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

Chapter 12 | 667

For an ideal gas PRT/v. Then

Thus,

which states that cvdoes not change with specific volume. That is, cvis not
a function of specific volume either. Therefore we conclude that the internal
energy of an ideal gas is a function of temperature only (Fig. 12–11).

(b) For an incompressible substance, vconstant and thus dv0. Also
from Eq. 12–49, cpcvcsince ab0 for incompressible substances.
Then Eq. 12–29 reduces to

Again we need to show that the specific heat cdepends on temperature only
and not on pressure or specific volume. This is done with the help of
Eq. 12–43:

since v constant. Therefore, we conclude that the internal energy of a
truly incompressible substance depends on temperature only.

a

0 cp

0 P

b

T

Ta

02 v

0 T^2

b

P

 0

duc dT

a

0 cv

0 v

b

T

 0

a

0 P

0 T

b

v



R

v

¬and¬a

02 P

0 T^2

b

v

c

01 R>v 2

0 T

d

v

 0

EXAMPLE 12–9 The Specific Heat Difference of an Ideal Gas

Show that cpcvRfor an ideal gas.

Solution It is to be shown that the specific heat difference for an ideal gas
is equal to its gas constant.
Analysis This relation is easily proved by showing that the right-hand side
of Eq. 12–46 is equivalent to the gas constant Rof the ideal gas:

Substituting,

Therefore,
cpcvR

Ta

0 v

0 T

b

2

P

a

0 P

0 v

b

T

Ta

R

P

b

2

a

P

v

bR

v

RT

P

Sa

0 v

0 T

b

2

P

a

R

P

b

2

P

RT

v

Sa

0 P

0 v

b

T



RT

v^2



P

v

cpcvTa

0 v

0 T

b

2

P

a

0 P

0 v

b

T

AIR

LAKE

u = uu = u((TT))

cv = cv (T)

cp = cp(T)

u = u(T)

c = c(T)

FIGURE 12–11

The internal energies and specific

heats of ideal gases and

incompressible substances depend on

temperature only.