Chm 118
Fall 2015, Exercise Set 1
Preliminaries, Quantum Mechanics, and Atomic Structure
Mr. Linck
Version: 4.3.
July 27, 2015
1.1. Simple Lewis Structures
Give the Lewis structure of H2 O, SCl2 , NH3 , ClF, and CH3 OH; convince yourself that all
formal charges are zero in these compounds. Use line structures and not “dot” structures.
Include lone pairs.
1.2. Lewis Structure Patterms
Examine the Lewis structures given in Figure 1, correct any that are wrong, and determine
patterns that exist among the different Lewis structures that allow you to rapidly discern
incorrect structures.
1.3. Simple Lewis Structures
Use you concepts from the preceeding problem to draw the Lewis structure of urea, (NH2 )2 CO.
1.4. Counting Electrons
In chemistry, when we use the word radical, we mean a material composed of main group elements with an odd number of electrons. Which of the following compounds are “radicals?”
CH3 , PF5 , HCO2 , NH–2 .
1.5. Connectivity and Lewis Structure
Is OF2 likely to have an oxygen atom or a fluorine atom in the center? Why? HINT: Use,
as always, formal charge.
1.6. Connectivity and Lewis Structures
The Lewis structure of CO2 is not C=O=O, where some electrons are missing: you fill them
in. What’s wrong with this Lewis structure? What is the correct Lewis structure? Why is
the one we call “correct” correct?
1.7. Connectivity and Lewis Structure
Is the structure of NOF as written, or is it ONF, or perhaps OFN?
1.8. Violation of the Octet
In Table 1 are the properties of several compounds that violate the octet. See if you can find
some consistent “rule” that gives you some abiity to predictd what bonding characteristics
these compounds have that will account for why they violate the octet rule and other
compounds do not.
1.9
2
Figure 1: Some Lewis Structures
Cmpd
PF5
PBr5
AsCl5
SF6
SF5 Cl
SCl4
SO2
SeF6
SeCl4
TeF4
TeI4
Color
colorless
red-yellow
colorless
colorless
colorless
colorless
colorless
colorless
colorless
colorless
black
Properties
gas, mp −84.5o C
dec, 84o C
dec. −50o C
gas, mp −50.8o C
gas, mp −64o C
s, dec. − − 30o C
gas bp −10o C
gas, mp −46.6o C
solid, sublimes with dec 196o C
solid, mp 130o C
solid, dec with boiling at 283o C
Cmpd
PCl5
AsF5
SbF5
SF4
SF5 Br
SO3 ,
SO3
SeF4
SeBr4
TeBr4
Color
colorless
colorless
colorless
colorless
colorless
colorless
colorless
colorless
orange
yellow
Properties
l, bp 150o C
gas, bp −52.8o C
l, bp (with d) 140o C
gas, bp −40o C
gas, mp −79o C
mp 16.8o C
s, bp 44.4o C
gas, bp 102o C
solid, liquid only, mp 123o C
dec with boiling at 414o C
Table 1: Properties of Compounds that Violate the Octet.
Chm 118
Exercise Set 1
1.18
Bond
H-H
F-H
I-H
C-F
3
Bond Energy
436
565
497
489
Bond
C-H
S-H
C-C
F-F
Bond Energy
416
362
331
158
Bond
N-H
Cl-H
C-N
Si-C
Bond Energy
391
429
305
306
Bond
O-H
Br-H
C-O
C-I
Bond Energy
463
365
358
214
Table 2: Bond Energies in kJ/mole for Various Single Bonds.
1.9. Lewis Structures With More Than Eight Electrons
Some more compounds that exist (i.e., can be put into a bottle) but violate the Lewis “octet
2–
rule” on the high side are SO2–
4 (with some anion), IF5 , ClF3 , SiF6 (with some cation).
Compounds that violate the Lewis “octet rule” and (therefore?) do not exist are: SI4 ,
2–
–
SiH2–
6 , S5 (tetrahedral form), PH5 , F3 , NF5 (all negative ions with some cation). See if you
formulation from the last problem works. If your rule needs modification, do so.
1.10. Lewis Structures With More Than Eight Electrons
Give the Lewis structures of Cl3 PO, SF4 , and SF4 O.
1.11. Lewis Structure with More Than Eight Electrons
Write Lewis structures for SiF2–
6 , XeF2 , XeOF4 .
1.12. Lewis Structure and Chemical Behavior
The bond energy is the energy required to break a molecule into two fragments. For instance
the energy to break the H-H bond in dihydrogen is 436 kJ/mole. Why is it harder to break
the N-N bond in dinitrogen than it is to break the H-H bond in dihydrogen?
1.13. Bond Energy Trends
Data for the single bond energy for several species are given in Table 2. What patterns do
you see? Do you believe enough in your patterns to doubt any data?
1.14. Bond Energy Trends
Energies for some double and triple bonds are in Table 3. What trends do you see?
1.15. Using Bond Energies
Use the bond energies in the Tables 2 and 3 to compute the energy of the process
H2 CCH2 + H2 → H3 CCH3
Is the process energetically downhill? Why?
1.16. Lewis Structure and Chemical Properties
The N-N bond length in NH2 NH2 is about 1.5Å whereas that in N2 is 1.1Å. Comment.
1.17. Lewis Structure and Chemical Properties
The P-P bond length in PH2 PH2 is about 2.2Å and that in elemental white phosphorous,
which exists as tetrameric units, is the same. Deduce a structure for P4 and comment on
the observation concerning the bond lengths. HINT: Be inventive and make use of the facts:
the bond lengths in the two materials are the same, therefore the bonds are of the same
kind.
Chm 118
Exercise Set 1
1.25
4
Double Bonds
C-C
Triple Bonds
C-C
Bond
Bond Energy
615
C-N
616
C-O
729
O-O
498
811
C-N
892
C-O
1077
N-N
945
Table 3: Bond Energies in kJ/mole for Various Double and Triple Bonds.
1.18. Lewis Structure and Chemical Properties
If P4 is heated to 1200o C it is 50% dissociated into 2 moles of P2 . Is that surprising? HINT:
Take a look at problem 33.
1.19. Lewis Structure and Chemical Properties
The bond length found in P2 —see the last problem—is 1.893Å. Explain the difference
between that and the bond length in P4 —see problem 17.
1.20. Energy and Chemical Forms
The energy (actually enthalpy—heat at constant pressure) required to take P4 (g) to 2 moles
of P2 (g) is 289 kJ/mole of P4 . That required to take a mole of P2 (g) to 2 moles of P(g) is
485.3 kJ/mole of P2 . What is the energy required to take one mole of P4 (g) to four moles
of P(g)? What property of energy (enthalpy) did you use? Do you know the name of that
type of process? HINT: It is hard to write and pronouce in the possessive as “X’s law”;
better done (although almost no one does) as the “law of X”.
1.21. Bonding Approximations
All very simple bonding models discriminate between an ionic model and a covalent model.
How do you see the difference?
1.22. Bonding Approximations
The great natural philosopher of rural Arkansas, Boniface Beebe, once wrote: “The easiest
way to model the bonding in diamond is with an ionic model.” Comment. In the same
publication, Professor Beebe also said “The ionic interaction between lithium ion and fluoride is stronger than that between between magnesium ion and fluoride because lithium is
smaller.” Comment.
1.23. Ionic versus Covalent Model of Bonding
Consider a bond between two elements. For what pairs of elements would an ionic model
make sense? For what pairs would a covalent model make sense? HINT: Use the periodic
table.
1.24. Making Bonding Sense of Chemical Facts
Compounds of the formula CaP and SrP, which are diamagnetic, have been isolated. Describe the bonding within these compounds. HINTS: A diamagnetic compound is one in
which there are no unpaired electrons. Also, there is most likely a dominant ionic bond
between the element on the left of the periodic table (charged positively) and that on the
right (charged negatively). Finally, you might want to work with polyatomic anions.
Chm 118
Exercise Set 1
1.35
5
1.25. Energy and Chemical Forms
The compound P4 at room temperature is a white solid (which is why it is called “white”
phosphorous). Will it take more or less energy to convert P4 (s) to four moles of P(g) than
it did the P4 (g) in the problem 20? How do you reach your conclusion?
1.26. Lewis Structure and Chemical Properties
Make a guess about the geometrical structure and bonding in P2 H2 . What would you
estimate is the length of the P-P bond (if one exists)? Is this compound isoelectronic with
any carbon containing species?
1.27. Stability and Lewis Structure
Draw a Lewis structure for cyclic O3 . What, if anything, does this Lewis structure suggest
about the stability of cyclic O3 ? HINT: Answer this only on the basis of the Lewis structure
model: that is, electron pairs, octet rules, formal charges; not hybridization (if you have
heard of that) or other esoteric phenomena.
1.28. Lewis Structure and Existence
Which of the following molecules are unlikely to be “found in a bottle?” SFn where n runs
from 2 to 6. Give your reasoning.
1.29. Magnetic Properties
Which of the species in the last problem are unlikely to be diamagnetic? HINT: See problem 24.
1.30. Magnetic Properties and Structure
If you had a compound of empirical formula SF that was diamagnetic, how would you
formulate the molecular structure? That is, give a valid Lewis structure.
1.31. Lewis Structure and Chemical Behavior
If someone asked you if SBr4 could be synthesized and characterized, what would you say?
HINT: “Characterized” in chemistry means properties, such as melting point, boiling point,
vapor pressure, of the substance have been measured.
