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Chemistry LibreTexts

Group Work 7: Angular Momentum

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    31878
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    Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

    Go to the http://winter.group.shef.ac.uk/orbitron/AOs/1s/index.html (or search for "orbitron 1s). Click on the equations, wavefunction, and electron density links.

    Q1: General Properties

    Using the Orbitron equation and the equations at the end of this worksheet, why does the 1s equation lead to a spherically symmetric orbital?

     

     

    Now explore other s atomic orbitals (use the tool bar on the left). Click on the equations, wavefunction, and electron density links. What similarities do you note for these orbitals? Why do the equations lead to these similarities?

     

     

    What differences do you note for the s orbitals? Why do the equations lead to these differences?

     

     

    Now explore the 2p atomic orbitals. Click on the equations, wavefunction, and electron density links. How is the wavefunction similar or different from the 1s wavefunction? What factors in the wavefunction equation lead to these similarities or differences?

     

     

    Look at other p orbitals. Click on the equations, wavefunction, and electron density links. What similarities are there among the p orbitals? What parts of the equations are responsible for the similarities?

     

     

    What differences are there among the p orbitals? What parts of the equations are responsible for the differences? 

     

     

    Q2: Nodes

    A wavefunction has a node when its value is zero at a location that is not one of its limits. We can find a node by setting the function equal to zero and solving for the variable(s).

     

     

     

    Which of the orbitals you have explored above have nodes? Describe the nodes for each orbital, that is, what is its shape and where is it located?

     

     

     

    Do s orbitals have nodes? Do p orbitals have nodes?

     

     

     

    In general, how many nodes does each wavefunction have? Can you develop a formula that predicts the number of nodes? 

     

     

     

    H atom wavefunction Spherical harmonics
    \(\psi _{100}=\dfrac{1}{\sqrt{\pi}}(\dfrac{1}{a_0})^{3/2}e^{-r/a_0}\) \(Y_{0,0}(\theta,\phi)=Y_0^0(\theta,\phi)=\dfrac{1}{(4\pi)^{1/2}}\)
    \(\psi _{200}=\dfrac{1}{\sqrt{32\pi}}(\dfrac{1}{a_0})^{3/2}(2-\dfrac{r}{a_0})e^{-r/2a_0}\) \(Y_{1,0}(\theta,\phi)=Y_1^0(\theta,\phi)=(\dfrac{3}{4\pi})^{1/2}\cos\theta\)
    \(\psi _{210}=\dfrac{1}{\sqrt{32\pi}}(\dfrac{1}{a_0})^{3/2}\dfrac{r}{a_0}e^{-r/2a_0}\cos\theta\) \(Y_{1,1}(\theta,\phi)=Y_1^1(\theta,\phi)=(\dfrac{3}{8\pi})^{1/2}\sin\theta e^{i\phi}\)
    \(\psi _{21\pm 1}=\dfrac{1}{\sqrt{64\pi}}(\dfrac{1}{a_0})^{3/2}\dfrac{r}{a_0}e^{-r/2a_0}\sin\theta e^{\pm i\phi}\) \(Y_{1,-1}(\theta,\phi)=Y_1^{-1}(\theta,\phi)=(\dfrac{3}{8\pi})^{1/2}\sin\theta e^{-i\phi}\)
    \(\psi _{300}=\dfrac{1}{81\sqrt{3\pi}}(\dfrac{1}{a_0})^{3/2}(27-18\dfrac{r}{a_0}+2\dfrac{r^2}{a_0^2})e^{-r/3a_0}\) \(Y_{2,0}(\theta,\phi)=Y_2^{0}(\theta,\phi)=(\dfrac{5}{16\pi})^{1/2}(\cos^2\theta -1)\)
    \(\psi _{310}=\dfrac{1}{81}(\dfrac{2}{\pi})^{1/2} (\dfrac{1}{a_0})^{3/2}(6\dfrac{r}{a_0}-\dfrac{r^2}{a_0^2})e^{-r/3a_0}\cos\theta\) \(Y_{2,1}(\theta,\phi)=Y_2^{1}(\theta,\phi)=(\dfrac{15}{8\pi})^{1/2}\sin\theta\cos\theta e^{i\phi}\)
    \(\psi _{31\pm 1}=\dfrac{1}{81\sqrt{\pi}}(\dfrac{1}{a_0})^{3/2}(6\dfrac{r}{a_0}-\dfrac{r^2}{a_0^2})e^{-r/3a_0}\sin\theta e^{\pm i\phi}\) \(Y_{2,2}(\theta,\phi)=Y_2^{2}(\theta,\phi)=(\dfrac{15}{32\pi})^{1/2}\sin^2\theta e^{2i\phi}\)
    \(\psi _{320}=\dfrac{1}{81\sqrt{6\pi}}(\dfrac{1}{a_0})^{3/2}\dfrac{r^2}{a_0^2}e^{-r/3a_0}(3\cos^2\theta -1)\) \(Y_{2,-2}(\theta,\phi)=Y_2^{-2}(\theta,\phi)=(\dfrac{15}{32\pi})^{1/2}\sin^2\theta e^{-2i\phi}\)
    \(\psi _{32\pm 1}=\dfrac{1}{81\sqrt{\pi}}(\dfrac{1}{a_0})^{3/2}\dfrac{r^2}{a_0^2}e^{-r/3a_0}\sin\theta\cos\theta e^{\pm i\phi}\) \(Y_{3,0}(\theta,\phi)=Y_3^{0}(\theta,\phi)=(\dfrac{15}{32\pi})^{1/2}\sin^2\theta\)
    \(\psi _{32\pm 2}=\dfrac{1}{162\sqrt{\pi}}(\dfrac{1}{a_0})^{3/2}\dfrac{r^2}{a_0^2}e^{-r/3a_0}\sin^2\theta e^{\pm 2i\phi}\)