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

volume. That is,Psatf(Tsat). Therefore, the partial derivative (P/T)vcan
be expressed as a total derivative (dP/dT)sat, which is the slope of the satu-
ration curve on a P-Tdiagram at a specified saturation state (Fig. 12–9).
This slope is independent of the specific volume, and thus it can be treated
as a constant during the integration of Eq. 12–18 between two saturation
states at the same temperature. For an isothermal liquid–vapor phase-change
process, for example, the integration yields

(12–20)

or

(12–21)

During this process the pressure also remains constant. Therefore, from
Eq. 12–11,

Substituting this result into Eq. 12–21, we obtain

(12–22)

which is called the Clapeyron equation after the French engineer and
physicist E. Clapeyron (1799–1864). This is an important thermodynamic
relation since it enables us to determine the enthalpy of vaporization hfgat a
given temperature by simply measuring the slope of the saturation curve on
a P-Tdiagram and the specific volume of saturated liquid and saturated
vapor at the given temperature.
The Clapeyron equation is applicable to any phase-change process that
occurs at constant temperature and pressure. It can be expressed in a general
form as

(12–23)

where the subscripts 1 and 2 indicate the two phases.

a

dP

dT

b

sat



h 12

Tv 12

a

dP

dT

b

sat



hfg

Tvfg

dhT dsv dP¬¬S

g

f

dh

g

f

T dsShfgTsfg

a

dP

dT

b

sat



sfg

vfg

sgsfa

dP

dT

b

sat

1 vgvf 2

Chapter 12 | 659

P

T T

= const.

LIQUID

SOLID

VAPOR

∂––P

( )∂T (^) sat
FIGURE 12–9
The slope of the saturation curve on a
P-Tdiagram is constant at a constant
Tor P.
EXAMPLE 12–5 Evaluating the hfgof a Substance from
the P-v-TData
Using the Clapeyron equation, estimate the value of the enthalpy of vaporiza-
tion of refrigerant-134a at 20°C, and compare it with the tabulated value.
Solution The hfgof refrigerant-134a is to be determined using the Clapeyron
equation.
Analysis From Eq. 12–22,
hfgTvfga
dP
dT
b
sat
→
0