1.32. Lewis Structure and Chemical Behavior
If you argued that the compound in the previous question could be made, what would you
say to someone who asked you if SI4 could be synthesized and characterized?
1.33. Lewis Structure and Chemical Behavior
For your information, Holleman-Wiberg reports that SCl4 exists only at low temperature.
When warmed, it decomposes into SCl2 and Cl2 . It seems to me that most (I will later
argue ALL) things decompose at a high enough temperature. If you know why, say why.
Otherwise, remember the fact and hold on tightly to the seat of your chair until we get
there.
1.34. Lewis Structure and Chemical Behavior
Given the facts for SCl4 in the last problem, what do you think about the “existence” of
SBr4 ? about SI4 ?
Chm 118
Exercise Set 1
1.46
6
Cmpd
SOF2
SOCl2
SOBr2
Physical State
gas
liquid
solid
rSO , Å
1.420
1.444
1.45
rSX , Å
1.583
2.076
2.27
OSX angle
106.2
107.3
108.
XSX angle
92.2
96.2
96.
Table 4: Structural Properties of Thionyl Halides.
1.35. Lewis Structure and Existence
There are compounds that exist, are stable, but violate the Lewis “octet rule” on the
low side. Examples are BF3 , BeCl2 , AlCl3 . Compounds such as (CH3 )3 C+ exist but are
quite unstable. Verify that these compounds violate the octet rule. What seems to be the
underlying condition(s) for which a violation of the Lewis rule on the low side is possible.
Suggest a reason why you can “bottle” BF3 but not (CH3 )3 C+ ?
1.36. VSEPR
Give the VSEPR structure of AlCl3 , PH3 , CF4 , H2 S, H3 O+ , and NH–2 .
1.37. Lewis and Geometrical Structure of Main Group Compounds
Write Lewis structures for BH–2 , SnCl2 , S2 O2–
3 and (CH3 )2 Be. Use VSEPR to determine
structures.
1.38. Lewis and Geometrical Structure of Main Group Compounds
Write Lewis structures for SiH4 , AsCl3 , SCl2 and IF+
4 . Use VSEPR to determine structure.
1.39. Geometrical Structure
What will be the shape of GeH–3 ? HINT: By shape, I mean the arrangement of nuclei in
space.
1.40. Structure and Properties
What is the approximate shape of SOF2 ? HINT: This is a simple VSEPR question.
1.41. Interpretation of Data
What trends do you see in the data given in Table 4?
1.42. Geometrical Structure
What will be the shape of SF+
3?
1.43. Using Lewis Structures to Understand Chemistry
When SF4 is treated with AlCl3 a pair of ions is formed. Write the reaction and justify
what is happening in the process.
1.44. Geometrical Structure of Main Group Compounds
Find approximate bond angles for CH2 NOH, B(OH)3 , and HNO3 .
1.45. Lewis Structure
Write a Lewis structure for B2 Cl4 .
Chm 118
Exercise Set 1
1.57
7
Central Atom
PX3
AsX3
SbX3
BiX3
X=F
-946
-960
-899
-900
X = Cl
-320
-305
-260
-379
X = Br
-199
-197
-382
-276
X= I
-46
-58
-100
-150
Table 5: Heat of Formation for Various Group 13 Trihalides.
1.46. Lewis Structure and Geometrical Structure
What is the VSEPR structure of B2 Cl4 ? What do the Lewis and VSEPR structures say
about the question: “Is B2 Cl4 twisted or planar?”
1.47. Lewis Structure
In view of your answer to the last problem, comment on what a Lewis structure is trying
to tell you and what it has no ability to tell you.
1.48. Trends in Bonding
What is a heat of formation? Examine the data in the first row of Table 5 and find the
trend.
1.49. Trends in Bonding
What trend do you see down the X = F column of Table 5? HINT: Don’t over interpret.
1.50. Trends in Bonding
What trend do you see down the X = Cl column of Table 5?
1.51. Symmetry Operation
What is a symmetry operation?
1.52. Proper Symmetry Operations
A proper symmetry operation is one that you could do with your fingers on a molecular
models: Rotations about an axis is the example. To specify such an operation requires two
facts. What do you think they are?
1.53. Proper Symmetry Operations
A proper symmetry operation of rotation by 90 degrees about the z axis changes the point
{ x y z } to the point { -y x z }. What does a rotation about the z axis of 180 degrees do
to the point { x y z }? See Figure 2 for an illustration. NOTE: The first is called a C4
operation and the second is called a C2 . Can you suggest a reason why?
1.54. Proper Symmetry Operations
Find a proper symmetry operation in NO–2 .
1.55. Proper Symmetry Operations
Find a proper symmetry operation in NH3 . Is there a second proper operation?
1.56. C2 Symmetry Operations
Demonstrate that ethylene, C2 H4 , has a C2 symmetry operation; does it have more than
one? HINT: You must know where the nuclei are in space to answer this type of question.
Chm 118
Exercise Set 1
1.61
8
Figure 2: Rotation of 90o about the z axis (vertical axis) in S2+
4 . The sulfur atoms are
labeled with numbers so that the motion can be seen.
Figure 3: Improper symmetry operation, σ, on S2+
4 . The sulfur atoms are labeled with
numbers so that the motion can be seen.
1.57. Improper Symmetry Operations, I
There are three kinds of improper symmetry operations, which are operations that cannot
be done with your fingers on a model, but must be imagined. The first is a plane of
symmetry. You must specify the plane (or use the your flattened hand to show where it is).
If the plane is the yz plane, then a reflection plane, usually labeled σ changes the point {x
y z} to {-x y z}. Is there a plane of symmetry in NO–2 ?
1.58. Improper Symmetry Operations, I
What is the plane of symmetry as illustrated in Figure 3? The vertical axis could be called
the z axis. It takes one other axis to define the plane. HINT: You could use the labels on
the sulfurs to help define your plane.
1.59. Improper Symmetry Operations, I
Is there a plane of symmetry in NH3 ? Specify where it is relative to a drawing (or, if you
prefer, in words).
1.60. Improper Symmetry Operations, I
What is the plane of symmetry as illustrated in Figure 4? The vertical axis could be called
the z axis. It takes one other axis to define the plane. HINT: You could use the labels on
the sulfurs to help define your plane.
Chm 118
Exercise Set 1
1.67
9
Figure 4: Another improper symmetry operation, σ, on S2+
4 . The sulfur atoms are labeled
with numbers so that the motion can be seen.
Figure 5: Improper symmetry operation, i, on the trans form of H2 O2 . The atoms are
labeled with numbers so that the motion can be seen.
1.61. Improper Symmetry Operations, I
Is there a plane of symmetry in C2 H4 ? Label it so a reader will understand, either in a
picture or by words.
1.62. Proper Symmetry Operations
What is a proper symmetry operation? What is an improper one?
1.63. Proper Symmetry Operations
What are the proper symmetry operations found in H2 O? In the square planar compound
PtCl2–
4 ?
1.64. Proper Symmetry Operations
What are the proper symmetry elements found in staggered ethane? HINT: There are a
couple of these that are hard to see.
1.65. Proper Symmetry Operations
What are the proper symmetry elements found in eclipsed ethane? HINT: There are a
couple of these that are hard to see.
1.66. Improper Symmetry Operations II
The second kind of improper symmetry operation is the center of inversion, usually denoted
“i”. You do not need to specify anything about this operation; it takes the point {x y z}
to {-x -y -z}. A center of inversion is illustrated in Figure 5. Is there a center of inversion
in PtF2–
4 , a square planar compound? HINT: Also see problem 69
Chm 118
Exercise Set 1
1.72
10
Figure 6: Improper symmetry operation, S4 , on the B2 Cl4 . The chlorine atoms are labeled
with numbers so that the motion can be seen.
Figure 7: The structure of diborane, B2 H6 . The small spheres are hydrogen atoms.
1.67. Improper Symmetry Operations II
Is there an i in PtCl2–
4 , a square planar compound?
1.68. Improper Symmetry Operations II
Is there an i in NH3 ?
1.69. Improper Symmetry Operations III
The third kind of symmetry operation is called a rotation/reflection, and is labeled Sn . In
this process you first rotate by an angle of 360/n, then reflect in a plane perpendicular to
that axis of rotation. This procedure is illustrated in Figure 6. Look at that process and
then show that an S2 is identical to i.
1.70. Improper Symmetry Operations
What are the improper symmetry elements found in H2 O? in NH3 ? In the square planar
compound PtCl2–
4 ?
1.71. Improper Symmetry Operations
What are the improper symmetry elements found in ethane?
Chm 118
Exercise Set 1
1.83
11
1.72. Symmetry Operations
Does B2 H6 , called diborane, whose structure is indicated in Figure 7, have a plane of
symmetry? Does it have a C2 axis of rotation? Is anything wrong with the Lewis structure
of this molecule?
1.73. Symmetry Operations
Find all the symmetry operations in diborane. HINT: For further help with symmetry
operations in a number of molecules, see the web page:
symmetry.otterbein.edu/galley/index.html
1.74. Symmetry Operations
Find the symmetry operations in H3 PO. HINT: You may need a Lewis structure and a
VSEPR analysis on this to get the structure; if so, do so, as you always should.
1.75. Symmetry Operations
Comment on the sentence taken from The Little Book of Symmetry, published in 1879 by
Boniface Beebe: “The following molecules all contain a C3 axis of rotation: CH4 , SF6 , NH3 ,
CH2 Cl2 .
