Physical Chemistry Third Edition

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:

EK ε 1 +ε 2 +ε 3 + ··· +εNκ 1 +κ 2 +κ 3 + ··· +κN (9.2-16)

whereεiis the energy of particle numberiandκiis 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

i 1

κi (9.2-17)

IfNjis the number of the molecules in a particular statej, thenpjis the fraction of

molecules in that state:

pj

Nj

N

(9.2-18)

The probability that a randomly chosen molecule will be in statejis also equal topj.

We can write the average molecular energy as a sum over states instead of a sum over

molecules:

〈ε〉

∑∞

j 1

pjεj (9.2-19)

whereεjis the energy of a particle in statej. 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.

PROBLEMS

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.

9.4a.The frictional force on a spherical object moving through

a fluid is approximately described by the formula

F−fv

wherefis called thefriction coefficientand wherevis

the velocity of the object. Obtain a formula for the

velocity and position of an object described by this

equation moving in thexdirection with initial position

x(0)x 0 and initial velocityvx(0)v 0. Ignore any

force due to gravity.