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

An alternative form of this relation is obtained by using the cyclic relation:

Substituting the result into Eq. 12–45 gives

(12–46)

This relation can be expressed in terms of two other thermodynamic proper-
ties called the volume expansivityband the isothermal compressibilitya,
which are defined as (Fig. 12–10)

(12–47)

and

(12–48)

Substituting these two relations into Eq. 12–46, we obtain a third general
relation for cpcv:

(12–49)

It is called the Mayer relationin honor of the German physician and physicist
J. R. Mayer (1814–1878). We can draw several conclusions from this equation:

1.The isothermal compressibility ais a positive quantity for all sub-
stances in all phases. The volume expansivity could be negative for some
substances (such as liquid water below 4°C), but its square is always positive
or zero. The temperature Tin this relation is thermodynamic temperature,
which is also positive. Therefore we conclude that the constant-pressure spe-
cific heat is always greater than or equal to the constant-volume specific heat:

(12–50)

2.The difference between cpand cvapproaches zero as the absolute
temperature approaches zero.
3.The two specific heats are identical for truly incompressible sub-
stances since vconstant. The difference between the two specific heats is
very small and is usually disregarded for substances that are nearlyincom-
pressible, such as liquids and solids.

cp cv

cpcv

vTb^2

a

a

1

v

a

0 v

0 P

b

T

b

1

v

a

0 v

0 T

b

P

cpcvTa

0 v

0 T

b

2

P

a

0 P

0 v

b

T

a

0 P

0 T

b

v

a

0 T

0 v

b

P

a

0 v

0 P

b

T

 1 Sa

0 P

0 T

b

v

a

0 v

0 T

b

P

a

0 P

0 v

b

T

Chapter 12 | 665

20 °C

100 kPa

1 kg

21 °C

100 kPa

1 kg

20 °C

100 kPa

1 kg

21 °C

100 kPa

1 kg

(a) A substance with a large β

(b) A substance with a small β

∂––v

( ) ∂T (^) P
∂––
( ) ∂T (^) P
v
FIGURE 12–10
The volume expansivity (also called
the coefficient of volumetric
expansion) is a measure of the change
in volume with temperature at
constant pressure.
EXAMPLE 12–7 Internal Energy Change of a van der Waals Gas
Derive a relation for the internal energy change as a gas that obeys the van
der Waals equation of state. Assume that in the range of interest cvvaries
according to the relation cvc 1 c 2 T, where c 1 and c 2 are constants.
Solution A relation is to be obtained for the internal energy change of a
van der Waals gas.