Spring Chemistry Homework #4 Answer Key

Introduction to Quantum Chemistry

Oxtoby, Nachtrieb and Gillis: Chapter 15:
 

33. The Heisenberg Uncertainty Principle states:

Here, Dx is the uncertainty in position and Dp is the uncertainty in momentum. Since the problem asks for the minimum uncertainty in velocity (p = mv), we will treat the above as an equality. So,

 

Thus, the velocity of this electron cannot be known to a precision more than ± 58,000 m/s.

b) The mass of a helium atom is given by:

So the increased mass of the helium atom decreases the uncertainty of the velocity (but the uncertainty in momentum remains the same in a and b)

35. For the one dimensional particle in the box, the energy of the particle is given by:

 

where,

h = Plank's constant
n = quantum level
m = mass of the electron
L = length of the box

 

For n = 1, 2 and 3, the results are:

E₁ = 3.37 × 10^-18 J
E₂ = 1.35 × 10^-17 J
E₃ = 3.03 × 10^-17 J

To excite an electron from the ground state (E1) to the first excited state (E2) requires a photon with energy equal to E₂ - E₁, or 1.01 × 10^-17 J. To find the wavelength, we do the following:

 

a) n = 3, l = 2, m = 1, m_s = 0 Forbidden (m_s cannot equal 0)
b) n = 2, l = 0, m = 0, m_s = -½ Allowed (this is an electron in a 2s orbital)
c) n = 7, l = 2, m = -2, m_s = ½ Allowed (this is an electron in a 7d orbital)
d) n = 3, l = -3, m = -2, m_s = -½ Forbidden (l must be in the range: 0, 1, …n-1)

40/42.
 

	n	l	orbital	total nodes	radial nodes	angular nodes
a	3	2	3d	2	0	2
b	7	4	7g	6	2	4
c	5	1	5p	4	3	1

 

44. The radial portion of the wavefunction for the 3p orbital is:

This term will equal zero only when s = 6. The definition of s, as given in Table 15.2 is:

Therefore, when Z = 1, and a_o = 0.529 Å,

b) For the 3s orbital,
 

This function will equal zero only when s equals the roots of the quadratic term of the function. These are s = 3.8, 14.2. So the radii at which nodes occur are: 2.0 Å and 7.5 Å.

DeKock & Gray: Chapter 1:

13. For an electron in a 3d orbital, the following quantum numbers are possible:

n = 3 (only)
l = 2 (only)
m = -2, -1, 0, 1, 2