Physical Chemistry , 1st ed.

11.51.P0.68% for a 1selectron.

11.53.Radial, angular, and total nodes respectively: (a)1, 0,
and 1 for  2 s(b)2, 0, and 2 for  3 s(c)1, 1, and 2 for  3 p
(d)0, 3, and 3 for  4 f.

11.57. 2 px 4 ^1  2 ^2 Za

3

 (^3) 
1/2
ZareZr/asin  cos 
11.60. r 1.5afor  1 s, where a0.529 Å
Chapter 12
12.2.(3d 2 ) 
1
1
62


Z
a
3
 (^3) 
1/2
Z
a
2
2
r^2 eZr/3asin^2 e^2 ior

1
1
62


Z
a
3
 (^3) 
1/2
Z
a
2
2
r^2 eZr/3asin^2 e^2 i
12.3.e/eannihilation  1.637  10 ^13 J or 9.860 
1010 J/mol
12.4.Yes, and spin functions are orthogonal.
12.5.(b) ms0; ms2, 1, 0, 1, or 2; ms^32 , ^12 ,
^12 , or ^32 .
12.6. (a)negative (b)positive (c)negative (d)negative
(e)positive
12.7.Hˆ
2


2
(^2 e1^2 e2^2 e3) 
4 
3

e
0
2
r 1

4 
3

e
0
2
r 2


4 
3

e
0
2
r 3

4 
e

2
0 r 12

4 
e

2
0 r 13

4 
e

2
0 r 23

not separable
12.8. (a)E5.883  10 ^17 J (b)E4.412  10 ^17 J
12.11.Li^1  2 [(1s 1 )(1s 2 ) (1s 2 )(1s 1 )]
12.12.H has only a single electron, so there is no antisym-
metry with respect to exchange to consider.
1 s 1 1 s 1  2 s 1 2 s 1 
12.13. (a)^1 s^21 s^2 ^2 s^22 s^2 
Be
1
24 
  1 s
3 1 s 3 ^2 s 3 2 s 3 
 1 s 4 1 s 4  2 s 4 2 s 4 
1 s 1 1 s 1  2 s 1 2 s 1  2 px,1
1 s 2 1 s 2  2 s 2 2 s 2  2 px,2

B

 1

1

 20

  1 s 3 1 s 3  2 s 3 2 s 3  2 px,3 

1 s 4 1 s 4  2 s 4 2 s 4  2 px,4

 1 s 5 1 s 5  2 s 5 2 s 5  2 px,5 

(The last column could be 2px, or 2py or , or 2pz or .
Thus, there are six possible determinants for a B atom.) (b)C
has six different possible determinants, as does F.

12.14.C has 720 terms in its antisymmetric wavefunction;
Na has 39,916,800, and Si has 87,178,291,200 terms.

12.16. (a)excited (b)ground (c)excited (d)excited

12.17. (a)Li (1s^22 p^1 ) will have six possible arrangements:
1 s^22 px^1 , 1s^22 px^1 , 1s^22 py^1 , 1s^22 py^1 , 1s^22 pz^1 , and 1s^22 pz^1 .

12.18.The correction to the energy won’t be an exact cor-

rection even if the integral can be solved analytically because

the wavefunctions in the integral are for the ideal system, not

the real system.

12.19. Eperturb  3 c/(4 2 ), where cis the anharmonicity

constant.

12.20.A correction like cx^3 makes the integrand an odd func-

tion, making the numerical value of the integral exactly zero.

12.21.a 3  15 kma^2 /(16^2 h^2 )

12.24. (a)no (unless both A& Bequal zero, which is a triv-

ial wavefunction anyway) (b)no (c)no (d)yes (e)no (f)no

(g)no. Most of these trial functions do not satisfy the bound-

ary conditions for a particle-in-a-box.

12.34.The Born-Oppenheimer approximation is more applic-

able to Cs 2 , whose nuclei move more slowly than those in H 2.

12.35. E2(H(^111 S^12 S 2

12

H

)

^12 )

12.40.For example, B: (g 1 s)^2 (u*1s)^2 (g 2 s)^2 (u*2s)^2 (u 2 px,

u 2 py)

12.42.No, F 22 should not exist according to MO theory.

Chapter 13

13.2. (a)E, C 2 (principal axis), 4C 2   (perpendicular to princi-

pal axis), h, 2v, i, S 2 (b)E, C 2 , 2v(c)E, several ’s

13.3. (b)C2v

13.5. (a)Complete group (b)Complete group (c)Incom-

plete group: Emissing (d)Incomplete group: C^23 (the inverse

of C 3 ) missing

13.6.C 2 , C 3 

13.7. (a)Sn(b)i

13.8. (a)4 classes, order  4 (b)8 classes, order  8 (c) 12

classes, order  24 (d)3 classes, order  4 (e)2 classes, or-

der  2 (f)5 classes, order  24

13.10. (a)S^34 , which is classed with the other S 4 symmetry

operations in the Tdcharacter table (b)C 2 , which is its own

inverse

13.11. (a)v (b)h(c)S 6 (d)S 4 (e)C^34

13.12.Symmetry elements do not necessarily follow the com-

mutative law (this is more apparent for larger groups).

13.13.Porphine has D2hsymmetry as a molecule; substitut-

ing metal ions for the two H’s inside the porphine ring, and

the symmetry becomes D4h.

0

0

 1

0

 1

0

 1

0

0

0

0

 1

sin 

cos 

0

cos 

sin 

0

0

0

1

 2 ^3 

^12 

0

^12 

 23



0

0

0

1

 1

1

0

1

1

0

Chapter 13 811

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