Physical Chemistry Third Edition

9.3 The Velocity Probability Distribution 403

b.Find a valuev′xsuch that there is a 95% probability that a molecule will have a value ofvx

between−v′xandv′xfor N 2 molecules at 298.15 K. Find the ratio of this value to the standard

deviationσvx.

c.Repeat this calculation for N 2 molecules at 1000.0 K.

The three-dimensional probability distributiong(v) can be written as

g(v)f(vx)f(vy)f(vz)

(

m

2 πkBT

) 3 / 2

e−m(v

(^2) x+v (^2) y+v (^2) z)/ 2 kBT
g(v)

(

m

2 πkBT

) 3 / 2

e−mv

(^2) / 2 kBT
(9.3-40)
This function is called theMaxwell probability distributionor theMaxwell–Boltzmann
probability distribution.In terms of the molecular kinetic energy,
g(v)

(

m

2 πkBT

) 3 / 2

e−ε/kBT (9.3-41)

which is called theBoltzmann probability distribution. We do not prove it at this time,

but this probability distribution applies to all forms of molecular energy, including

potential energy and the vibrational, rotational, and electronic energies. We write

(probability of a state of energyε)∝e−ε/kBT (9.3-42)

where the symbol∝means “is proportional to.”

There are several important physical facts about the Boltzmann probability

distribution:

1. At a finite temperature, molecular states of higher energy are less probable than
   states of lower energy. States with energy much larger thankBTare quite improbable
   compared with states of energy equal to zero.


2. A molecular state of high energy will be more probable at a high temperature than
   at a low temperature.


3. As the temperature approaches infinity, all states approach equal probability.


4. If the temperature approaches zero, only the state of lowest energy will be populated.

EXAMPLE 9.7

Find the ratio of the probabilities of the two following velocities of neon atoms at 300 K:

first velocity:vx500ms−^1 ,vy−400ms−^1 ,vz250ms−^1

second velocity:vx200ms−^1 ,vy350ms−^1 ,vz−275ms−^1

Solution

Neon has several isotopes. We use the average molecular mass,

m

0 .020183 kg mol−^1

6. 022 × 1023 mol−^1

 3. 35 × 10 −^26 kg