1.76. Naming a Group of Symmetry Operations
The set of symmetry operations present in a water molecule is called C2v . That is just a
name for the set, E, C2 , σxz , and σyz . We say a water molecule has C2v symmetry. What
are the symmetry operations present in NO–2 ? What is the symmetry of NO–2 ?
1.77. Lewis Structures and Resonance
Presumably you said that the two oxygen atoms are equivalent in the nitrite ion in the last
problem (because it is true). Draw the Lewis structure. What do you have to invoke to
make the two oxygen atoms equivalent?
1.78. Lewis Structures Requiring Resonance
Give the Lewis structures for NO–3 , CH3 C(O)O– (where the (O) means off the chain of
elements, in this case the chain is C-C-O ), CO2–
3 .
1.79. Stability and Lewis Structure
Draw the Lewis structure for bent, but non-cyclic, O3 . You will have to use resonance.
Why is this compound reactive?
1.80. Symmetry Operations
What are the symmetry operations present in cyclic O3 ? HINTS: See problem 27. This
group of symmetry operations is called D3h .
1.81. Symmetry Operations
What are the symmetry operations present in non-cyclic O3 ?
1.82. Stability and Lewis Structure
Write a Lewis structure for the ions NCS– and CNS– where those arrangements are meant
to imply the arrangement of atoms in space. Which appears to be more stable? Why?
Chm 118
Exercise Set 1
1.92
12
Figure 8: Napthalene framework
1.83. Isoelectronic Compounds
Give two compounds that are isoelectronic with O3 . If they are isoelectronic, what happens
to their Lewis structures?
1.84. Stability and Lewis Structure
How is SO2 related to O3 ? Why is SO2 relatively nonreactive (at least as an oxidant)
compared to ozone? HINT: Don’t try to answer this problem without knowing what an
oxidant is.
1.85. Lewis Structures Requiring Resonance
Draw the Lewis structure of benzene, cyclic C6 H6 , including important resonance structures.
1.86. Symmetry Operations
What are the symmetry operations in benzene? HINT: Lots of them. Find at least ten.
1.87. Lewis Structures Requiring Resonance
Draw the Lewis structure of napthalene, C10 H8 , which is similar to benzene with two rings
fused on an edge. See Figure 8, which does not have all bonds drawn in it. Also note there
is a hydrogen atom at positions 1 to 8.
1.88. Lewis Structures and Properties
Measurement indicates that the C1-C2 bond length in napthalene is 1.36Å whereas the
C2-C3 bond length is 1.42Å. Give an explanation.
1.89. Lewis Structures and Properties
What other bonds in napthalene have the same length as the C1-C2 bond? Why is this so?
HINT: Think symmetry.
1.90. Equivalent Atoms
If you had trouble with the last problem, here is a hint: Can you rotate napthalene so that
the C1-C2 bond is where the C3-C4 bond currently is? What does that imply about those
two bond lengths?
1.91. Equivalent Atoms
Are there any other equivalent bonds in napthalene?
Chm 118
Exercise Set 1
1.100
13
Figure 9: Structures
Figure 10: Structures
1.92. Resonance
Are structures 5 and 6 resonance structures of each other? If so, which is “more important”?
1.93. Lewis Structures Requiring Resonance
Draw the Lewis structure of NO2 Cl.
1.94. Resonance
Structures 1 and 2 are resonance structures of each other? Right?
1.95. Geometrical Structure of Main Group Compounds
Find the VSEPR structure of ClO–4 , NCS– , CO2 , CO2–
3 , C2 H2 , Cl2 CCCCl2 .
1.96. What VSEPR Says and What It Doesn’t Say
Your answer to the last part of the last problem is ambiguous. Why?
1.97. Lewis and Geometrical Structure of Main Group Compounds
Write Lewis structures for NO–2 and NO+
2 . Use VSEPR to determine structures. What
would you anticipate for the structure of NO2 . You have to deviate from the “normal”
rules to get an answer; justify your method of thinking.
1.98. Lewis Structure
Write Lewis structures for ClO–4 , O2–
2 , Cl2 CO.
1.99. Symmetry Operations
What proper symmetry operations are present in (a) SO3 , (b) planar C2 O2–
4 , and (c) transPtCl2 (NH3 )Br– . HINT: In the last case, the molecule is planar and you should ignore the
hydrogen atoms on the nitrogen for symmetry purposes (in this case).
Chm 118
Exercise Set 1
1.110
14
Molecule
ClO+
2
ClO2
ClO–2
Bond Length, Å
1.408
1.475
1.57
Bond Angle
119o
165o
110o
Table 6: Structural Data for Various Chlorine Dioxides.
1.100. Symmetry Operations
Find the improper symmetry operations in the molecules of problem 99.
1.101. Structures
The structures of species ClOn2 for n = -1, 0, and 1 are given in Table 6. Do these values
make sense? Why or why not?
1.102. Lewis Structure
2–
Write Lewis structures for SO2–
2 and SO3 .
1.103. Acids and Bases
What is an acid? What is a base? HINT: There are several different definitions of an acid
(base). Let’s use the easiest one for now.
1.104. Acids and Bases
Why would you expect that the materials in problem 102 might be called “bases”?
1.105. A Trivial, but Useful, Look at Bases, and hence, Acids
Imagine being very, very small and hitchin’ a ride on a proton, a hydrogen ion. You look
out at the world and see a negative charge nearby; will you be attracted to that charge?
What if there were two negative charges, one on a flourine atom and the other on a carbon
atom. To which would you be most attracted? HINT: Think about which of those atoms
would most likely be willing to share its charge with your hydrogen. What if there were
two negative charges, one of -2 and the other of -1. To which would you be more attracted?
What if a -1 charge was on an atom next to a flourine atom in one case and a -1 charge on
an atom next to a boron atom in the other; to which would you be more attracted?
1.106. Lewis Structure and Basicity
We use Lewis structures to understand chemistry. On the basis of your Lewis structures
2–
+
from problem 102, which species (SO2–
2 or SO3 ) should react most readily with H ?
1.107. Basicity
Which material would you expect to be the strongest base, H2 O (going to what?), OH–
(going to what?), or O–2 (and you know the extra question)?
1.108. Acidity
Which material would you expect to be the strongest acid, H3 O+ (going to what?), H2 O
(going to what?), or OH– ?
1.109. Acidity and Basicity
How can H2 O be an acid and a base?
Chm 118
Exercise Set 1
1.124
15
1.110. Rules for Acidity
From the last several problems, formulate a rule for the strength of an acid given the charge
on the compound.
1.111. Acidity
Is NH+
4 or NH3 the strongest acid?
1.112. Acidity
Is H3 PO4 or H2 PO–4 the strongest acid?
1.113. Acidity
Is HCl or H2 O the strongest acid?
1.114. Lewis Structure and Basicity
Which species in problem 102 is the stronger base? HINT: For any acid or base strength
problem, write the chemical expression for the acid losing a proton to become a base, or the
base gaining a proton to . . . .
1.115. Reactions
Using Table 8, compute the approximate equilibrium constant for the reaction of CH–3 with
H2 O. In organic chemistry you will learn that you can prepare a reagent containing “CH–3 ”.
What would happen if you allowed water to contact that reagent?
1.116. Reactions
The pKa of acetylene, C2 H2 , is 25. Is the base NH–2 capable of removing a proton from
acetylene? If you mixed equal number of moles of acetylene and NH–2 together, what species
would dominate in the mixture? If you then added a mole of water, what would you get?
1.117. Acidity and Basicity
Which material is the stronger acid, HSO–2 or HSO–3 ? Why?
1.118. Acidity and Basicity
Which material is the stronger acid, H2 SO2 , H2 SO3 , or H2 SO4 ? Why?
1.119. Lewis Structure, Resonance and Basicity
Which would you expect is the strongest base, CH3 O– or CH3 C(O)O– ? Why?
1.120. Equilibrium Constant
What is an equilibrium constant?
1.121. Equilibrium Constant
Write the equilibrium constant for the reaction of Cu2+ with NH3 to form Cu(NH3 )2+ , all
species in water.
1.122. Equilibrium Constant
Write the equilibrium constant for the reaction of Cu2+ with NH3 to form Cu(NH3 )2+
2 , all
species in water.
Chm 118
Exercise Set 1
1.132
16
Acid
H3 PO4
HClO4
HClO2
H3 BO3
HIO3
pKa
2.23
-7.0
2.95
9.14
0.77
Acid
H3 AsO4
HClO3
HOCl
HNO2
H2 SO4
pKa
2.22
-1.30
7.53
3.37
-2.0
Table 7: Some Acid Dissociation Constants for Oxyacids.
1.123. Equilibrium Constant
2+
and
How is the equilibrium constant for the reaction of Cu(NH3 )2+
2 to form Cu(NH3 )
Cu2+ (all in water) related to the equilibrium constants of the last two problems?
1.124. Equilibrium Constant
Write the equilibrium constant for the reaction of Cu2+ (aq) with NH3 to form Cu(NH3 )2+
4 .
1.125. Equilibrium Constant Manipulation
2+
The equilibrium constant for the dissociation of Cu(NH3 )2+
4 into Cu (aq) and NH3 has a
2+
2+
−15
value of 5 x 10 . Find the ratio of Cu (aq) to Cu(NH3 )4 if the concentration of NH3
is 1.0 M. What if the concentration of NH3 is 0.10 M?
1.126. Acid Dissociation Constant
What is the special name for the equilibrium constant for the reaction
HX(aq) = H+ (aq) + X− (aq)?
1.127. The “p” in pK
What is a pKa ?
