9.2 A Model System to Represent a Dilute Gas
393
Exercise 9.6
Show that any constant can be added to a potential energy without changing the forces.
Since the potential energy equals zero, the energy of the model system is equal to
its kinetic energy:
E K ε 1 + ε 2 + ε 3 + · · · + ε N κ 1 + κ 2 + κ 3 + · · · + κ N
(9.2-16)
where ε i is the energy of particle number i and κ i is its kinetic energy. The average
(mean) molecular energy is given by the sum of the molecular energies divided by the
number of molecules:
κ ε
E
N
K
N
1
N
N
i1
κ i
(9.2-17)
If N j is the number of the molecules in a particular state j, then p j is the fraction of
molecules in that state:
p j
N j
N
(9.2-18)
The probability that a randomly chosen molecule will be in state j is also equal to p j .
We can write the average molecular energy as a sum over states instead of a sum over
molecules:
ε
∞
j1
p j ε j
(9.2-19)
where ε j is the energy of a particle in state j. There are infinitely many states that are
possible, as indicated by the limits on the sum. We will omit these limits on sums from
now on and understand that the sums are over all possible states.
P R O B L E M S
Section 9.2: A Model System to Represent a Dilute Gas
9.3 a. Estimate the number of coordinate states that a neon
atom might occupy if it is known only that it is contained
in a volume of 1.00 L. That is, ignore the different
velocities that the atom might have. Assume that the
atom obeys classical mechanics and that the precision of
measurement of its position is 1.00 nm (1.00 × 10 −9 m).
That is, if the center of the atom is anywhere in a cubical
region 1.00 nm on a side, this counts as one state.
b. Estimate the number of coordinate states occupied by
1.00 mol of neon atoms by the criterion of part a.
Assume that the atoms have zero size so that any
number of atoms could occupy the same cubical
region.
9.4 a. The frictional force on a spherical object moving through
a fluid is approximately described by the formula
F −f v
where f is called the friction coefficient and where v is
the velocity of the object. Obtain a formula for the
velocity and position of an object described by this
equation moving in the x direction with initial position
x(0) x 0 and initial velocity v x (0) v 0 . Ignore any
force due to gravity.
393
Exercise 9.6
Show that any constant can be added to a potential energy without changing the forces.
Since the potential energy equals zero, the energy of the model system is equal to
its kinetic energy:
E K ε 1 + ε 2 + ε 3 + · · · + ε N κ 1 + κ 2 + κ 3 + · · · + κ N
(9.2-16)
where ε i is the energy of particle number i and κ i is its kinetic energy. The average
(mean) molecular energy is given by the sum of the molecular energies divided by the
number of molecules:
κ ε
E
N
K
N
1
N
N
i1
κ i
(9.2-17)
If N j is the number of the molecules in a particular state j, then p j is the fraction of
molecules in that state:
p j
N j
N
(9.2-18)
The probability that a randomly chosen molecule will be in state j is also equal to p j .
We can write the average molecular energy as a sum over states instead of a sum over
molecules:
ε
∞
j1
p j ε j
(9.2-19)
where ε j is the energy of a particle in state j. There are infinitely many states that are
possible, as indicated by the limits on the sum. We will omit these limits on sums from
now on and understand that the sums are over all possible states.
P R O B L E M S
Section 9.2: A Model System to Represent a Dilute Gas
9.3 a. Estimate the number of coordinate states that a neon
atom might occupy if it is known only that it is contained
in a volume of 1.00 L. That is, ignore the different
velocities that the atom might have. Assume that the
atom obeys classical mechanics and that the precision of
measurement of its position is 1.00 nm (1.00 × 10 −9 m).
That is, if the center of the atom is anywhere in a cubical
region 1.00 nm on a side, this counts as one state.
b. Estimate the number of coordinate states occupied by
1.00 mol of neon atoms by the criterion of part a.
Assume that the atoms have zero size so that any
number of atoms could occupy the same cubical
region.
9.4 a. The frictional force on a spherical object moving through
a fluid is approximately described by the formula
F −f v
where f is called the friction coefficient and where v is
the velocity of the object. Obtain a formula for the
velocity and position of an object described by this
equation moving in the x direction with initial position
x(0) x 0 and initial velocity v x (0) v 0 . Ignore any
force due to gravity.