End of chapter questions
1. k b T is often used to measure the strength of a
given interaction, where k b is Boltzmann’s
constant (1.38065 × 10
–23 J K
–1 ). Coulombic
interactions are generally among the strongest
and the farthest reaching of the interactions
that we have discussed. (a) Using the information given in Example 5.1, determine at what
distance Na
+ and Cl
– would have to be separated in order for V(r) Na–Cl to be equal in magnitude to k b T at room temperature (298K) in the
gas phase (ε = 1). (b) Perform the same calculation in water and toluene (use the dielectric
constants in Table 5.1) and discuss how this
could have an impact on the solubility of NaCl
in these two solvents.
2. (a) Imagine that a Ca
2+ ion interacts with an H 2 O
molecule in the absence of any other molecules at a distance of 0.4 nm. Plot the interaction potential energy as a function of θ. Assume
that the only interaction between the two
species is an ion–dipole interaction and that
the distance remains fixed. (b) At what angles
does the magnitude of the potential energy
reach a maximum? Why does this make sense?
What do the positive and negative values of V
represent?
3. Obtain an expression for the force between an
ion and its induced dipole in an ion-induced
dipole interaction. Is the force attractive or
repulsive?
4. (a) Assume that two dipolar molecules (each
with dipole moment µ) are in the same plane.
Show that for all distances r, the interaction
potential energy between the dipoles is smaller
(more negative) if the two dipoles are oriented
in a line rather than antiparallel to each other.
(b) The answer from 4(a) is slightly misleading
because it might lead us to believe that the
dipoles always prefer to orient themselves in a
line rather than anti-parallel. If the dipolar
molecules are anisotropic in shape (oblong in
the direction of the dipole), explain why the
molecules might prefer to orient themselves in
an antiparallel orientation rather than in a line.
5. Explain in your own terms why, in the absence
of other interactions, a dipole-induced dipole
interaction will always be attractive. Use diagrams if appropriate.
6. At what value of r (in terms of σ) does the
Lennard-Jones potential reach its minimum
value? Mathematically, what does σ represent?
7. The Onsager model treats the relationship
between the dipole moment of a polar molecule within a liquid as the interaction of that
dipole embedded in a spherical cavity (spherical molecule) within a dielectric composed of
an infinite number of identical, freely rotating
dipoles, such that the dipole induces a field on
the dielectric and the average field of the
dipoles in the dielectric partially counteracts
the field from the dipole. This assumption of no
interactions between individual molecules
allows the dielectric response of a material to
be estimated from single-molecule properties.
However, consider the molecules listed in Table
5.1, where the Onsager model provides very
good estimates in some cases and fails
for others (particularly water). For each molecule listed in Table 5.1, (a) list what are likely to
be the strongest intermolecular interactions it
CHAPTER 5: Intermolecular Interactions and Self-Assembly
178
1. k b T is often used to measure the strength of a
given interaction, where k b is Boltzmann’s
constant (1.38065 × 10
–23 J K
–1 ). Coulombic
interactions are generally among the strongest
and the farthest reaching of the interactions
that we have discussed. (a) Using the information given in Example 5.1, determine at what
distance Na
+ and Cl
– would have to be separated in order for V(r) Na–Cl to be equal in magnitude to k b T at room temperature (298K) in the
gas phase (ε = 1). (b) Perform the same calculation in water and toluene (use the dielectric
constants in Table 5.1) and discuss how this
could have an impact on the solubility of NaCl
in these two solvents.
2. (a) Imagine that a Ca
2+ ion interacts with an H 2 O
molecule in the absence of any other molecules at a distance of 0.4 nm. Plot the interaction potential energy as a function of θ. Assume
that the only interaction between the two
species is an ion–dipole interaction and that
the distance remains fixed. (b) At what angles
does the magnitude of the potential energy
reach a maximum? Why does this make sense?
What do the positive and negative values of V
represent?
3. Obtain an expression for the force between an
ion and its induced dipole in an ion-induced
dipole interaction. Is the force attractive or
repulsive?
4. (a) Assume that two dipolar molecules (each
with dipole moment µ) are in the same plane.
Show that for all distances r, the interaction
potential energy between the dipoles is smaller
(more negative) if the two dipoles are oriented
in a line rather than antiparallel to each other.
(b) The answer from 4(a) is slightly misleading
because it might lead us to believe that the
dipoles always prefer to orient themselves in a
line rather than anti-parallel. If the dipolar
molecules are anisotropic in shape (oblong in
the direction of the dipole), explain why the
molecules might prefer to orient themselves in
an antiparallel orientation rather than in a line.
5. Explain in your own terms why, in the absence
of other interactions, a dipole-induced dipole
interaction will always be attractive. Use diagrams if appropriate.
6. At what value of r (in terms of σ) does the
Lennard-Jones potential reach its minimum
value? Mathematically, what does σ represent?
7. The Onsager model treats the relationship
between the dipole moment of a polar molecule within a liquid as the interaction of that
dipole embedded in a spherical cavity (spherical molecule) within a dielectric composed of
an infinite number of identical, freely rotating
dipoles, such that the dipole induces a field on
the dielectric and the average field of the
dipoles in the dielectric partially counteracts
the field from the dipole. This assumption of no
interactions between individual molecules
allows the dielectric response of a material to
be estimated from single-molecule properties.
However, consider the molecules listed in Table
5.1, where the Onsager model provides very
good estimates in some cases and fails
for others (particularly water). For each molecule listed in Table 5.1, (a) list what are likely to
be the strongest intermolecular interactions it
CHAPTER 5: Intermolecular Interactions and Self-Assembly
178