B. E. Douglas, D. H. McDaniel, J. J. Alexander, Concepts and Models of Inorganic Chemistry, 3rd edition, John Wiley & Sons, New York, 1994, page 52.

Problem 1.8

Using Table 1.1 and Equation (1.23), calculate the values of r for which 2p and 4d hydrogenlike orbitals have radial nodes.
 
 

Answer:

From Equation (1.23):


for a radial node.

This means L(x) = 0.
 

From Table 1.1:

For a 2p orbital, L(x) = -6, i.e. there are no radial nodes.
 

For a 4d orbital, L(x) = 720x - 5760 = 0, or x = 8.

Using Z = 1 (hydrogen atom), n = 4, a0 = 0.529 Å, and x = 2Zr/na0 gives

r = na0x/2Z = (4)(0.529)(8)/(2)(1) = 8.46 Å.