Download Homework Problem Set 7 - Illinois State Chemistry

Chemistry 380.37
Fall 2015
Dr. Jean M. Standard
November 9, 2015
Homework Problem Set 7
1.
Which molecule would be expected to have the higher diffusion coefficient in the gas phase, CH 4 or H2?
Explain.
2.
In the liquid phase, which molecule would be expected to have the higher diffusion coefficient, methanol or
octanol? Explain.
3.
The gas phase diffusion coefficient of methane is 1.78×10 m2s–1. Assuming a mean free path of 80 nm,
calculate the average speed of methane molecules in m/s.
4.
For a molecule undergoing collisions and moving through a liquid solution, the average distance squared from
–5
the molecule's initial position, r 2 , is linearly related to the time,
r 2 = Dt,
where D is the diffusion coefficient. In contrast, for a molecule that is allowed to freely translate in space with
no collisions, determine how the distance squared varies with time. Assume classical behavior and compare
with the diffusional behavior in liquid solution.
5.
Consider a bond involving atoms A and B represented using a harmonic force field of the form
U =
1
2
(
k s,AB rAB − rAB,eq
)
2
,
where rAB is the instantaneous bond length and rAB,eq is the equilibrium bond length. The equilibrium A-B
bond length is 1.00 Å, and the€stretching force constant is 500 kcal mol–1 Å–2. The atoms are arranged so that
they have the cartesian coordinates given below (in Å).
€
€
Atom
A
B
x
–0.50
0.37
y
–0.23
0.54
a.) What is the instantaneous A-B bond length?
b.) What is the potential energy of the system in units of kcal mol–1?
c.) Determine the x-, y-, and z-components of the force on atom A.
d.) Determine the x-, y-, and z-components of the force on atom B.
e.) Are the atoms moving closer together or farther apart?
z
0.00
0.00
2
6.
Consider the simple one-dimensional harmonic potential as a representation of the motion of a bond in a
molecular system,
U ( x) =
1
2
k x2.
The coordinate x represents the bond displacement, x = r − req . Assume that the force constant k for the bond is
720 kcal mol–1 Å–2. The reduced mass
€ is 12.0 g/mol. The bond displacement coordinate has an initial value of
0 Å. The initial velocity is 8400 m/s.
€
a.) What is the potential energy?
b.) What is the kinetic energy?
c.) What is the total energy?
7.
Consider the potential energy function studied in problem 6 with the same parameters. Since the potential
energy function described in problem 6 is harmonic, the bond undergoes simple harmonic motion as a function
of time. The exact analytical solutions of Newton's equations for a harmonic oscillator with an initial position of
zero and an initial velocity v(0) are
x(t) =
v(0)
sin(ω t )
ω
v(t) = v(0) cos(ω t ) .
The parameter ω is the angular velocity and is defined by the relation
€
ω =
k
,
m
where m is the mass. The angular velocity ω is related to the harmonic frequency of oscillation ν o ,
€
νo =
ω
.
2π
€
a.) Using the same parameters as given€in problem 6, determine the harmonic frequency of vibration for this
system.
b.) Using the analytic solution to Newton's equations for the harmonic oscillator, plot the bond displacement
coordinate x as a function of time for this system.
c.) Using the analytic solution to Newton's equations for the harmonic oscillator, plot the velocity as a function
of time for this system.
8.
Again consider the simple one-dimensional potential used in problems 6 and 7. Assume that the force
constant k is 720 kcal mol–1 Å–2. The reduced mass is 12.0 g/mol. The initial position is 0 Å and the initial
velocity is 8400 m/s.
a.) Using the Verlet leapfrog algorithm, carry out two time steps to determine the position and velocity of the
bond at times t1 and t2. Use a step size of 1 femtosecond.
b.) Compare your results from part (a) to the exact solution obtained in problem 7.
9.
The Maxwell-Boltzmann distribution for speeds is
# m &
F (v) = 4 π v 2 %
(
$ 2π k BT '
3/ 2
# mv 2 &
.
exp% −
(
2k
T
$
B '
In this equation, m is the mass of a particle, kB is the Boltzmann constant, T is temperature, and v is the speed.
The speed v is defined€to be the magnitude of the velocity vector,
[
v = v 2x + v 2y + v 2z
1/ 2
]
,
where v x , v y , and v z are the components of the velocity vector.
€
The most probable speed v mp corresponding to the maximum in the distribution is given by the equation
€ €
€
€
" 2k T %
v mp = $ B '
# m &
1/ 2
.
a.) Plot the Maxwell-Boltzmann distribution function F (v) (y-axis) versus v (x-axis) for a collection of atoms
at 300 K. Assume that one mole
€ of atoms weighs 12.0 g.
F (v) versus v for a
b.) Using the same mass as in part (a), plot the
€ Maxwell-Boltzmann distribution function
collection of atoms at 1000 K. Discuss the effect of temperature on the distribution of atomic speeds.
c.) Calculate the most probable speed for atoms weighing 12.0 g/mol at 300 K and
€ 1000 K. Express your
answers in m/s.
d.) From the most probable speeds calculated in part (c), determine the most probable kinetic energy of one
mole of atoms at 300 K and 1000 K. Express your answers in J/mol.
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10. The Equipartition Theorem states that each degree of freedom contributes 2 k BT to the internal energy of a
1
particle (or 2 R T to the internal energy of one mole of particles).
a.) Use the equipartition theorem to calculate the internal energy of a mole of atoms at 300 K. Express your
answer in J/mol.
b.) Repeat the calculation of the internal energy for a mole of atoms at 1000 K. Express your answer in J/mol.
c.) Compare your results from parts (a) and (b) to the kinetic energy determined in part 9(d). Are the values
similar?
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