The Third Law of Thermodynamics

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The Third Law of Thermodynamics

(Lecture 10)

1

^st

semester, 2021

Advanced Thermodynamics (M2794.007900) Song, Han Ho

(*) Some materials in this lecture note are borrowed from the textbook of Ashley H. Carter.

(2)

è

The third law of thermodynamics is concerned with the behavior of systems in equilibrium near 0 K.

Entropy, by definition, is given by

In many engineering applications, we only need to know the change in entropy.

However, for example, if we want to calculate the change in Gibbs function,

knowing the absolute entropy or S₀ could be important!

à The third law enables us to determine S

₀

!

Statements of the Third Law

K 0 at entropy :

where ₀

0 S0 S

T S =

ò

^T

d

Q^r +

The Third Law of Thermodynamics

VdP SdT

dG = - +

(3)

è

Let’s derive the Nernst and Planck statements of the third law.

By definition,

For an isothermal process,

In the vicinity of absolute zero,

From his experiments, Nernst also postulated that,

Or,

Statements of the Third Law

( )

or 2

T

H T

T G T

T G H TS H

G

P P

-

÷ = ø ç ö

è æ

¶

÷ ¶ ø ç ö

è æ

¶ + ¶

= -

=

Gibbs-Helmholtz equation

( )

T P

T G H

G úûù

êëé

¶ D + ¶

D

= D

H G @ D D

( )

₀_and_lim

( )

₀

lim0 0 =

úûù êëé

¶ D

= ¶ úûù êëé

¶ D

¶

®

® T P

T P T

H T

G

( )

^lim ^lim

( )

⁰

lim ₁ ₂

0 1

2 0

1 2

0 ú = - =

û ê ù

ë

é ÷

ø ç ö

è æ

¶ - ¶

÷ø ç ö

è æ

¶

= ¶ úûù êëé

¶ -

¶

®

® S S

T G T

G T

G G

P T T P

T P

(4)

è

Continue on.

This is the Nernst formulation of the third law. In words,

“All reactions in a liquid or solid in thermal equilibrium take place with no change of entropy in the neighborhood of absolute zero.”

Planck extended Nernst’s hypothesis and proposed,

Let , then,

Statements of the Third Law

( ) 0

lim

₁ ₂

0

- =

®

S S

T

T P T P

T

T T

H T

T G H T

G ÷

ø ç ö

è æ

¶

= ¶

÷ø ç ö

è æ

¶

= ¶

®

®0 ( ) lim0 ( ) and lim0 lim0

lim

H G- º F

) ( 0 of

vicinity in the

0

and

0 T K

T ÷_P = =

ø ç ö

è æ

¶ F

= ¶ F

(5)

è

Continue on.

By L’Hopital’s rule,

Finally,

Statements of the Third Law

0

and

P P

P

T H T

T T

T H T T

T T H T

T H T

T G T

T G

T T G H TS

H G

÷ ø ç ö

è æ

¶ - ¶ F

=

÷ - ø ç ö

è æ

¶ F

® ¶

÷ ø ç ö

è æ

¶ - ¶

=

F

÷ - ø ç ö

è æ

¶ F

® ¶

÷ ø ç ö

è æ

¶ - ¶

=

F

÷ - ø ç ö

è æ

¶ - ¶

÷ ø ç ö

è æ

¶

® ¶

=

F

÷ - ø ç ö

è æ

¶

® ¶

÷ ø ç ö

è æ

¶

+

¶

=

-

=

-

º F

T T _P ^T

T

= F

÷ø ç ö

è æ

¶ F

¶

®

® lim

lim0 0

( )

⁰

lim lim

0

lim0 0 ÷ = 0 - =

ø ç ö

è æ

¶

= ¶

÷ = ø ç ö

è æ

¶

®

® S

T G T

H

P T P T

T

(6)

è

Continue on.

This is the Planck’s statement of the third law. In words,

“The entropy of a true equilibrium state of a system at absolute zero is zero.”

This statement is true for every pure crystalline solid. A certain glass structure has a nonvanishing entropy at absolute zero, with its disordered structure.

à Will learn in our discussion on statistical thermodynamics!