1.128. Lewis Structure and Chemical Properties
Table 7 shows the pKa for a number of “oxy acids”. Can you find a pattern that allows a
rough prediction of the acidity?
1.129. Using Concepts to Identify the Unusual
Given the information in Table 7, if you saw data that said that arsenous acid, H3 AsO3
has a pKa of 9.23 whereas phosphorous acid, H3 PO3 has a pKa of 2.00, what might you
conclude?
1.130. Using Concepts to Identify the Unusual
The pK of the two acids in the last problem really are what is listed there. They are true,
not intentionally wrong. Which of these materials is “normal”?
1.131. Using Models to Understand Chemistry
Phosphorous acid (see problem 129) forms only monoanions and dianions when reacted
with bases whereas arsenous acid form both of these and trianions. Can you account for
this and the acid strength of these species (for data see problem 129)? HINTS: You need a
different topology for one of them. To review, which one is odd?
Chm 118
Exercise Set 1
1.140
17
Group IV
CH4
55
SiH4
35
GeH4
25
Group V
NH3
35
PH3
27
AsH3
23
Group VI
H2 O
15.7
H2 S
7.1
H2 Se
3.8
H2 Te
2.6
Group VII
HF
3.2
HCl
-7
HBr
-8
HI
-9
Table 8: pKa Values for Binary Hydrogen-Element Compounds.
1.132. Using Models to Understand Chemistry
Predict the pKa of H4 SiO4 .
1.133. Acidity Trends
Data for binary acids are given in Table 8. What trends do you see in the data?
1.134. Determining the Cause of Acidity Trends
Can you suggest a reason why HF is a stronger acid than NH3 is? HINT: The problem is
considerably more complicated than this simple question is attempting to probe. See, for
instance, Schmid, R.; Miah, A. M., J. Chem. Ed., 2001, 78, 116-117. We will flirt with
these concepts below.
1.135. Using Models to Understand Chemistry
–
The compound GeH4 reacts with NH3 in liquid ammonia to form NH+
4 and GeH3 ; methane
does not undergo a similar reaction. Which is the stronger acid, germane or methane?
1.136. Using Models to Understand Chemistry
Could you have predicted any part of the results of the last problem? Explain.
1.137. Modeling a Solid Prepared from Group I and Group VII Elements
How would you describe the dominant bonding in a material such as LiF or NaCl? Why
do you choose that model?
1.138. Energy Analysis of a Chemical Process
Find a path from reactants to products for the reaction below that involves gas phase ions.
LiCl(s) = Li+ (aq) + Cl− (aq)
1.139. Energy Analysis of a Chemical Process
Find a path from reactants to products for the reaction below that involves gas phase ions.
2Ag(s) + Cl2 = 2AgCl(s)
Chm 118
Exercise Set 1
1.147
18
Ion
Li+
Na+
K+
Rb+
Cs+
Fe2+
Ni2+
∆H
-123
-97
-77
-71
-63
-459
-503
Ion
Be2+
Mg2+
Ca2+
Mn2+
Zn2+
Co2+
Cu2+
∆H
-594
-459
-381
-441
-489
-491
-502
Table 9: Enthalpies of Hydration (kcal/mole) for Various Cations.
1.140. Examination of Hydration Free Energy
The changes in free energy, the energy that can be converted into work other than that of
the pressure-volume kind, for the process
H+ (g) + X− (g) = H+ (aq) + X− (aq)
where X is a halogen, are -1537, -1409, -1382, and -1347 kJ/mole for X = F, Cl, Br, and I,
respectively. Plot these data versus the inverse of the ionic radius of the halide ion. Can
you guess why the plot is a straight line?
1.141. Hydration Free Energy
Think about the extrapolation to a zero value of 1/r for the plot in the last problem. What
physical property is that?. Give a meaning to that intercept. HINT: Think about what you
are plotting.
1.142. Hydration Enthalpies
In Table 9 are enthalpy values for the process
Mn+ (g) = Mn+ (aq)
What is going on molecularly to make these values negative? What trends in the data do
you see. Can you figure out a way to visualize the data? Can you actually make a real
plot?
1.143. Metal Ion Complexes as Acids
Metal ions in aqueous solution typically exist as aquated species, M(H2 O)n+
6 . From where
would a proton be taken if the aquated metal ion were to act as an acid?
1.144. Acid Strength of Metal Ion Complexes
If the metal ion, M(H2 O)n+
6 , donates a proton to some base, what are the ligands that
remain on the resulting metal ion “complex”?
1.145. Metal Ion Complexes as Acids
Write a chemical reaction that illustrates how a metal ion in aqueous solution, M(H2 O)n+
6 ,
acts as an acid.
1.146. Equilibrium Constant for Metal Ion Acid
Write the expression for the equilibrium constant for the process in the last problem
Chm 118
Exercise Set 1
1.158
19
1.147. Predicting Chemistry
+
Which would you expect to be the stronger acid, Cu(H2 O)+
6 or Ag(H2 O)6 ? Why?
1.148. Predicting Chemistry
3+
Which would you expect to be the stronger acid, K(H2 O)+
6 or Fe(H2 O)6 ? Why?
1.149. Metal Ion Complexes as Acids
How do you account for the fact that the dominant form of V(II) in dilute acidic aqueous
2+
solution is V(H2 O)2+
6 whereas that of V(IV) is VO(H2 O)5 ?
1.150. Predicting Chemistry
Predict the dominant form of V(II) in a more basic solution than that in the last problem.
1.151. The Acidity of HX Compounds
Show that the sum of the energy for the following set of reactions is equal to the energy of
the acidity process for HX.
HX(aq) = HX(g)
HX(g) = H(g) + X(g)
H(g) = H+ (g) + e− (g)
X(g) + e− (g) = X− (g)
H+ (g) = H+ (aq)
X− (g) = X− (aq)
1.152. The Acidity of HX Compounds
Refer to the last problem and show how the bond energy of HX might influence the acidity:
Would a strong bond make for a strong acid or a weak one?
1.153. The Acidity of HX Compounds
Referring to problem 151, show how solvation energies are important in determining the
acidity of HX compounds. If a given anion has a large (negative) solvation energy, what is
the acidity of that compound compared to one with a smaller (negative) solvation energy?
1.154. Chemical Species and Distribution Curves
Given the data in Figure 11, what is the fraction of the metal ion in the form of M(H2 O)2+
6
at a pH of 5 if the pKa = 5? What is the fraction if the pH = 4? Use the written chemical
equation for this process to see why your answers are what they are.
1.155. Chemical Species and Distribution Curves
In Figure 12 are distribution curves for several different values of the pKa of the metal.
Describe what this information conveys to you.
1.156. Oxidation Numbers
–
Give the oxidation states for the elements in the following compounds: ClO– , ClO+
2 , NO3 ,
–
NO2 , O2 , H2 O, MnO4 , HClO4 , SO2 , LiCl.
1.157. Oxidation Numbers
What is the oxidation number (oxidation state) of Cl in each of the following: HClO4 ; ClO–3 ;
ClF+
2 ; HOCl; AsCl3 .
Chm 118
Exercise Set 1
1.158
20
Figure 11: Distribution curve for a metal ion with a pKa of 5. The left axis is the fraction of
2+
material in the form of M(H2 O)2+
6 . At pH values to the left of the line, M(H2 O)6 becomes
the dominant species.
Figure 12: Distribution curves for a metal ion with a pKa of, from left to right, 2 (blue), 3
(magenta), 4 (tan), and 6 (green).
Chm 118
Exercise Set 1
1.170
21
1.158. Oxidation Numbers of Common Substances
There are exceptions (as usual), but see if you can convince yourself that the oxidation
numbers exhibited by common chemicals containing Cl are -I, 0, I, III, V, and VII. HINT:
Looking at a list of “compounds of chlorine” isn’t a bad place to start.
1.159. Oxidation Numbers of Common Substances
Try the same as the last problem for P. What oxidation states do you get?
1.160. Oxidation Numbers of Common Substances
Considering the last two problems, see if you can find a common pattern for “allowed”
oxidation numbers for the main group elements. HINT: You might check your hypothesis
on some other element.
1.161. Oxidation Numbers
What might oxidation number roughly measure?
1.162. Electronegativity
What is the definition of electronegativity?
1.163. Electronegativity
Which is more electronegative, F or F− ?
1.164. Electronegativity
Which is more electronegative, Cl(I), Cl(III), or Cl(V)?
1.165. Electronegativity
Which is more electronegative, F or S2+ ? HINT: I care less about your ability to know the
answer than I do about your ability to think straight about the issue, although I am pretty
sure if you look at the following ionization energies, you can come to a rigorous conclusion:
F 17.4 eV, S 10.36 eV, S+ 23.4 eV, S2+ 35 eV.
1.166. Electronegativity
Which is more electronegative, the F atom in CF4 or the F atom in GeF4 ? HINT: You
need to be very logical to do this problem unless you just take a 50/50 chance and guess,
which is educationally foolish.
1.167. Symmetry Operations
Find the proper symmetry operations in benzene, a planar C6 H6 molecule. HINT: Don’t
forget resonance in this case, which is very important for assignment of symmetry. Why?
1.168. Symmetry Operations
Find the improper symmetry operations in benzene, a planar C6 H6 molecule.
1.169. Symmetry Operations and Functions
Find the symmetry operations in a water molecule. Label the two (identical) hydrogen
atoms “k” and “l”. Now for each symmetry operation for H2 O calculate a number (it
will be, in this case, either a zero or a two) that is equal to how many of the labels are
“unmoved” by the symmetry operation. HINT: In this problem and those that follow, it
is critically important for you to recognize how many objects you have. Here we have two
objects that we are observing, the two hydrogen atoms.
