Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

666 | Thermodynamics

Analysis The change in internal energy of any simple compressible system

in any phase during any process can be determined from Eq. 12–30:

The van der Waals equation of state is

Then

Thus,

Substituting gives

Integrating yields

which is the desired relation.

u 2 u 1 c 11 T 2 T 12 

c 2

2

1 T^22 T 122 aa

1

v 1



1

v 2

b

u 2 u 1 

T 2

T 1

1 c 1 c 2 T 2 dT

v 2

v 1

a

v^2

dv

Ta

0 P

0 T

b

v

P

RT

vb



RT

vb



a

v^2



a

v^2

a

0 P

0 T

b

v



R

vb

P

RT

vb



a

v^2

u 2 u 1 

T 2

T 1

cv dT

v 2

v 1

cTa

0 P

0 T

b

v

Pd dv

EXAMPLE 12–8 Internal Energy as a Function of Temperature Alone

Show that the internal energy of (a) an ideal gas and (b) an incompressible

substance is a function of temperature only, uu(T).

Solution It is to be shown that uu(T) for ideal gases and incompressible

substances.

Analysis The differential change in the internal energy of a general simple

compressible system is given by Eq. 12–29 as

(a) For an ideal gas PvRT. Then

Thus,

To complete the proof, we need to show that cvis not a function of veither.

This is done with the help of Eq. 12–42:

a

0 cv

0 v

b

T

Ta

02 P

0 T^2

b

v

ducv dT

Ta

0 P

0 T

b

v

PTa

R

v

bPPP 0

ducv dTcTa

0 P

0 T

b

v

Pd dv