orbital, ℓ=0 so it has 0 planar nodes. This means the 3s orbital must have 2 radial nodes. For the p orbitals, ℓ=1, so each 3p orbital has 1 planar node with the 1 remaining node being a radial node. For the 3d orbitals, ℓ=2 so each has 2 planar nodes. This means that the 3d orbitals have no radial nodes. 6. Draw diagrams of the following hydrogen atom orbitals and indicate the type and location of each node: 2s, 3s, 3p x , 4d xy a. 2s orbital = spherical with 1 radial node b. 3s orbital = spherical with 2 radial nodes c. 3p x orbital: two lobes along the x axis with 1 planar node (along the yz plane), 1 radial node

d. 4d xy orbital: four lobes in the xy plane with 2 planar nodes (1 along the xz plane, 1 along the yz plane), and 1 radial node 7. Identify the number of planar nodes, radial nodes, and the total number of nodes in the following orbitals. From that information, label each orbital with the appropriate shell and subshell values. (a) (b) (c) (a) 0 planar nodes (so l =0=s) (b) 1 planar node (so l =1=p) (c) 2 planar nodes (so l =2=d) 4 radial nodes, 2 radial nodes 1 radial node 4 total nodes (so 4 = n-1, n=5) 3 total nodes (so 3=n-1, n=4) 3 total nodes (so n=4)  5s orbital  4p  4d

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8. How can an electron get from one lobe of a p orbital to the other without going through the point of zero electron density between them? It moves as a wave. If we think of the wave properties of electrons, a node is just a point of zero amplitude for the wave. Waves can pass through nodes and have amplitude on either side of the node even while having zero amplitude at the node.