Chm 118
Exercise Set 1
1.174
22
C2v
A1
A2
B1
B2
E
1
1
1
1
C2z
1
1
-1
-1
σxz
1
-1
1
-1
σyz
1
-1
-1
1
Table 10: Table of characteristic numbers for C2v symmetry.
1.170. Symmetry Operations and Functions
Again, for a water molecule: Draw a line between the two hydrogen atoms. Put a vector
from the midpoint of this line to hydrogen atom “k”. For each symmetry operation for H2 O
determine a number that logically tells you what happens to the vector. HINTS: Remember
that vectors have direction, that you can multiply a vector by 1 and get it back again; or
by -1 and reverse its sign, its direction. Also, pay attention to the number of objects issue;
in this case we have only one object, the vector.
1.171. Symmetry Operations and Functions
Still for a water molecule: Imagine one hydrogen atom moving toward the oxygen atom
along its O-H bond. Put in a little vector to denote the velocity of this motion. Imagine
the other hydrogen atom also moving toward the oxygen atom along its O-H bond with the
same velocity. Put in a vector for that. Since the velocities are the same, the vectors will be
equal in length. Now determine what happens to the pair of vectors (taken together: there
is the statement that we have only ONE object) under each of the symmetry operations
of the water molecule. HINT: Again, consider the pair of vectors as one object: Either
the pair goes to themselves, or the pair disappears, or the pair goes to minus themselves.
Incidentally, I think the middle option is impossible for a symmetry operation!
1.172. Symmetry Operations and Functions
Ugh, more water molecule: Imagine one hydrogen atom moving toward the oxygen atom
along the O-H bond. Put in a little vector to denote this velocity of this motion. Imagine
the other hydrogen atom moving away from the oxygen atom along its O-H bond with
the same speed, and, therefore, reversed velocity. Put in a vector for that velocity. Now
determine what happens to the pair of vectors (taken together! as one object) under each of
the symmetry operations of the water molecule. HINT: See the hint for the last problem.
1.173. Functions and Characteristic Numbers
A molecule’s shape determines the symmetry operations it has. Objects in the molecule,
the labels on the hydrogen atoms, the vectors, of the last few problems, determine the
characteristic numbers we are going to talk about. A table of the characteristic numbers
for C2v symmetry, that is, the symmetry of a water molecule, is in Table 10. The names
of the sets of characteristic numbers, A1 , . . . B2 , are the normal ones used; however, the
important point is that they are just names and for our purposes in this course it does not
matter if you use lower case or capital letters. Use a1 as a name, nothing more (although
that particular name is always given to the set of numbers that are all “ones”).
What is the label of the row that gives the characteristic numbers you got in problem 170?
Express your answer like a professional, “b2 symmetry”, for instance.
Chm 118
Exercise Set 1
1.181
23
C3v
A1
A2
E
E
1
1
2
2C3
1
1
-1
3σv
1
-1
0
Table 11: Table of characteristic numbers for C3v symmetry.
1.174. The Symmetrical Stretch in Water
What is the label of the row that gives the characteristic numbers you got in problem 171?
Since the motion described by those vectors had both hydrogen atoms vibrating against
the oxygen atom in a symmetric way, it is called a “symmetric stretch”. Professionally
speaking, “the symmetrical stretch in water is of [blank] symmetry”. Fill in the blank with
the name of one of the characteristic sets of numbers.
1.175. The Asymmetrical Stretch in Water
In problem 172, the two hydrogen atoms vibrate out of phase against the oxygen atom.
This is an antisymmetrical stretch. Can you speak professionally?
1.176. Functions and Characteristic Numbers
Put a z axis on the water molecule (conventionally along the C2 axis) and insert a vector
pointing in the positive z direction. What numbers in the C2v point group table characterize
that vector? HINT: Need I say again, “Express your answer like a professional”.
1.177. Functions and Characteristic Numbers
Put x and y vectors on the water molecule. It will be convenient if you choose those axes so
that the x vector is never turned into anything but ± itself and similarly for the y. What
numbers in the C2v point group table characterize each (that is, take x separately as one
object and then y separately as one object, and get two answers) of those vectors?
1.178. Symmetry Operations
Consider the NH3 molecule. Find the symmetry operations. Do you agree with those given
in the C3v table, Table 11? HINT: In that table the nomenclature 2C3 implies that there
are two different rotations of 120o , both of which have the same “number” characterizing
them within any given row.
1.179. Functions and Characteristic Numbers
There is an N-H stretch in NH3 that is of a1 symmetry. Use vectors on each of the three
hydrogen atoms to sketch what this motion looks like. Establish that your (one) “picture”
is of a1 symmetry. HINT: A “stretch” changes the length of the N-H bond, but not any
angles.
1.180. Number of Pictures
I contend that the number under the “E” symmetry operation is the number of “pictures”
that you are dealing with. See if you can offer a rationalization for this contention. HINT:
The identity operations does nothing to the ammonia molecule. How many objects must
you have to get two objects going into themselves when you do nothing to the molecule?
Seems like I’ve answered the question.
Chm 118
Exercise Set 1
1.187
24
D3h
0
A1
0
A2
0
E
00
A1
00
A2
00
E
E
1
1
2
1
1
2
2C3z
1
1
-1
1
1
-1
3C2
1
-1
0
1
-1
0
σh
1
1
2
-1
-1
-2
2S3
1
1
-1
-1
-1
1
3σv
1
-1
0
-1
1
0
(x, y)
z
Table 12: Characteristic numbers for D3h symmetry.
1.181. Structure and Symmetry
Sketch the structure of CH2 Cl2 and find the name of the set of symmetry operations that
this molecule has.
1.182. Finding a Stretch from its Name
Find the C-H stretch in CH2 Cl2 , last problem, that has b1 symmetry. HINTS: Clearly the
answer must be one of the sets of characteristic numbers that you have in front of you. Let
the plane defined by the carbon and the two hydrogens be the xz plane.
1.183. Finding a Bend from its Name
Find the bend of the two C-H bonds in the plane defined by HCH in CH2 Cl2 that has a1
symmetry. HINTS: Clearly the answer must be one of the sets of characteristic numbers
that you have in front of you. A bend changes an angle but not a bond length. Hence the
vectors must be perpendicular to bonds and, in this case, centered on the hydrogen atoms.
1.184. Finding a Bend from its Name
Find the bend (or some call it a wag) of the two C-H bonds in the plane defined by HCH
in CH2 Cl2 that has b1 symmetry. HINTS: Clearly the answer must be one of the sets of
characteristic numbers that you have in front of you. A bend changes an angle but not a
bond length. Hence the vectors must be perpendicular to bonds and, in this case, centered
on the hydrogen atoms. Also, as a “bend” it must change an angle.
1.185. Symmetry Operations
Consider the SO3 molecule. Find the symmetry operations. Do you agree with those given
in Table 12 for D3h symmetry? HINTS: To find symmetry operations you must know the
structure of the molecule in space. In the D3h table the nomenclature 2C3z implies that there
are two different rotations of 120o , both of which have the same “number” characterizing
them within any given row.
1.186. Symmetry of a Motion in SO3
In Figure 13 is a motion of the SO3 molecule. What is the symmetry of this (single) motion?
How would you describe what is happening to the SO3 molecule?
Chm 118
Exercise Set 1
1.191
25
Figure 13: A motion of SO3 . All arrows representing motion of oxygen atoms are equal in
length
1.187. Functions and Characteristic Numbers
Label the three (identical) oxygens of SO3 a, b, and c. Now for each symmetry operation
for SO3 determine how many of these oxygen atoms retain their label after the symmetry
operation has been performed. HINT: For instance, for a C2 (180o ) rotation about the SO(a) bond, O atoms b and c are not where they were, but O(a) is. So the number associated
with that rotation is “1”.
1.188. Functions and Characteristic Numbers
I contend that your answer for the last problem is made up of the sum of two of the rows of
the D3h table. That is, under each symmetry operation, if you add the numbers associated
with that operation for row X and row Y of the table, you will get your answer from the
last problem. See if you can figure out which those rows are. Speaking professionally, you
0
00
would say “The characteristic numbers generated by the three oxygen atoms are a2 and e
where you should substitute the correct answers.
And now on to quantum mechanics.
1.189. Stern-Gerlach Experiment
A silver atom is directed to a Stern-Gerlach apparatus arranged in the up/down direction.
What is the probability that the Ag atom will come out of the upper tube?
1.190. Stern-Gerlach Experiment
A silver atom that came out of the upper tube of the apparatus in the last problem is
directed to another Stern-Gerlach apparatus arranged in the up/down direction. What is
the probability that the Ag atom will come out of the upper tube?
Chm 118
Exercise Set 1
1.196
26
Figure 14: Two cosine waves
1.191. Stern-Gerlach Experiment
A silver atom that came out of the upper tube of the apparatus in problem 189 is directed
to a Stern-Gerlach apparatus arranged in the in/out direction. What is the probability that
the Ag atom will appear from the “out” tube?
1.192. Stern-Gerlach Experiment
Quantum mechanics is weird, but not illogical. Try this: A silver atom from the upper tube
of the Stern-Gerlach apparatus in problem 189 is directed into a Stern-Gerlach apparatus
that is tilted 0.001 degrees relative to the apparatus in problem 189; that is, it is almost an
up/down apparatus. What is the probability a Ag atom will come out of the upper tube
of the second apparatus? HINT: I am not looking for a numerical answer, but a statement
showing understanding.
