The Classical Statistical Treatment of an Ideal Gas

(1)

The Classical Statistical Treatment of an Ideal Gas

(Lecture 14)

1

^st

semester, 2021

Advanced Thermodynamics (M2794.007900)

(2)

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Let’s derive the thermodynamic properties of an ideal gas, based on the partition function.

For a dilute gas, Maxwell-Boltzmann distribution is applied for an equilibrium state.

We’ve already have the following equations:

We’ll deal with other thermodynamic properties!

Thermodynamic Properties from the Partition Function

The Classical Statistical Treatment of an Ideal Gas

( )

÷

ø ç ö è

= æ -

-

=

+ -

-

= -

=

+ -

+

=

Z kT N

N Z

kT

N Z

NkT TS

U F

N Z

T Nk S U

ln ln

~ ln

1 ln

ln 1 ln

ln

µ

(3)

è

Continue on.

1. Internal energy

2. Gibbs function (for a single component)

Thermodynamic Properties from the Partition Function

( )

V V

n j

kT j j

n j

j kT j n

j

j kT j V

n j

j kT n

j

kT j j n

j

j j

T NkT Z

T kT Z

Z e N

Z g U N

e kT g

kT dT e d

T g Z

e g Z

e Z g

N N U

j

j j

÷ø ç ö

è æ

¶

= ¶

÷ø ç ö è æ

¶

= ¶

=

®

÷= ø ç ö è - æ

÷ = ø ç ö è æ

¶

=

å

å å

å

=

-

=

-

=

-

=

-

=

-

=

ln

1 1

Here,

where

2 2

1

2 1 1

1 1

1

e

e e

e

e e

(

-

)

= çæ ÷ö

-

=

µ

~ N

(4)

è

Continue on.

3. Enthalpy

4. Pressure

Thermodynamic Properties from the Partition Function

The Classical Statistical Treatment of an Ideal Gas

( ) ( )

úû ê ù

ë

é ÷

ø ç ö

è æ

¶ + ¶

=

+ úû =

êë ù

é + - +

+ -

-

= +

=

®

- º

T V

T Z NkT

NkT U

N Z

T Nk T U N

Z NkT

TS G

H

TS H

G

1 ln

ln ln

ln

( )

T

Z Z

N Z

NkT V F

P F

ö æ¶

ö æ ¶

+ -

-

÷ = ø ç ö

è æ

¶ - ¶

=

ln 1

1 ln

ln

and

(5)

è

Partition function is a measure of the states available to the system and is the essential link between the microscopic systems and the thermodynamic properties.

Suppose the system with

Partition Function for a Gas

. all for

1 j

g

_j

=

) 0 (let

....

1

₀

1 0

2

1

+ + =

+

=

^- ^- ^-

=

å

ⁿ -ê ^kTê ê^kT

e

j

e

kT

e e

g Z

j

Z e g N N Z

Ne g

N ^kT _j _j ^kT

j j

j ej

e

or

- -

=

(6)

è

Continue on.

For a sample of gas in a container of macroscopic size, the energy levels are closely spaced and can be regarded as a continuum. Then, the density of states was given by,

Then,

Partition Function for a Gas

2 ) 4

(e e ₃p m³^/²e¹^/²de h

d V

g =

2 / 3 2 / 3 3

0

/ 2 / 1 2

/ 3 0 3

/

2

2 2

4

2 ) 4

(

ö æ

÷ø ç ö

è

= æ

=

ò

^¥ ^-

ò

^¥ ^-

mkT kT kT h m

V

d e

h m d V

e g

Z ^kT ^kT

p p p

e p e

e

e ^e ^e

for translational kinetic energy of bosons with zero spin

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è

Let’s evaluate the thermodynamic properties of a monatomic ideal gas with bosons of zero spin. à Consider translational kinetic energy only.

The partition function and the partial derivatives of its logarithm are,

Then,

Properties of a Monatomic Ideal Gas

1 2 3 ln

and 1 ln

ln 2 2 ln 3

2 ln ln 3

2

2 2

/ 3 2

T T

Z V

V Z

h V mk

T h Z

V mkT Z

V T

×

÷ = ø ç ö

è æ

¶

= ¶

÷ø ç ö

è æ

¶

® ¶

÷ø ç ö

è + æ

+

=

÷ ® ø ç ö

è

= æ p p

Z

NkT V PV

V NkT NkT Z

P

T

3 1

3 ln

or 1

ln

2

2æ¶ ö = æ × ö =

=

÷ = ø ç ö è

= æ

÷ø ç ö

è æ

¶

= ¶

(8)

è

Continue on.

For an entropy,

For specific entropy (per mole basis),

Properties of a Monatomic Ideal Gas

The Classical Statistical Treatment of an Ideal Gas

( )

ïþ ïý ü ïî

ïí

ì ú

û ê ù

ë + é

=

÷+ ø ç ö

è

æ +

=

ïþ ïý ü ïî

ïí

ì ú

û ê ù

ë + é

=

úû ê ù

ë

é ÷- +

ø ç ö

è + æ

+ +

×

= + -

+

=

3 2 / 3 0

0 3

2 / 3

2

ln 2 2 5

where ln

2ln 3 or

ln 2 2 5

1 2 ln

2ln ln 3

2ln 3 2

1 3 ln

ln

h Nk mk

S N S

T V Nk

S

Nh mkT Nk V

h N V mk

T T Nk

N NkT Z

T Nk S U

p p

p

Sackur-Tetrode equation

for the entropy of a monatomic gas

n S s

n Nk R

R c

n S s

s v R T

c

s =

_v

ln + ln +

₀

where = / ,

_v

= ( 3 / 2 ) , = / ,

₀

¹

₀

/