Chapter 20 THE KINETIC THEORY OF GASES 20.1

Brownian Motion (A New Way to Look at Gases) The modern trail to belief in atoms can be said to have started in 1828 when the Scottish botanist Robert Brown observed through his microscope that tiny grains of pollen suspended in water underwent ceaseless random motion. We now call this phenomenon

Brownian motion

. Brown also noted that this same “dancing” motion occurred when particles of finely powdered coal, glass, rocks, and various minerals were suspended in a fluid. The motion seemed to be  a fundamental property of matter.

20-2 Avogadro’s Number The French physical chemist Jean Baptiste Perrin (1870 – 1942) made quantitative measurements of effect. Einstein derived the following expression for [( 

x

) 2 ] av bombarded particle is a sphere of radius

a

if the suspended in a gas: [( 

x

) 2 ] av 

RT

3 

aN

A 

t

Perrin found

N

A  6  10 Nobel Prize in physics. 23 molecules/mol and received the 1926

20-3 Ideal Gases Let us look in a little more detail at the atomic or microscopic properties of the ideal gas.

1.

The ideal gas consists of particles, which are in random motion and obey Newton’s laws of motion

. 2.

The total number of molecules is “large.”

3.

The volume occupied by the molecules is a negligibly small fraction of the volume occupied by the gas

. 4.

No forces act on a molecules except during a collision, either with the container walls or with another molecule

. 5.

All collisions are (i) elastic and (ii) of negligible duration.

20-4 Pressure, temperature, and RMS Speed Consider

N

molecules of an ideal gas confined within a cubical box of edge length

L.

For a single molecule of mass

m

: 

mv x



mv x

  2

mv x F x

 2

mv x

2

L

/

v x



mv x

2

L p

 1

L

2

mv

1 2

x



mv

2 2

x

  

L m L

3 (

v

1 2

x



v

2 2

x

  )

Thus

p

   

v

1 2

x



v

2 2

x N

      (

v

2 ) av Because we have many molecules and because they are moving entirely at random, we have

(

v

2

x

)

av 

1 3 (

v

2

)

av So the pressure is

p

 1 3  (

v

2 ) av The

root-mean-square

speed of the molecules

v rms

 (

v

2 )

av

 3

p

  3

RT



20-5 Translational Kinetic Energy The average translational kinetic energy of a single molecule is

K

avg  ( 1 2

mv

2 ) avg  1 2

m

(

v

2 ) avg  1 2 2

mv

rms

K

avg  ( 1 2

m

) 3

RT

  3

RT

2

N

A We obtain

K

avg  3 2

kT

When we measure the temperature of a gas, we are also measuring the average translational kinetic energy of its molecules.

20-6 Mean Free Path The average value of the straight-line distance our chosen molecule travels between collisions is called the molecule’s

mean free path

 .

In time

t

our “fat” molecule would sweep out a cylinder of cross-sectional area 

d

2 , length

L

cyl =

vt

, and volume

V

cyl = area  length = ( 

d

2 )(

vt

).

The number of (point) molecules in the cylinder is then

N

cyl

N

(

V

cyl /

V

) =

N



d

2

vt

/

V

. The mean free path  = is the total distance covered by the moving molecule in time

t

divided by the number of collisions that it makes in that time:  

L cyl N cyl



vtV N



d

2

vt



V N



d

2 or  

kT



d

2

p

10-7 The Distribution of Molecular Speeds

10-7-1 Maxwell’s speed distribution law The Scottish physicist James Clerk Maxwell (1831 – 1879) first solved the problem of the distribution of speeds in a gas containing a large number of molecules. The

Maxwell speed distribution

 as it is called  for a sample of gas at temperature

T

containing

N

molecules, each of mass

m

, is

N

(

v

)  4 

N m

2 

kT

3 / 2

v

2

e



mv

2 / 2

kT

The interpretation of

N

(

v

) is that the (dimensionless) product

N

(

v

)

dv

gives the number of molecules having speeds in the range

v

to

v

+

dv

.