1.193. Stern-Gerlach Experiment
Quantum mechanics is weird, but not illogical. Try this: A silver atom from the upper tube
of the Stern-Gerlach apparatus in problem 189 is directed into a Stern-Gerlach apparatus
that is tilted 45 degrees (toward “in-out”) relative to the apparatus in problem 189; that
is, it is between a situation where the probability of the silver atom coming out the upper
tube is 1.0 and the situation where the apparatus is in a in/out position and the probability
is 0.5. What would you guess that the probability a Ag atom will come out of the “upper”
tube (or “out” tube, depending on your perspective) of the second apparatus? HINT: See
the last hint.
1.194. Waves and Interference
Figure 14 is a plot of the wave amplitude versus x for two cosine waves; the solid line is
that for Cos[x] and the dashed is for Cos[2x]. Is there interference at x=3.14 (i.e., π)? If
so, is it dominantly constructive or dominantly destructive? What are the values of x for
which interference is constructive?
1.195. Waves and Interference
I claim that Figure 15 is a plot of Cos[x] + Cos[2x] versus x. Where is interference constructive?
Chm 118
Exercise Set 1
1.201
27
Figure 15: The wave Cos[x] + Cos[2x]
1.196. Waves and Particles
How do waves that you have seen in life (those are for most people waves in water) differ
from a particle?
1.197. Operators
The rules of quantum mechanics (which are right because they give answers verified as
correct by experiment) involve “operating” on a function (often called a wave function) and
asking what happens to the function as a result of that “operation”. We are often looking
for an operation on a function that gives as a result the same function back multiplied by a
number; this is called the eigenfunction/eigenvalue property. That is, O y = 3 y (where O
is the operator, y is the function, and “3” is the eigenvalue) is an eigenfunction/eigenvalue
problem. Plot the function y = 2 x + 0.1 over the range of x from 0 to 1 to convince yourself
it is a function, a recipe for finding a value at a point in space. Now operate on that function
with O where this operator multiplies the function by 2. Plot the new function? Does it
obey the eigenfunction/eigenvalue requirement? That is, is 2 y = 2(2 x + 0.1) equal to c
y where c is a number? HINT: Duh!
1.198. Operators
Use the function y = 2x and let the operator take all values of x and divide them in half.
Does the new function meet the eigenfunction/eigenvalue requirement? If so, what is the
eigenvalue?
1.199. Operators
In quantum mechanics, some of the operators are differential operators, such as ∂/∂x. If
you operate with this operator on the function y = x2 , is it an eigenfunction/eigenvalue
problem?
1.200. Operators
Use the function y = e2x and let the operator be ∂/∂x. Does the function and this operator
meet the eigenfunction/eigenvalue requirement? If so, what is the eigenvalue?
Chm 118
Exercise Set 1
1.210
28
1.201. Operators and Quantum Mechanics
Take the operator
2
2
~ ∂
- 2m
∂x2
where ~ is Planck’s constant divided by 2π and m is the mass of the object, the parve, and
operate on the function ψ = A e−ax , where A and a are constants. Does this meet the
eigenfunction/eigenvalue requirement? If so, what is the eigenvalue?
1.202. Parve on a Pole
A particularly easy problem to solve is called “the parve on a pole.” This is a quantum
particle moving in one dimension between two boundaries with no potential energy between
those boundaries. The parve is stopped from being “outside” those boundaries. In your own
words, describe the parve on a pole situation. State where the parve is and what potential
energy it is subjected to at various positions in space.
1.203. The Kinetic Energy Operator in Quantum Mechanics
Take the operator from the problem 201 and operate on the function ψ = A Sin[ax], where
A and a are constants. Does this meet the eigenfunction/eigenvalue requirement? If so,
what is the eigenvalue? And if “if so”, congratulations! You have solved Schrödinger’s
equation for a parve on a pole.
1.204. Boundary Conditions
For a parve on a pole, the parve is forbidden from being outside the pole. Therefore the
value of the function, ψ must be zero at x = 0 and at x = L, where L is the length of the
pole. What condition does this put on the constant “a” in problem 203? HINT: We require
that at x = 0, the function ψ be zero. Is that satisfied for ψ = A Sin[a 0]? Now try the
harder part: it must also be true at x = L that ψ is equal to zero at x = L. Does this
restrict a to any particular value(s)?
1.205. Boundary Conditions and Quantum Mechanics
In addition to obeying Schrödinger’s equation, a legal wave function in quantum mechanics
must obey boundary conditions. Boundary conditions introduce what?
1.206. Parve on a Pole
The lowest energy wave function for a parve on a pole of length L is a sine wave running
between x = 0 and x = L and spanning half a wavelength in this interval. Use the de
Broglie relationship, p = λh , where p is the momentum, p = m v, and solve the equations
for the energy of this parve. HINTS: All educated people (that means you!) should know
deBroglie’s equation. Also, all the energy of this parve is kinetic energy and all educated
people should know the expression for kinetic energy in terms of velocity.
1.207. Parve on a Pole
A parve is trapped on a pole with zero potential energy. What happens to the lowest
allowed energy of the parve if the length of the pole is doubled?
1.208. Parve on a Pole
Sketch the lowest (n = 1) wave function for a parve on a pole.
Chm 118
Exercise Set 1
1.217
29
Figure 16: Figure to Schematically Illustrate Problem 216
1.209. Parve on a Pole
Sketch the n = 5 wave function for a parve on a pole.
1.210. Kinetic Energy and Momentum
For a parve on a pole, the only energy is kinetic energy, 12 mv2 (since the potential energy
p2
where p is the
has been set to zero). Show that kinetic energy can also be written as 2m
momentum, mv.
1.211. Wave Length and Kinetic Energy
What does the de Broglie postulate say about the relationship of wave length to momentum?
What does it then say about the relationship of wave length to kinetic energy?
1.212. Parve on a Pole
Given the de Broglie postulate in problem 206, what can you say about the kinetic energy
of a parve on a pole in the n = 5 level compared to the n = 1 level? Explain without a
calculation by thinking about the de Broglie postulate.
1.213. Parve on a Pole
Compute the energy an electron would have if it was trapped on a pole of length 4.2 Å
(about the length of a four carbon fragment) in the n = 1 level using the POP model.
HINTS: You need to look up some constants and expand units like the Joule in terms of
kg, m, and sec.
1.214. Parve on a Pole
What energy would it require to excite the electron from the n = 1 to the n = 2 level in
the system of the last problem?
1.215. Energy Change and Wavelength
What wavelength of light must be absorbed to provide the energy to excite the electron
from the n = 1 level to the n = 2 level in the system of the last two problems? HINT:
You should know (or learn and then know) the relevant equation, one of Einstein’s several
equations.
1.216. Parve on a Pole Model for Real Systems
This follows from the last problem, but is somewhat more rigorous. We model a conjugated
double bond system (alternating single and double bonds in a carbon framework) by a pole
of length 1.4Å times the number of C-C bonds plus one; see Figure 16 for an example.
(The “plus one” takes into account that the parve can move a little bit outside the ends
of the chain). We fill POP energy levels with a pair of electrons for each double bond.
Show that as the chain lengthens, the energy of transition from the highest filled molecular
orbital (the HOMO) to the lowest unoccupied molecular orbital (LUMO) decreases. HINT:
You can show this in general, which requires more abstract algebra, or by brute force and
analogy, which requires only a calculator.
Chm 118
Exercise Set 1
1.224
30
1.217. Parve on a Pole Model to Lea rn about a Real System
This (and the problems that follow) is a simple exercise to establish why the electron in an
atom does not just collapse into the nucleus under the pull of the positive charge. We use in
the development two different models, but the gist of the argument is valid. First we worry
about the kinetic energy of the electron. Assume that a hydrogen atom (one electron and
one proton) has a radius of about 1Å. Assume we can model this as a parve on a pole of a
length equal to the diameter of the hydrogen atom. What energy will this parve have in its
ground state assuming that the nucleus does nothing, isn’t there? Compute your answer in
kJ/mole.
1.218. Parve on a Pole Model to Learn about a Real System
Now we compute the potential energy of attraction of the electron for the nucleus. We can
estimate this from Coulomb’s law
V =
1 Ze Znucl
4πo
r
where the first fraction is just a unit conversion factor from charge, coulombs, to joules, and
has a value of 8.987×109 J m C−2 , and the Z’s are the charge on the electron and nucleus.
The latter have the same magnitude, but Ze is negative: -1.602×10−19 C. Evaluate the
potential energy (in kJ/mole) for an r of 1Å.
1.219. Parve on a Pole Model to Learn about a Real System
Find the total energy for the system using the kinetic energy from problem 217 and the
potential energy from the last problem. Is our “hydrogen atom” stable? That is, does it
have negative energy?
1.220. Parve on a Pole Model for Real Systems
What if we crush the atom so that the length of the pole (the diameter of the atom) is
equal to 0.5Å? What will the kinetic energy be now? Is it more postive now that we have
trapped the electron on a smaller pole? What happens to the potential energy? Is it more
negative? Why?
1.221. Parve on a Pole Model to Learn about a Real System
Repeat the last problem with a “hydrogen atom” at a radius of 0.25Å. Positive or negative?
Stable or not?
1.222. Parve on a Pole Model to Learn about a Real System
Clearly the energy is a function of distance and at some value will presumably be a minimum,
the “stable distance” for an electron in our hydrogen atom model. Articulate why you
cannot “crush” a hydrogen atom.
