Physical Chemistry Third Edition

404 9 Gas Kinetic Theory: The Molecular Theory of Dilute Gases at Equilibrium

first velocity,

v^2 

(

500 m s−^1

) 2

+

(

−400 m s−^1

) 2

+

(

250 m s−^1

) 2

472500 m^2 s−^2

ε

kBT



mv^2

2 kBT



(3. 3510 −^26 kg)(472, 500 m^2 s−^2 )

(2)(1. 3807 × 10 −^23 JK−^1 )(300 K)

 1. 91

second velocity,

ε

kBT

 0. 963 (calculation similar to first velocity)

probability ratio

e−^1.^91

e−^0.^963

 0. 388

Exercise 9.11

Find the ratio of the probabilities of the two velocities for argon atoms at 300.0 K:

first velocityvx650 m s−^1 , vy780 m s−^1 , vz990 m s−^1

second velocityvx300 m s−^1 , vy290 m s−^1 , vz430 m s−^1

Equation (9.3-40) represents the probability density in the three-dimensional veloc-

ity space of Figure 9.4. A point in this velocity space represents the velocity of one

molecule. If we haveNmolecules, we can represent their velocities by a set ofN

points in the same velocity space. The density (number of points per unit volume) of

this swarm of points at the point representing the velocityv′is proportional to the

probability densityg(v′). Figure 9.9 schematically represents the swarm of points for

a system of a few hundred molecules.

vz

vy

vx

Figure 9.9 Points Representing the
Velocity States of Molecules in a
System.

PROBLEMS

Section 9.3: The Velocity Probability
Distribution

9.7a.Find the probability of drawing the ace of spades

from one randomly shuffled deck of 52 cards and

drawing the eight of diamonds from another deck of

52 cards.

b.Find the probability of drawing the ace of spades and

the eight of diamonds (in that order) from a single deck

of 52 cards.

c.Find the probability of drawing the ace of spades and

the eight of diamonds (in either order) from a single

deck of 52 cards.

9.8Compute the odds for each possible value of the sum of the

two numbers that can show when two dice are thrown.

9.9Find the mean value and the root-mean-square value of

sin(x) for 0<x< 2 πradians, assuming a uniform

probability distribution.

9.10 a.Find the mean value and the root-mean-square value of

sin^2 (x) for 0<x< 2 πradians, assuming a uniform

probability distribution.

b.Find the mean value and the root-mean-square value of

sin^2 (x) for 0<x< 4 πradians, assuming a uniform

probability distribution. Comment on the relationship

between your answers in parts a and b. What can you