Ch 6, Mastering Tutorial 6e (6.6) questions

[Zellmer, Robert]

I got a few questions from a student concerning some of the Tutorial Mastering homework.

The following two questions come from Tutorial 6e (section 6.6).

Question 1 (Part D):

How would the dx2−y2 orbital in the n=5 shell compare to the dx2−y2 orbital in the n=3 subshell?
Which of the following are True?


  1.  The contour of the orbital would extend further out along the x and y axes.
  2.  The value of ℓ would increase by 2.
  3.  The radial probability function would include two more nodes.
  4.  The orientation of the orbital would be rotated 45o along the xy plane.
  5.  The m_ℓ value would be the same.
A.  True.   The most probable distance (radius) for the 5d orbital is further out than the 3d.
            Also, the dx2−y2 looks like two p orbitals, one on the x-axis and one on the y-axis.

B. False.  The “ℓ” value is always the same for an s orbital no matter what shell it’s in.
     It’s the same for a p orbital (subshell) no matter what shell it’s in.  This applies
     for all orbitals with the same value of ℓ.  For a d-orbital ℓ = 2, not matter what
     shell it’s in.

C. True.  See below.

D. False.          A dx2−y2 orbital looks the same in terms of shape, no matter the shell.  It
                        doesn’t turn into a dxy orbital.

E. True.  The ℓ value tells us the shape and the m_ℓ value tells us the direction (orientation)
               of the orbitals.  For ℓ = 2, m_ℓ = -2, -1, 0, +1, +2.  So, a dx2−y2 has the same m_ℓ
               no matter which shell it’s in.

For C above it involves nodes.  So does another question later in this e-mail.  So, I’ve
included a discussion below.

total number of nodes for an orbital = n-1 (where n is the principal quantum #).
# spherical (radial) nodes = n – ℓ – 1 , where ℓ = the angular quantum # for the orbital.
# angular (planar) nodes = ℓ.

So, here’s a little table dealing with nodes:

1s has 1-1 = 0 total nodes.

n = 2, 2-1 = 1 total node

2s: # spherical (radial) nodes = 2 – 0 – 1 = 1
      # planar nodes = ℓ = 0 (ℓ =0 for an s orbital)
2p: # spherical nodes = 2 – 1 – 1 = 0
      # planar nodes = ℓ = 1 (ℓ =1 for a p orbital)


n = 3,   3 -1 = 2 total nodes

3s : # spherical (radial) nodes = 3 – 0 – 1 = 2
       # planar nodes = ℓ = 0 (ℓ =0 for an s orbital)
3p:  # spherical nodes = 3 – 1 – 1 = 1
       # planar nodes = ℓ = 1 (ℓ =1 for a p orbital)
3d:  # spherical nodes = 3 – 2 – 1 = 0
       # planar nodes = ℓ = 2 (ℓ =2 for a d orbital)
            (for the d_z^2 there are two canonical nodes
                which are angular nodes, not planar)

n = 4,  4 – 1 = 3 total nodes

4s: # spherical (radial) nodes = 4 – 0 – 1 = 3
      # planar nodes = 0 (ℓ =0 for an s orbital)
4p: # spherical nodes = 4 – 1 – 1 = 2
      # planar nodes = ℓ = 1 (ℓ =1 for a p orbital)
4d: #spherical nodes = 4 – 2 – 1 = 1
      # planar nodes = ℓ = 2 (ℓ =2 for a d orbital)
4f: #spherical nodes = 4 – 3 – 1 = 0
     # planar nodes = ℓ = 3 (ℓ =3 for an f orbital)


Question 2 (Part B):

The probability of finding an electron at a point in an atom is referred to as
the probability density (ψ2). The spatial distribution of these densities can be
derived from the radial wave function R(r) and angular wave function Y(θ,ϕ),
then solving the Schrödinger equation for a specific set of quantum numbers.
Which of the following statements about nodes and probability density are
accurate?

The probability of finding an electron at the center of a d orbital is greater
than zero.   FALSE
The 2s orbital does not have any nodes.  FALSE
The 4f orbitals have three nodes.  TRUE
The probability of finding an electron at the center of a p orbital is zero.  TRUE
The 3p orbitals have two nodes. TRUE


Some of this is answered above in question one

The probability of finding the electron at the center of any orbital (at the
nucleus) is zero.  This answers both the 1st and 4th statements,
as they’re covering the same concept.

The 3p has 1 spherical (radial) node.  It also has 1 planar node.  That’s
at total of 2 nodes.

The 2s has 1 node, a spherical (radial) node.
The 3p orbital has two notes, 1 radial and 1 planar

The 4f orbital has 3 nodes, 0 spherical (radial) nodes and 3 angular nodes.

Dr. Zellmer
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