1.223. Parve on a Table
Given that you know what the wave function for the ground state of a POP looks like,
make a guess about what the wave function for the ground state of a parve on the surface
of a square table looks like. If you are a reasonable artist, you might be able to draw the
shape of the wave function in the third dimension given the surface is defined by x and y;
you certainly should be able to move your hands over a table to outline roughly the shape
of the wave function for a parve on a surface.
Chm 118
Exercise Set 1
1.235
31
1.224. Parve on a Table
Given the kinetic energy of a parve moving in the x direction doesn’t influence the kinetic
energy of a parve moving in the y direction, make a guess about the equation for the energy
of a parve on a square table. HINT: Two dimensions, two quantum numbers.
1.225. Parve in a Room
Make a guess for the wave function for a parve in a room, a three dimensional problem.
You will not be able to draw it (since that requires four dimensions), but should be able to
describe it. How many quantum numbers will be necessary for this wave function?
1.226. Electron Configuration in Atoms
Write the electron configuration for Li, K, B, C, N, O, S, Ni, F, Cl, Br. Use a periodic table
and appropriate abbreviations for inner shell electrons. This is not a busy work problem,
although it is review of what you learned elsewhere. You should learn to assign electronic
configurations quickly and accurately.
1.227. Electron Configuration in Ions
Write the electron configuration for Li+ , B2+ , C+ , N− , O+ , S2− , Ni2+ , Cl3+ . You should
learn to assign electronic configurations quickly and accurately.
1.228. Isoelectronic Species
What atom is N isoelectronic with? What anion? What cation? What other anion? What
other cation?
1.229. Isoelectronic Species
What is the relationship between F− and O2− ? Between F− and Ne?
1.230. Isoelectronic Species
What is the relationship between CH3 and F?
1.231. Use of Isoelectronic Relationships
Given your answer to the last problem, if CH3 Cl existed, should you seriously consider the
possibility of the existence of FCl? Why or why not?
1.232. Electron Configuration in Ions
Here we use “oxidation state nomenclature” for materials with deficient (or excess) elections.
Write the electron configuration for Be(II), C(II), Cr(II), Cr(III). Cr(IV), Cr(V), Cl(V),
Br(-I). You should learn to assign electronic configurations quickly and accurately. HINT:
Transition metal ions are “d” electron systems, not 4s1 or 4s2 systems.
1.233. Ionization Energy
What is an ionization energy? What would be a “second ionization energy”?
1.234. Relative Ionization Energies
Look up some first ionization energies for elements in various parts of the periodic table;
don’t pick atoms that are “neighbors”. Do you see any “gross” relationship between first
ionization energy and periodic position?
Chm 118
Exercise Set 1
1.244
32
1.235. A Simple Model of Relative Ionization Energies
Imagine we have two electrons a given distance from a positive charge of magnitude +2.
We reach in with tweezers and pull one of the electrons out. As we do so, it looks back
(ahh, those pesky electrons, looking at their own world) and “sees” a net charge of plus
one. Why? The energy it costs us to remove this electron is reflected by what charge it
“sees” as we pull it out. Now we remove the second. What charge would it see? What
would be a rough guess of the relative energy cost to remove the second electron compared
to the first?
1.236. A Simple Model of Relative Ionization Energies
What very simple model would account for these facts? The first two ionization energies of
Be are 9.3 and 18.2 eV.
1.237. Relative Ionization Energies
Imagine we have three electrons a given distance from a positive charge of magnitude +3.
We reach in with tweezers and pull one of the electrons out. As we do so, it looks back (as in
problem 235) and “sees” a net charge of one. The energy it costs us to remove this electron
is reflected by what charge it “sees” as we pull it out. Now we remove the second. What
charge would it see as it looks back? What is the relative energy for removal of the second
compared to the first? Now we pull out the third. What would be the relative energy for
removal of the third compared to the second?
1.238. Ionization Energy Trends
What very simple model would account for these facts? The first three ionization energies
of P are 10.5, 19.7, and 30.2 eV.
1.239. Ionization Energy Trends
What very simple model would account for these facts? The second, third, and fourth
ionization energies of S are 23.4, 35.0, and 47.3 eV. Isn’t our model amazing?
1.240. Building a Model to Relate Ionization Energy Trends
With the past several problems in hand, how do you have to modify the simple model to
account for these facts: The first two ionization energies of Be are 9.3 and 18.2 eV whereas
the third is 153.8 eV?
1.241. Ionization Energy Trends
And here is another problem, at least if you are critical of the model: Closely examine the
first five ionization energies of C (the values are 11.2, 24.4, 47.9, 64.5, 392.0 eV) and see
how you fit those data into the arguments we have been developing. Does our simple model
work at all? Does it fail at some value or another? What fix, patch, brace can you apply?
HINT: The data scream “shells”, even “sub-shells.”
1.242. Ionization Energy Trends
What very simple model would account for these facts: The first three ionization energies
of Sb are 8.64, 16.5, and 36.3 eV.
1.243. Quantum Numbers in the Hydrogen Atom
How many quantum numbers are there in the hydrogen atom? Why?
Chm 118
Exercise Set 1
1.256
33
1.244. Quantum Numbers in the Hydrogen Atom
Give an example of the three quantum numbers for a hydrogen atom.
1.245. Quantum Numbers in the Hydrogen Atom
When we describe electrons in atoms we use symbols like 2s, 2pz , 3px , etc. What do these
symbols mean? HINT: This is a test of your previous knowledge more than an exercise in
looking something up; however, as we go through this part of the course you should ask
yourself this question again and again.
1.246. Quantum Numbers of the Hydrogen Atom
In a hydrogen atom how many different spatial orbitals are there when the quantum number
n = 3?
1.247. Energy of the Hydrogen Atom
What is the energy of an electron in a hydrogen atom if it has a principle quantum number
of 3? HINT: For a hydrogen atom the energy of an electron is given by En = -13.59/n2 in
electron volts.
1.248. Light and Quantum Levels
What wavelength of light is necessary to excite a hydrogen atom from the n = 2 to the n =
3 quantum level?
1.249. Light and Quantum Levels
What wavelength of light will be emitted when a hydrogen atom falls from the n = 3 to the
n = 2 quantum level?
1.250. Light and Quantum Levels
What wavelength of light will be emitted when a hydrogen atom falls from the n = 4 to the
n = 2 quantum level?
1.251. Quantum Numbers of the Hydrogen Atom
In a hydrogen atom, how many different orbitals are there when the quantum numbers are
n = 4, ` = 3?
1.252. Quantum Numbers of the Hydrogen Atom
How many different orbitals are there when the quantum numbers are n = 4, ` = 4? HINT:
Trick question.
1.253. Quantum Numbers of the Hydrogen Atom
How many different orbitals are there when the quantum numbers are n = 4, ` = 2?
1.254. Quantum Numbers of the Hydrogen Atom
How many different orbitals are there when the quantum numbers are n = 4, ` = 2, m` = 1?
1.255. Quantum Mechanics of the Atom
How many electrons can have quantum numbers n = 4, ` = 2, m` = 1?
Chm 118
Exercise Set 1
1.268
34
1.256. Spectra in Atoms
In an emission spectrum, hot atoms (or molecules) have an electron fall from an excited
level to a lower lever, liberating the energy as a photon. In atomic H, emission is seen in
the Lyman series at 121.6 nm, 102.6 nm, 97.3 nm, etc. Use your knowledge of the energy
levels of a hydrogen atom as a function of n to deduce what n value the final state has in
this series.
1.257. Spectra in Atoms
Light from the sun occurs over a range of wavelengths, but there are some colors missing.
Among others, wavelengths of 656.3 nm, 486.1 nm, 434.0 nm are missing. Show that these
lines are part of the Balmer series in which a hydrogen atom jumps from the n = 2 level to
a higher n value.
1.258. Spectra in Atoms
Given the sun is primarily hydrogen and is hottest at its core, much cooler at a large radius,
account for how the process in the last problem happens. Give details.
1.259. Photon Energy
Learn the approximate energy of a visible photon in kcal/mole or kJ/mole.
1.260. Bonds and Photon Energy
How does the energy you found in the last problem compare to that of a typical chemical
bond, say that of C-H?
1.261. Bonds and Photon Energy
The bond energy between two oxygen atoms in hydrogen peroxide, H2 O2 , is about 35
kcal/mole. Why is it that peroxide compounds are known to decompose in the presence of
visible light?
1.262. Bond Energy
Look up the energy of a “hydrogen bond”? DNA strands are held together with hydrogen
bonds. Why are such strands so stable when the hydrogen bond is so weak?
1.263. Picturing Hydrogen Atom Wave Functions
Why is it difficult to “draw” hydrogen atom wave functions? How do we get around the
problem?
1.264. Radial Wave Function
Sketch the radial wave function for a 3s electron.
1.265. Radial Probability Distribution
Sketch the radial probability at a distance from the nucleus versus distance for a 3s electron.
1.266. Angular Wave Function
Sketch the angular wave function for a 3s electron.
1.267. Nodes in the Wave Function
How many radial nodes does a 3s wave function have? A 3d? What, if any, is the relationship
between the number of radial nodes and the quantum numbers?
Chm 118
Exercise Set 1
1.280
35
1.268. Nodes in the Wave Function
How many angular nodes does a 3dxy wave function have? A 2pz ? Describe the surfaces
that defines these nodes. Do the same for a 3dz 2 wave function. What is the relationship
between the number of angular nodes and the ` quantum number?
