The Chemistry Maths Book, Second Edition

280 Chapter 9Functions of several variables

EXAMPLE 9.27Change in entropy in thermodynamics

The change in the entropy when a thermodynamic system goes from a state with

pressure and temperature(p

1

, T

1

)to a state with pressure and temperature(p

2

, T

2

)

can be obtained from calorimetric measurements of the heat capacity and thermal

expansivity,

in the following way. Treating the entropy as a function of pand T, the total differential

of entropy is

Then, because entropy is a function of state, the change on going from one state to

another is independent of the path:

Choosing the pathC 1 = 1 C

1

1 + 1 C

2

shown in Figure 9.11,

∆S

1 → 2

1 = 1 ∆S

1 → 3

1 + 1 ∆S

3 → 2

where∆S

1 → 3

is at constantp 1 = 1 p

1

:

∆S

1 → 3

1 = 1 S(p

1

, T

2

) 1 − 1 S(p

1

, T

1

)

and∆S

3 → 2

is at constantT 1 = 1 T

2

:

∆S

3 → 2

1 = 1 S(p

2

, T

2

) 1 − 1 S(p

1

, T

2

)

Now (Example 9.22), it follows from the differential enthalpy dH 1 = 1 TdS 1 + 1 Vdpthat

(dividingdHbydTat constantp)

==

∂

∂













=

ZZ

C

2

dS

S

p

dp

p

p

TT

1

2

2

==

∂

∂













=

ZZ

C

1

dS

S

T

dT

T

T

pp

1

2

1

∆SSpTSpT

S

T

dT

S

C

pT

12 → 2 2 1 1

=,−,=

∂

∂











 +

∂

()()Z

∂∂

































p

dp

dS

S

T

dT

S

p

dp

pT

=

∂

∂











 +

∂

∂













C

H

TV

V

T

p

pp

=

∂

∂













,=

∂

∂













α

1

0 T

1

T

2

p

1

p

2

• 
  
  
  
  • 
    
  
  
  • 
    
  
  
  
  

1

2

3

c

1

c

2

T

p

.............

...

...

..

...

............................

..

...

..

.

..

...

..

.

..

...

..

.

..

..

..

...

..

.

..

...

..

.

..

...

..

.

..

..

.........................................................................................................

.

.....

....

...

...

.....

.....

.

.....

....

...

...

.....

.....

.

....

.....

...

...

.....

....

.

.

....

.....

...

.

...

..........

.........

..........

..........

....

...

......

......

.....

......

.....

......

......

...

......................................................................................................................................................................................................................................................................................................

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

......

....

.....

......

....

.....

......

.....

....

......

.....

....

Figure 9.11