Another statement of the third law is (unattainability principle):

“It is impossible to reduce the temperature of a system to absolute zero using a finite number of processes.”

Statements of the Third Law

0 lim

0

=

®

S

T

(7)

è

Let’s prove the equivalence of the Nernst postulate and unattainability principle.

In S-T diagram,

According to the Debye law for heat capacity of a solid,

Thus, S increases with T to the certain power.

Equivalence of the Statements

The Third Law of Thermodynamics

0 0

0 0 0

or

0K at entropy :

where T S

S CdT

S T S

S Q

T

T r

+

=

+

=

ò ò d

T3

C_V µ

(8)

è

Continue on.

Consider a cooling process from T₁ to T₂ by an adiabatic reversible process.

Then,

If , then, there must be certain so that

Equivalence of the Statements

( ) ( )

( ) ò

ò

ò ò

+ -

-

=

+

= +

=

1 2

0 1

01 0 2 02

0 1

0 2 01 02

1 1

2

2 @ @

T T

T C dT S

T S C dT

T C dT T S

C dT S

T S

(

S₀₂ -S₀₁

)

> ⁰ T₁ =T₁'

( )

0 ' or

) ' to ing correspond :

' (where

0 or

2

1 2

'

0 2

'

0 1

01 02

2

1

=

= -

ò

T

T T T

C dT

T C dT S

S

T

(

^-

)

^>

(9)

è

Continue on.

Now consider a cooling process from T₂ to T₁ by an adiabatic reversible process.

Then,

If , then, there must be certain so that

This violates the unattainability principle! So .

Equivalence of the Statements

( ) ( )

( ) ò ò

ò ò

= +

- -

+

= +

=

1 2

0 1

0 2

02 01

0 1

0 2 01 02

1 1

2

2 @ @

T T

T C dT T

C dT S

S

T C dT T S

C dT S

T S

"

2

2 T

T =

( )

0

"

or

)

"

to ing correspond :

"

(where

0 or

1

2 1

"

0 1

"

0 2

02 01

1

2

=

= -

ò

T

T T T

C dT

T C dT S

S

T

(

^S₀₁^-^S₀₂

)

^> ⁰

(

S02 -S01

)

>⁰

(10)

è

Continue on.

Finally, to satisfy the unattainability principle, and

Consequently, the Nernst postulate should be satisfied.

An infinite series of isothermal and adiabatic processes are required to reach absolute zero!

Equivalence of the Statements

01

02

S

S =

( S

₀₁

- S

₀₂

) > 0 ( S

₀₂

- S

₀₁

) > 0

(11)

1. Expansivity

2. Slope of the phase transformation curves

à This has been verified for all known sublimation curves, for the vaporization curve of Helium II, and for the fusion curve of solid helium.

Consequences of the Third Law

1 0 lim lim

1 1

0

0 ú =

û ê ù

ë

é ÷

ø ç ö

è æ

¶ - ¶

=

÷ø ç ö

è æ

¶ - ¶

÷ = ø ç ö

è æ

¶

= ¶

®

® T T

T

T P

P S V

T V V

b

b Maxwell relation

T

P P

S T

V ÷

ø ç ö è æ

¶ - ¶

÷ = ø ç ö è æ

¶

Nernst postulate

V S dT

dP

D

= D Clausius-Clapeyron equation

0 lim

lim0 0 =

D

= D

®

® V

S dT

dP

T T

Nernst postulate

(12)

3. Heat Capacity

Integrating,

In the vicinity of absolute zero, left-hand-sides become zero by Nernst postulate.

To satisfy the equation, C_V and C_P should approach zero, at least as rapidly as T (in the denominator).

Consequences of the Third Law

P P

P

V V

V

T T S T

C H

T T S T

C U

÷ø ç ö

è æ

¶

= ¶

÷ø ç ö

è æ

¶

= ¶

÷ø ç ö

è æ

¶

= ¶

÷ø ç ö

è æ

¶

= ¶

Gibbs equation

VdP TdS

dH

PdV TdS

dU

+

=

-

=

ò

^- ⁼

=

- ^T _V ^T _P

T C dT S

T S C dT S

S 0 0 or 0 0

0 lim

and 0

lim0 = 0 =

®

® V T P

T C C