1
30 April, 2007
Michael F. Brown
CHEMISTRY 481 (Biophysical Chemistry)
Problem Set 12  STUDY GUIDE
To be turned in by: NEVER
Background reading:
Chapter 9
Chapter 10
Chapter 11.1–11.3, 11.7
Chapter 13.1–13.3, 13.10
Chapter 15
Back of Chapter Problems related to the homework (optional):
Problems
9.3–9.5, 9.7–9.8, 9.11–9.15, 9.17, 9.19–9.21
Problems 10.12, 10.13, 10.1810.20, 10.22, 10.2610.28
Problems 11.10, 11.11,11.13, 11.27, 11.17
Problems 15.3, 15.5, 15.8, 15.11, 15.30
Problem 1
.
One of the major applications of classical mechanics in biochemistry involves
solving Newton’s Laws of motion for macromolecules.
This method is called
molecular
dynamics
.
It can be used to investigate the atomic motions of proteins, nucleic acids (DNA and
RNA), and the lipids in membranes.
Let us consider the molecular dynamics of membrane
lipids.
The vibrations of the bonds joining the various atoms are modeled as a
classical
harmonic oscillator
.
Consider a representative C–H bond of a lipid in a membrane bilayer.
For simplicity, the
bond vibrations are modeled in terms of the relative motion of the two atoms.
In a center of
mass coordinate frame, the equation of motion is given by:
d
2
x
dt
2
+
ω
0
2
x
=
0
Here
ω
0
=
k
/
μ
and
μ
is called the
reduced mass
, which is defined by:
μ
=
m
C
m
H
m
C
+
m
H
Let us assume that the force constant
k
= 450 N m
–1
for the case of a CH bond.
a) What is the
natural frequency
ν
0`
/ s
–1
of the harmonic oscillations of the C–H bonds?
b) In one type of experiment, hydrogen (H) is replaced chemically by deuterium (D).
Do you
expect the
frequency of the bond oscillations
to increase, decrease, or remain unaltered upon
substitution of D for H?
Why?
c) What is the
natural frequency
ν
0
/ s
–1
of the C–D bond oscillations?
(Assume the force
constant
k
is the same as for a C–H bond.)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
d) Calculate the
force
needed to produce vibrations of a C–H bond with an amplitude (
A
) of 10.0
pm.
Problem 2
.
The
wavefunction
corresponding to a hydrogenic 1
s
orbital is given by
ψ
(
r
)
=
1
4
π
e
−
r
/
a
0
where the Bohr radius
a
0
= 52.9 pm.
(
Hint
: the above wavefunction is not normalized, if you
cannot normalize it then proceed to parts (b)–(e) without the normalization constant.)
a) Find the
normalized
wave function.
b) Calculate the
expectation value
of the radius, given by
+
r
,
.
c) Calculate the
expectation value
of the radius squared, given by
+
r
2
,
.
d) Calculate the
variance
of the radius, defined by
+
r
2
,
–
+
r
,
2
.
e) Calculate the
root mean square
radius, defined by
+
r
2
,
1/2
.
Problem 3
.
An important pigment molecule found in plants is
β
carotene:
Assume that the electronic properties of
β
carotene can be considered in terms of a
particleina
box
, in which
L
= 2.0 nm.
a)
Write the
Schrödinger equation
and state the
eigenfunctions
and
eigenvalues
.
Be sure to
define all symbols.
b)
Given the free electron model, what is the
wavelength
of light absorbed by
β
carotene?
Problem 4
.
A biochemist is interested in determining the amount of secondary structure
(hydrogen bonding) in a newly discovered protein from human immunodeficiency virus (HIV).
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '06
 brown
 Physical chemistry, pH, Schrodinger Equation, Nuclear magnetic resonance, spherical polar coordinates

Click to edit the document details