If we add up (integrate) the numbers of molecules in each differential speed range

dv

from

v

= 0 to

v

  , we must obtain

N

, the total number in the system. That is, it must be true that

N

   0

N

(

v

)

dv

The speed distribution curve for oxygen molecules at

T

= 80 K is both broadened and flattened as the temperature is increases to 300 K

20-7-2 Average, RMS, and most probable speeds Average speed

v

avg

v avg

   0

vP

(

v

)

dv v avg

 8

RT

  8

kT



m

Average of the square of the speeds (

v

2 ) avg (

v

2 )

avg

   0

v

2

P

(

v

)

dv

(

v

2 )

avg

 3

RT



Root-mean-square speed

v

rms

v rms

 3

RT

  3

kT m

The most probable speed

v P v P

 2

RT

  2

kT m

20-7-3 Experimental Verification of the Maxwell Speed Distribution

v



L

  Apparatus used by Miller and Kusch in 1955.

20-8 The Molar Specific Heats of an Ideal gas

20-8-1 Internal energy

E

int The internal energy of a sample of

n

moles of a

monatomic

gas,

E

int , is

E

int  ( 

N

A )( 3 2

kT

) We can rewrite this as

E

int  3 2 

RT

Thus, the internal energy

E

int of an ideal gas is a function of the gas temperature only; it does not depend on any other variable.

20-8-2 Molar specific heat at constant volume From A to B in

p

-

V

diagram (there are in two isotherms), the heat is

Q

 

C V



T

so that 

E

int  

C V



T



W

 

C V



T

Check the TABLE 20-2 on page 468, it was found that:

C V

 3 2

R

for monoatomic gas. Thus

E

int  

C V T

or 

E

int  

C V



T

20-8-3 Molar specific heat at constant pressure From A to C in

p

-

V

diagram, the heat is

Q

 

C p



T

The work done by the gas is

W



p



V

 

R



T

However, the internal change is 

E

int  

C V



T

By the first law of thermodynamics, we find

C p



C V



R

20-9 Degrees of Freedom and Molar Specific Heats

As Table 20-2 shows, the prediction that C

V

= (3/2)R agrees with experiment for monatomic gases but fails for diatomic and polyatomic gases. Why?

From such models, we would assume that all three types of molecules can have translational motions and rotational motions.

In addition, we would assume that the diatomic and poly atomic molecules can have oscillatory motions, with the atoms oscillating slightly toward and away from one another, as if attached to opposite ends of a spring. To keep account of the various ways in which energy can be stored in a gas, James Clerk Maxwell introduced the theorem of the

equipartition of energy:

Every kind of molecules has a certain number

f of degrees of freedom

, which are independent ways in which the molecule can store energy. Each such degree of freedom has associated with it  on average mole).  an energy of per molecule, (or per 2 1 2

For

rigid

molecules, Monoatomic molecule:

f

= 3 Diatomic molecules:

f

= 5 Polyatomic molecule:

f

= 6

E

int 

f

2 

RT C V



f

2

R

3  1 2

kT

( 3  2 )   1 2

kT

  ( 3  3 )   1 2

kT

 

20-10 Hint of Quantum Theory

We can improve the agreement of kinetic theory with experiment by including the oscillations of the atoms in a gas of diatomic or polyatomic molecules.

20-11 The Adiabatic Expansion of an Ideal Gas

In an adiabatic process the system is well insulated so that no heat enters or leaves, in which case

Q

= 0. The first law becomes, in differential form, 

C V dT

 

pdV

The equation of state of the (ideal) gas in differential form is

pdV



Vdp

 

RdT

Replacing

R

with its equal,

C p



C V

, yields 

dT



pdV



Vdp



R pdV C p



Vdp



C V

We get

dp p

  

C p C V

 

dV V

 0 Take the ratio as  the equation becomes

dp p

  

dV V

Integrating the equation yields

ln

p

 

ln

V



constant

pV

 

a constant

TV

  1 

constant

Problems: 1. 20-11 (on page 479), 2. 20-17, 3. 20-22, 4. 20-33, 5. 20-37, 6. 20-43, 7. 20-48, 8. 20-58, 9. 20-61.