Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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Question
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For the H-atom, how does the Atomic model as described by Bohr connects to the atomic model
as described by Heisenberg/Schroedinger?
![206
Chapter 6 The Hydrogen Atom
expression for Î, in spherical coordinates is so much simpler than for θ or Î, (cf.
Equation 6.37). Clearly the rotating system does not know x from y from z and, in
fact, this inability to distinguish between the three directions explains the (21 + 1)-fold
degeneracy.
6-4. Hydrogen Atomic Orbitals Depend upon Three
Quantum Numbers
Up to now we have solved Equation 6.9, giving the angular part of the hydrogen atomic
orbitals. Now we will solve Equation 6.8, giving the radial part of the hydrogen atomic
orbitals. Equation 6.8 with ß set equal to 1 (1+1) can be written as
dR [ħ²1(1+1)
e²
-2m² dr (dr) + [²+¹) - [25-B]R(r) =
ER(r)=0
2m¸r²
E
Equation 6.43 is an ordinary differential equation in r. It is somewhat tedious to solve,
but once solved, we find that for solutions to be acceptable as the wave functions, the
energy must be quantized according to
me4
8h²n²
R₁(r) = -
meª
32л²εħ²n²
E =
If we introduce the Bohr radius from Section 1-8, a = = εh²/лm₂ е² = 4πεħ²/me²,
then Equation 6.44 becomes
1/2
31
(n-1-1)!
2n[(n+1)!]³
e²
8πε an?
It is surely remarkable that these are the same energies obtained from the Bohr model of
the hydrogen atom. Of course, the electron now is not restricted to the sharply defined
orbits of Bohr but is described by its wave function, (r, 0, 0).
In the course of solving Equation 6.43, we find not only that an integer n occurs
naturally but that n must satisfy the condition that n ≥ 1 + 1, which is usually written as
0<l<n-1
n = 1, 2, ...
(6.46)
because we have already seen that the smallest possible value of 1 is zero. (Equation 6.46
might be familiar from general chemistry.) The solutions to Equation 6.43 depend on
two quantum numbers n and I and are given by
2
nao.
n = 1, 2, ...
230
n = 1, 2, ...
1+3/2
re
(6.43)
2r
nao
(6.44)
(6.45)
(6.47)
where the L2+(2r/na) are called associated Laguerre polynomials. The first few
associated Laguerre polynomials are given in Table 6.4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3e671be-5699-41c7-ad14-09ef8165fba8%2Fbb65b36a-d506-4daf-a8b4-531f647820fd%2F14cntus_processed.jpeg&w=3840&q=75)
Transcribed Image Text:206
Chapter 6 The Hydrogen Atom
expression for Î, in spherical coordinates is so much simpler than for θ or Î, (cf.
Equation 6.37). Clearly the rotating system does not know x from y from z and, in
fact, this inability to distinguish between the three directions explains the (21 + 1)-fold
degeneracy.
6-4. Hydrogen Atomic Orbitals Depend upon Three
Quantum Numbers
Up to now we have solved Equation 6.9, giving the angular part of the hydrogen atomic
orbitals. Now we will solve Equation 6.8, giving the radial part of the hydrogen atomic
orbitals. Equation 6.8 with ß set equal to 1 (1+1) can be written as
dR [ħ²1(1+1)
e²
-2m² dr (dr) + [²+¹) - [25-B]R(r) =
ER(r)=0
2m¸r²
E
Equation 6.43 is an ordinary differential equation in r. It is somewhat tedious to solve,
but once solved, we find that for solutions to be acceptable as the wave functions, the
energy must be quantized according to
me4
8h²n²
R₁(r) = -
meª
32л²εħ²n²
E =
If we introduce the Bohr radius from Section 1-8, a = = εh²/лm₂ е² = 4πεħ²/me²,
then Equation 6.44 becomes
1/2
31
(n-1-1)!
2n[(n+1)!]³
e²
8πε an?
It is surely remarkable that these are the same energies obtained from the Bohr model of
the hydrogen atom. Of course, the electron now is not restricted to the sharply defined
orbits of Bohr but is described by its wave function, (r, 0, 0).
In the course of solving Equation 6.43, we find not only that an integer n occurs
naturally but that n must satisfy the condition that n ≥ 1 + 1, which is usually written as
0<l<n-1
n = 1, 2, ...
(6.46)
because we have already seen that the smallest possible value of 1 is zero. (Equation 6.46
might be familiar from general chemistry.) The solutions to Equation 6.43 depend on
two quantum numbers n and I and are given by
2
nao.
n = 1, 2, ...
230
n = 1, 2, ...
1+3/2
re
(6.43)
2r
nao
(6.44)
(6.45)
(6.47)
where the L2+(2r/na) are called associated Laguerre polynomials. The first few
associated Laguerre polynomials are given in Table 6.4.
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