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

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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
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