Lecture 8 the kinetic theory of gases

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CHAPTER 21 THE KINETIC THEORY OF

THERMODYNAMICS

The kinetic theory of Gases

1 The number of molecules in the gas is large, and the average separation between them is large compared with their dimensions In other words,

the molecules occupy a negligible volume in the container We model the molecules

as particles

2 The molecules obey Newton’s laws of motion, but as a whole they move randomly By “randomly,” we mean that any molecule can move in any

direction with any speed

3 The molecules interact only by short-range forces during elastic collisions

That is consistent with the ideal gas model, in which the molecules exert no

long-range forces on each other

4 The molecules make elastic collisions with the walls These collisions lead

to the macroscopic pressure on the walls of the container

5 The gas under consideration is a pure substance; that is, all molecules are identical

3

2 P

Pressure (N/m2)

particle density (m^-3)

Average Translational Kinetic Energy of 1 particle

KINETIC THEORY

The pressure that a gas exerts is caused by the

collisions of its molecules with the walls of the

container

We try to derive the equation that relates

the macroscopic quantity P and the

microscopic quantity of the gas – the

average Translational Kinetic Energy 

4

KINETIC THEORY

Consider a collection of N molecules of an ideal gas in a container of volume V=L 3 Consider molecule i of mass m, velocity vi, It collides the wall

The impulse – momentum theorem gives

x

xi particle

wall

particle wall

i i

e

mv

2 F

F

v m '

v

m



























v

v’

L

2 xi particle

wall

xi

e L

mv F

v

L 2















mv

; L

N n

n 3

2 P

2

mv L

N 3

2 L

F P

e 3

v L

Nm F

3

v v

v v

e

v L

m N L

mv F

e L

mv F

F

e L

mv F

v

L 2

e mv 2 F

F

v m ' v m

2 3

o

o

2 3

2

wall particle

x

2 wall

particle

2 2

z

2 y

2 x

x

2 x i

2 xi wall

particle

x

2 xi particle

wall wall

particle

x

2 xi particle

wall xi

x

xi particle

wall

particle wall

i i











































































































3

2 P

Pressure

(N/m2)

particle density (m^-3)

Average Translational Kinetic Energy of 1 particle

Temperature is a direct measure of

average molecular translational kinetic energy

T

k 2 3

T k

n V

T k

nN V

nRT P

n 3

2 P

B

B o

B A

o















Theorem of equipartition of energy

The average translational kinetic energy per molecule is

T

k 2

3 v

m 2

1 v

m 2

1 v

m 2

1 v

m 2

1

B

2 z

2 y

2 x





Each degree of freedom contributes ½ kBT to the energy

of a system

It is also right for degrees freedom associated translatioal, rotation, and vibration of molecules

Translational motion has 3 degree of freedom

Degree of Freedom i

6 N

3 i

; 3 i

; 3 i:

Gas Manyatomic

1 i

; 2 i

; 3 i

: Gas Diatomic

3 i

: Gas Monoatomic

n oscillatio

rotation

transl

n oscillatio

rotation

transl

transl







 

Internal Energy

nRT 2

i T

k

nN

2

i

U

Const Gas

Universal :

k N

R

mol of

number

:

n

nN

N

T

k 2

i N

U

particle 1

of energy particle

of number energy

Internal

B A

B A

A B















nRT 2

i

U 

Equation of state for an ideal gas

PV=nRT

R is called the universal gas constant R=8.31 J/(mol.K)

P pressure is expressed in pascals (1 Pa = 1 N/m2)

V volume in cubic meters

PV has units of joules

20.5 The First Law of Thermodynamics

• The first law of thermodynamics states that when a

system undergoes a change from one state to another, the change in its internal energy is

• where Q is the energy transferred into the system by heat and W is the work done on the system

• Although Q and W both depend on the path taken from

the initial state to the final state, the quantity U does not depend on the path

W Q



Work done on the system















2

1

V

V

PdV W

PdV dy

S P s

d F

Appications of the 1rst Law of TD

Isothermal process of an Ideal Gas

1 2

1

2 V

V

V

V ln nRT W

Q

V

V ln nRT

dV V

nRT PdV

W

0

U

W Q

U

const PV

: const

T

2

1





















  

  





isobaric process P=const

R 2

2

i

C

T nC

Q

T

R 2

2

i n T

nR T

nR 2

i V

P U

Q

V P Q

U

T nR V

P PdV

W

W Q

U

const T

/ V : const

P

p

p





























































isovolumetric process V=const

R 2

i C

T nC

T

R 2

i n Q

U

0 PdV

W

W Q

U

: const

V

V

V

























 



Adiabatic process Q=0

const P

T

const TV

const T

PV

const PV

T

nR 2

i W

U

0

Q

W Q

U

1

1































The molar specific heat at constant volume :

The molar specific heat at constant pressure

The ratio of specific heats

R 2

i

CV 

R 2

2

i

Cp  

i

2 i C

C

V

P  





R C

C

1

R C

1

R C

V P

P

V



  











ADIABATIC PROCESS

Const

PV

C ln P

ln

V

ln

0 P

dP V

dV

i

2

i

0

VdP 2

i PdV

2

2

i

PdV )

VdP PdV

(

2

i

) VdP PdV

( 2

i nRdT

2

i

dU

nRdT VdP

PdV

nRT

PV

nRdT

2

i

dU

PdV

dU

0 Q : process Adiabatic

PdV Q

dU



















































const P

T

const TV

const T

PV

const PV

1

1































SUMMARY