Activity: Orbital Shape
 Page ID
 338424
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left#1\right}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives
 Formulate a model for the number of radial nodes and number of angular nodes as a function of the principal and angular momentum quantum numbers, n and l, respectively.
 Apply the model to predict:
 the number of radial and angular nodes for any atomic orbital.
 the trend in how the number of radial and angular nodes changes as the principal quantum number n increases for a series of orbitals with fixed angular momentum quantum number l (e.g., 3d, 4d, 5d).
 the trend in how the number of radial and angular nodes changes as the angular momentum quantum number l increases for a series of orbitals with fixed principal quantum number n (e.g., 4s, 4p, 4d).
 Observe how orbital size changes with the principal and angular momentum quantum number.
 Rationalize, in terms of nodal structure (number of radial and angular nodes), how changes in the principle and angular momentum quantum numbers will affect the atomic orbital size and shape.
 Identify periodic trend in 2s2p energy splitting, rationalize how the energy splitting affects sp mixing (hybridization), and apply trends and rationale to make predictions about energy cost of hybridization and degree of sp mixing for second row elements B, C, N, O and F.
 Determine shielding constant for any atomic orbital from the effective nuclear charge, identify trends in degree of shielding of atomic orbitals for a given element, predict order of shielding constant for orbitals of a given element, identify periodic trends in the degree of shielding for valence atomic orbitals or different elements, rationalize the origin of sp mixing in terms of degree of shielding and penetration.
Significance
Molecules are composed of atoms that are chemically bonded to one another. The properties of molecules are therefore determined by the properties of the constituent atoms and chemical bonds that are formed between them. The electron distribution of atoms is built up from atomic orbitals, and the constructive overlap of atomic orbitals on different atoms are important to form chemical bonds in molecules. Therefore, an understanding of the molecular properties and chemical bonding starts with understanding the properties of atomic orbitals.
Context
Quantum numbers are used to describe electrons in atoms. There are four quantum numbers:
n – principal quantum number. n=1,2,3…
l – orbital angular quantum number. l=0,1,2,…n1
m – magnetic quantum number. m=l, (l1), … 0, … l1, l
s – spin quantum number. s = ½, ½
The current activity concerns the n and l quantum numbers, and how they determine the size and shape of atomic orbitals. Closely related to this is the concept of a nodal surface.
Activity – Part I
Open webpage https://tools.elearning.rutgers.edu/orbitalexplorer/tools/tool1_dev2.php to load the tools needed for the following activities.
 Practice using the Orbital Explorer.
a. Visualize an atomic orbital of a certain atom by selecting the corresponding quantum numbers and element. In this step (step 1), use the 3p orbital of iodine as an example. Visualize the 3p orbital of iodine by selecting n=3, l=1(p), m_{l}=0 (3pz), Iodine for the element and click Submit. The plot of radial distribution function (RDF) and the 3D model of the contour surface of a 3p orbital will be displayed under the menu. The orbital can be rendered as isosurface or density pointillist. In this activity, you will use “Isosurface” but you are encouraged to explore the "Density Pointilist" render type.
b. Read the RDF plot and answer the following questions.
i. R(90% ) is the radius of the sphere that enclose 90% of the electron density of an orbital and is a parameter often used to describe atomic size. What is the r(90%) of an Iodine 3p orbital in Bohr (a_{0})? __ ________a_{0}
ii. A node of an atomic orbital is a 2dimentional surface where the electron density becomes zero (not including r=0 and r=∞). A radial node is a sphere of certain radius r that has zero electron density. From a radial distribution plot, you could read whether or not there is a radial node and where the node is located.
Does 3p orbital have any radial node? (yes or no) ________
If yes, how many radial nodes are there? ________
iii. At what radius (radii) is the radial node(s) located? r = ______ a_{0}
Draw a vertical line at the location of each radial node by using the function in the Addition Labels. If there is only one radial node, label the vertical line as “radial node”; If there are more than one radial node, label the vertical line closer to the nucleus as “1st^{ }radial node” and the line further as “2nd radial node”, “3rd radial node” etc. Set the vertical line(s) in green and locate the label(s) at the top of the line.
iv. At what radius is the electron density maximum in an iodine 3p orbital? r=________ a_{0}
Draw a vertical line to indicate it and label the vertical line as “maximum radial density”. Set the line in magenta and locate the label at the top of the line. Export the plot with the vertical lines you drew in step iii and iv, and insert it in the space below.
c. Examine the orbital contour surface in the Atomic Orbital 3D viewer. An angular node is a surface characterized by a constant θ or φ (in spherical coordinates) that has zero electron density. Angular nodes are not reflected in the RDF plot but is in the orbital’s 3D model. You could visualize the angular nodal surfaces by checking the “show angular node(s)” checkbox under the Atomic Orbital 3D viewer. You could turn the orbital around with your mouse to examine the orbital from different angles. Answer the following questions:
i. How many angular nodes are there in a 3p orbital? __________
ii. Take a screenshot or export an image file from the 3D viewer by right clicking the JSmol window, selecting file and then exporting as PNG file.
2. Analysis for ns orbitals as a function of principal quantum number.
a. In the main menu, select quantum numbers for the orbitals listed in the table below, select Iodine as the element and click Submit to display the radial distribution function (RDF) and the atomic orbital 3D viewer. Examine the RDF plot for the number of radial nodes, and the radius that encloses 90% electron density (r_{90%}). Examine the 3D orbital shape for the number of angular nodes. You could turn the orbital around with your mouse to examine the orbital from different angles. Sum the number of radial and angular nodes to get the total number of nodes for a certain orbital. Fill in the table below.
Table 1. Observations of ns orbitals for Iodine atom.
Orbital 
n 
l 
m_{l} 
# Radial Nodes 
# Angular Nodes 
Total # Radial and Angular Nodes 
r_{90%} (Bohr) 
1s


2s


3s


4s


5s

b. Formulate a model relating the n, l and m_{l} quantum numbers to the number of radial, angular and total nodal surfaces for the ns orbitals:
i. number of radial nodes = ________
ii. number of angular nodes = ___________
iii. Total number of nodes = ___________