Solutions to Homework 8 – chem 344 Sp’ 2014

1.

2.

3.

4.

5.

6.

a. Allowed, b. Allowed,S = 0 & J = 1, c. Forbidden by spin ,S = 1

7.

All different

orbitals means

they could all be

parallel spins

Since electrons are in different orbitals any

combination is possible paired or unpaired spins

Equivalent to min J lowest for less than half-filled

8.

9.

10. P15.6) Show that the total energy eigenfunctions 210(r,,) and 211(r,,) are

orthogonal. Do you have to integrate over all three variables to show that the functions are

orthogonal?

∫

√ ∫

∫

∫

This integral is zero because 3

2

0 0

sincos sin 0 0 0.

3d

It is sufficient to evaluate the integral over

11. P15.17) Ions with a single electron such as He+, Li

2+, and Be

3+ are described by the H

atom wave functions with Z/a0 substituted for 1/a0, where Z is the nuclear charge. The 1s

wave function becomes r 1 Z a0

3 2eZr a0 . Using this result, compare the

mean value of the radius r at which you would find the 1s electron in H, He+, Li

2+, and

Be3+

.

∫

∫

∫

∫

Using the standard integral: ∫

rZ

a

a

Z Za

4 6

16

3

2

3

0

3

0

4

4 0

2 30 0 0 0

3 3 1 3; ; ;

2 4 2 8H He Li Ber a r a r a r a

12. P15.20) Core electrons shield valence electrons so that they experience an effective

nuclear charge Zeff rather than the full nuclear charge. Given that the first ionization

energy of Li is 5.39 eV, use the formula in Problem P15.18 to estimate the effective

nuclear charge experienced by the 2s electron in Li.

From the previous problem,

2

213.60 eV

ZI

n ;

2 4 5.39eV1.26

13.60 eV 13.60 eVeff

n IZ

13. P15.24) The force acting between the electron and the proton in the H atom is given by F

= –e2/40r

2. Calculate the average value F for the 1s and 2pz states of the H atom in

terms of e, 0, and a0.

0 0

0 0

2*

210

22/ / 2

3 210 0 0 0 0

2 2 22 / 2 /0

3 3 2100 0 0 0 0 00

1

4

1 1sin

4

4 4

4 4 2 2

s

r a r a

s

r a r a

s

eF d

r

eF d d e e r d r

a r

ae e eF e dr e

a a a

0

0

0

2*

220

222/2 2

3 220 0 00 0 0

2 3/2

520 0 00

2/2

320 0 0

1

4

1 1cos sin

4 32

1 cos

4 16 3

1

4 24

Using the standard integr

pz

r a

pz

r a

pz

r a

pz

eF d

r

e rF d d e r d r

a a r

eF r e dr

a

eF r e dr

a

1

0

2 23

05 220 0 0 0

!al

12

4 24 48

n r

n

pz

nr e

e eF a

a a

14. P15.30) You have commissioned a measurement of the second ionization energy from two

independent research teams. You find that they do not agree and decide to plot the data together

with known values of the first ionization energy. The results are shown here:

The lowest curve is for the first ionization energy and the

upper two curves are the results for the second ionization

energy from the two research teams. The uppermost

curve has been shifted vertically to avoid an overlap with

the other new data set. On the basis of your knowledge of

the periodic table, you suddenly know which of the two

sets of data is correct and the error that one of the teams

of researchers made. Which data set is correct? Explain

your reasoning.

The data set shown by the dashed (purple) line is correct,

the red one is incorrect. Although the alkali atoms have

the lowest ionization energies, they must have the highest ionization energy for the second

ionization potential because the singly charged positive ions have the rare gas filled shell

configuration. The experimenters that produced the data set shown by the gray (red) line had

assigned atomic numbers that were too low by one.

15. P15.33) An approximate formula for the energy levels in a multielectron atom is

En Zeff

2e

280a0n

2, n 1, 2, 3, . , where Zeff is the effective nuclear charge felt

by an electron in a given orbital. Calculate values for Zeff from the first ionization energies

for the elements Li through Ne (SEE www.webelements.com). Compare these values for

Zeff with those listed in TABLE 15.1. How well do they compare? Using:

)mol 10 (6.022C) 10 (1.602177

m) 10 (5.291772)m J C 10 (8.85418781 8

81-23219-

-11-1-12-122

2

00

2

ion

A

ioneff

E

Ne

anEZ

,

the effective charges based on the first ionization energies for the elements Li to Ne are calculated as:

Element Li Be B C N O F Ne

Eionization in kJ mol-1

520.2 899.5 800.6 1086.5 1402.3 1313.9 1681.0 2080.7

Zeff 1.26 1.66 1.56 1.82 2.07 2.00 2.26 2.52

A comparison with the data in Table 15.1 shows that the approximation reproduces the effective charges reasonably well for the second-row elements with only 2s electrons, however, fails to predict the charges for elements with 2p electrons.

Extra Problems

1.

2.

3.

4.

5.

6. a. Allowed, b. Allowed, c. Forbidden

7. P15.8) How many radial and angular nodes are there in the following H orbitals?

a. 2 px

r,, 0 radial node and 1 angular nodes

b. 2s(r) 1 radial node and 0 angular node

c. 3dxz

r ,, 0 radial node and 2 angular nodes

d.

3dx2 y2

r ,, 0 radial node and 2 angular nodes

8. P15.16) Calculate the mean value of the radius r at which you would find the electron

if the H atom wave function is 210(r,,).

0

0 0 0

2

2 5

5

0 0 0 0

35 5 5

5 5 5

0 0 00 0 0

1cos sin

32

2 cos 2 2 1

32 3 32 3 24

r

a

r r r

a a a

r d d r e dra

r r e dr r e dr r e dra a a

Using the standard integral: ∫

6

0 05

0

15! 5

24r a a

a

9. P15.18) The energy levels for ions with a single electron such as He+, Li

2+, and Be

3+ are

given by En = –Z2e

2/80a0n

2, n = 1, 2, 3, .... Calculate the ionization energies of H, He

+,

Li2+

, and Be3+

in their ground states in units of electron-volts (eV). The ionization potential is the negative of the orbital energy.

2

2–19

2 2 2

2 12 1 2 1 11 19

0 0

2

2

1.602 10 C1eV

8 8 8.854 10 J C m 5.292 10 m 1.602 10 J

13.60 eV

Z

Z e nIa n

ZI

n

2 3H He Li Be13.60eV; 54.42eV; 122.4eV; 217.7eVI I I I