BioPHYSICAL chemistry

12.14

12.15

The limits are r=1.058 or x=2 and infinity so the probability is

equal to:

0 −(−e−^4 (2 × 42 + 2 × 4 +1)) =0.75

12.16

The limits are r=0.5 Å or x=1 and r=1.058 Å or x=2, so the

probability is:

− 4 e−^4 (2 × 4 + 4 +1) + 4 e−^2 (2 + 2 +1) =−0.44 +0.68 =0.24

12.17 The most probable radial position of the electron is simply the peak

position of this term. We can find the peak by setting the derivative

equal to zero:

12.18

=

3

2

a 0

4

44

0

3

0

2

0

32

0

0

0 2

a

rera r a xex x ae

∞

− −

∞

−

∫∫==

/ ddxx⎛−− − −xxx

⎝

⎜⎜

⎞

⎠

⎟⎟

32 ∞

(^20)

3

4

3

4

3

8

ψψτ

π

*drr π d

a

= errra/

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

− =

∞ ∞

∫∫

1

4

0

2

0 0

0 2 4 /

0

3

0

(^20)
a
rera r
∞
−
∫ d

0

8

1

0

3

2

0

0

=−/^0

⎛

⎝

⎜⎜

⎞

⎠

− ⎟⎟ =

a

er

r

a

ra or ra

04

2 14

0

3

2

0

3

= /^0 (

⎛

⎝

⎜⎜

⎞

⎠

− ⎟⎟=

d

d

d

r d

r

a

e

ar

π ra r

π

(^22) e−^2 ra/ (^0) )

04 = ()^2

d

d

*

r

πψψr

letxra/ then d d

a

== 0 ∫err ex−ra/ −x

0

3

(^4202242) xxex x=− x()+ +
∫
− (^22221)

1

4

4

0

3

2

2

0

3

002 2

π

π

a

err

a

−−ra erra

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

∫ //dd= ∫ dr

letxra/ then / d

a

== 0 ∫err ex−ra −x

0

3

(^4202242) ddxex x=− x()+ +
∫
− (^22221)

1

4

4

0

3

2

2

0

3

0 2 02

π

π

a

err

a

−ra er−ra

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

∫ / dd= ∫ / dr

ψψψpppx=− ()()+−− =xf r

1

2

468 ANSWERS TO PROBLEMS

9781405124362_5_end.qxd 4/29/08 9:17 Page 468