1.269. Nodes in the Wave Function
Is there any relationship between the total number of nodes in a wave function and some
quantum number or combination thereof?
1.270. Radial Wave Function
Give the radial wave function for the electron in a hydrogen atom described by the quantum
numbers (n, `, m` ), (1 0 0). How about (2 0 0)? (3 0 0)?
1.271. Radial Wave Function
Give the radial wave function for a hydrogen atom electron described by the quantum
numbers (3 0 0), (3 1 1), and (3 2 2).
1.272. Radial Probability Distribution
Give the radial probability at a distance from the nucleus versus distance for the electron
in a hydrogen atom described by the quantum numbers (2 1 0). Compare your answer with
(3 1 0).
1.273. Angular Wave Function
What is the difference between a 2px and a 2py wave functioni?
1.274. Angular Wave Function
Show how a 4dxy orbital is shaped.
1.275. Angular Wave Function
Describe the “shape” of a fxyz orbital. HINT: My intent is that you use the “name” to
figure out the shape; looking up the shape on the web is ok, but then you need to memorize
it.
1.276. Angular Wave Function
Sketch the angular wave function for each of the following atomic orbitals: s, px , py , pz ,
dxy , dxz . dyz , dz 2, dx 2−y 2. HINT: Try taking advantage of the name.
1.277. Angular Wave Function
Describe the difference between the dxy orbital and the dx 2−y 2 orbital.
1.278. Measurement in Quantum Mechanics
If you had 100 hydrogen atoms in their ground state and you measured the position of the
electron in the first of them, what might you get for an answer? HINT: I am looking for a
general description rather than a numerical answer.
1.279. Measurement in Quantum Mechanics
If you had 100 hydrogen atoms in their ground state and you measured the position of the
electron in all of them, what would your answers be?
Chm 118
Exercise Set 1
1.293
36
1.280. The Pauli Principle
“Electrons of the same spin avoid each other.” But as the great natural philosopher of rural
Arkansas, Boniface Beebe, has queried: “Don’t all electrons repel each other?” Help him
understand.
1.281. Quantum Numbers in an Atom
How many electrons can have quantum numbers n = 4, ` = 3, m` = 1, ms = 1/2?
1.282. The Pauli Principle
How can you justify four quantum numbers in a hydrogen atom when there are only three
dimensions?
1.283. Penetration
When we say that a 2s electron penetrates better than a 2p, what do we mean?
1.284. Radial Wave Functions and Penetration
Which penetrates the most, 2s or 2p in B?
1.285. Radial Wave Functions and Penetration
In Al, which penetrates the most, 2p relative to 2s penetrating the n=1 shell or 3p relative
to 3s penetrating the n=2 shell? HINT: Drawings would be nice so that you can answer
the question with reason.
1.286. Radial Wave Functions and Penetration
Which penetrates the most, 2p or 3p in H? HINT: Careful!
1.287. Electronic Configuration
What is the electronic configuration of B? of N? of S?
1.288. Electronic Configuration
Why is the electron configuration of Li 1s2 2s1 rather than 1s2 2p1 ?
1.289. Electronic Configuration
I contend the periodic table is an electron configuration map. Comment.
1.290. Electronic Configuration
According to the Pauli principle, which would be most stable? C (1s2 2s2 2p1x (↑)2p1y (↓))
or C(1s2 2s2 2p1x (↑)2p1y (↑))? How would the energies of these two compare to the energy
of C(1s2 2s2 2p2x )? HINT: Strictly speaking, these do not describe legitimate multielectron
wave functions, but they will suffice to illustrate the point at this stage in your education.
1.291. Electronic Configuration
Which is most stable? N(1s2 2s2 2p1x (↑)2p1y (↑)2p1z (↑)) or N(1s2 2s2 2p1x (↑)2p1y (↑)2p1z (↓))? HINT:
See the last problem.
1.292. Electronic Configuration
Which is most stable? O(1s2 2s2 2p2x 2p1y (↑)2p1z (↑)) or O(1s2 2s2 2p1x (↑)2p2y 2p1z (↓))? HINT: Be
careful; see hints of last two problems.
Chm 118
Exercise Set 1
1.305
37
1.293. Degeneracy and Electronic Configuration
How many ways can you compose a system which is equivalent to:
O(1s2 2s2 2p2x 2p1y (↑)2p1z (↑))
and has the same net spin? Speaking professionally, we say the state is “blank” fold spatially
degenerate.
1.294. Ionization Energy
How do the IE of the atoms vary between H and Ar?
1.295. Ionization Energy
What reason(s) could you suggest for the higher ionization energy of Li(1s2 2s) than Na([Ne]3s)?
1.296. Ionization Energy
Explain why the ionization energy (generally) increases across a row in the periodic table.
1.297. Ionization Energy
Explain why the ionization energy (generally) decreases down a column in the periodic
table. HINT: There are two opposing factors.
1.298. Ionization Energy
What are the four factors than influence ionization energy?
1.299. Successive Ionization Energy and Electron Configuration
Successive ionization energies for Si are 8.15, 16.34, 33.46, 45.13, and 166.73, all in electron
volts. Take each value and divide it by the charge on the species after the electron is removed
(for example, divide 8.15 by 1 since the charge on the ion after the first ionization energy
is +1). Explain the resulting data, both similarities and differences. HINT: Don’t expect
exact agreement but be startled at how nicely the numbers arrange during this procedure;
even though I (and you) have seen it before, I always am.
1.300. Ionization Energy
How can you rationalize the relative IE’s of Mg and Al?
1.301. Ionization Energy
Which has the lower second IE, Ca or Mg?
1.302. Ionization Energy
How can you rationalize the relative IE’s of P and S?
1.303. Ionization Energy
The first IE of sodium is less than that of magnesium. The second IE of sodium is greater
than that of magnesium. Comment.
1.304. Ionization Energy
The IE of Rb[Kr 5s] is 4.176 eV whereas that of the excited state of H, H[5s] is 0.544 eV.
Account for the difference. HINT: The second case is the ionization of an excited state of
the hydrogen atom.
Chm 118
Exercise Set 1
1.317
38
1.305. Ionization Energy
The IE of the excited state of Rb[Kr 5f] is 0.547 eV whereas that of the excited state of H,
H[5f] is 0.544 eV. Account for the lack of a substantial difference.
1.306. Ionization Energy
Make an educated guess about the IE of Rb[Kr 5d].
1.307. Relative Energy of Orbitals
What is the energy ordering for the 3s, 3p, and 3d orbitals in K? What causes that order?
1.308. Relative Energy of Orbitals
The energy of separation of the 2s and the 2p orbitals in atoms of the Li-Ne row of the
periodic table are: C : 8.8 eV; N: 12.4 eV; O: 16.5 eV; F: 21.6 eV. You can read in Boniface
Beebe’s treatise “Electrons in Atoms and Consequences Thereof”, Jones Brothers Farm and
Supply Co., 1877, p. 482, that the separation between the 2s and the 2p orbitals in Ne is
22.1 eV. Comment.
1.309. Relative Energy of Orbitals
Make an estimate of the separation energy for Ne; state your assumptions. HINT: See the
last problem.
1.310. Relative Energy of Orbitals
In view of the last problem, what can you say about the s/p separation as you go from C
to Ne?
1.311. Electron Configuration
What is the electron configuration of V?
1.312. Electron Configuration
What is the electron configuration of V2+ ?
1.313. Electron Configuration and Charge
How do you account for your answers to the last two problems?
1.314. Periodicity and Chemical Properties
Using the periodic table as your guide, what would you say about the ionization energy of
materials that are generally called metals?
1.315. Electron Configuration
Someone has said “The chemistry of transition metal ions is 3d chemistry.” Explain.
1.316. VOIE
VOIE is the ionization energy of an atom in which special kinds of electron/electron repulsion (such as we see in N(1s2 2s2 2p1x (↑)2p1y (↑)2p1z (↑)) as opposed to (1s2 2s2 2p2x 2p1y (↑)) have
been averaged out. What trend would you expect for the 2p VOIE of the elements between
B and Ne? REMARK: It is useful to remember that the VOIE of H is about the same as
that of a hybridized carbon atom.
Chm 118
Exercise Set 1
1.321
39
1.317. VOIE and Charge Polarization
Given the remark in the last problem, what would you guess is the direction of polarization
of charge in a C-H bond?
1.318. Chemical Change Producing Light
Barium nitrate is added to fireworks to produce a green color. Excited barium atoms
emit at wavelengths between 487nm and 578 nm. What energy must the chemistry of
the explosion of the fireworks have in order to excite the barium atom (in kJ/mole or
kcal/mole)? Incidentally, from where do you imagine the barium atoms come if we start
with Ba(NO3 )2 ?
1.319. Photoelectron Spectroscopy
In photoelectron spectroscopy an atom is irradiated with light of short wavelength. This
has sufficient energy to remove an electron from either the highest filled shell, or some of
those lower in energy (more stable). The kinetic energy of that electron is measured. If we
had a sample of a rubidium salt in a PES apparatus, would the kinetic energy of an electron
from the 4p or the 4s levels have the highest kinetic energy? Why?
1.320. Photoelectron Spectroscopy
See problem 319 for a description of the PES apparatus. Outline a method using PES in
which you could measure the ionization energy of an atom.
1.321. Periodicity and Ionization Energy
The first IE of Ti and Zr are about the same, 6.8 eV; the fourth IE of Ti is greater than
that of Zr. Comment.
Chm 118
Exercise Set 1