Take-Home Quiz on Hydrogenic Radial Distribution Functions

These exercises are to be done using Mathcad.  The radial distribution functions can be found in the sample Mathcad document.  pdf version.  Questions 1-6 all refer to a hydrogen atom.  That is z = 1.   

!.  Verify that the radial distribution function for a 3p orbital is normalized.

2.  Prepare plots of the radial distribution functions for 1s, 2s, and 3s orbitals on the same graph.

3.  What is the average distance of a 3s electron from the nucleus?  Compare it to the values for 1s and 2s electrons.

4.  Locate the nodes in the 4s radial distribution function.  Verify your results graphically.

5.  What is the probability of finding a 4s electron between the nucleus and the first node?

6.  What is the average distance of a 4s electron from the nucleus? Compare it to the values obtained in question 3.

7.   What is <r> for a 1s electron in the hydrogenic ions He⁺ and Li²⁺? Compare these values with <r>_1s for a hydrogen atom.

8.  For a many electron atom, the orbital energies depend on l as well as n, and they increase in the order: ns<np<nd.  Calculate the probability of finding a 3s, 3p and 3d electron, respectively,  within a distance of 1 bohr of the nucleus.  Use your results to explain why the orbital energies increase in the order 3s<3p<3d.  Use the words "penetration" and "shielding" in your discussion.