8.4 The ClausiusClapeyron Equation (application of
1^{st} and 2^{nd} laws of
thermodynamics)
Until now we have only considered ideal gases and we would like to
show that the properties
,
,
, etc. are true state
variables and that the 1^{st} and
2^{nd} laws of thermodynamics hold when the working
medium is not an ideal gas (i.e. a twophase medium). An elegant
way to do this is to consider a Carnot cycle for a twophase medium.
To state the fact that all Carnot engines operated between two given
temperatures have the same efficiency is one way of stating the
2^{nd} law of thermodynamics. The working fluid need
not be an ideal gas and may be a twophase medium changing phases.
The idea is to run a Carnot engine between temperatures
and
for a twophase medium and to let it undergo a change in
phase. We can then derive an important relation known as the
ClausiusClapeyron equation, which gives the slope of the
vapor pressure curve. We could then measure the vapor pressure
curve for various substances and compare the measured slope to the
ClausiusClapeyron equation. This can then be viewed as an
experimental proof of the general validity of the
1^{st} and 2^{nd} laws of
thermodynamics!
Figure 8.8:
Carnot cycle devised to test the validity of
the laws of thermodynamics

Consider the infinitesimal Carnot cycle
shown in
Figure 8.8. Heat is absorbed between states
and
. To vaporize an arbitrary amount of mass,
, the
amount of heat

(8..1) 
must be supplied to the system. From the 1^{st} and
2^{nd} laws of thermodynamics the thermal efficiency
for a Carnot cycle can be written as
Hence, for the infinitesimal cycle considered above,

(8..2) 
The work along
and
nearly cancel such that the net work is
the difference between the work along
and
, and
can be
viewed as the area enclosed by the rectangle
:
Substituting Equations (8.1) and
(8.3) into (8.2) one
obtains
Rearranging terms yields the ClausiusClapeyron equation,
which defines the slope of the vapor pressure curve:

(8..4) 
The beauty is that we have found a general relation between
experimentally measurable quantities from first principles
(1^{st} and 2^{nd} laws of
thermodynamics).
In order to plot the ClausiusClapeyron relation and to compare it
against experimentally measured vapor pressure curves, we need to
integrate Equation (8.4). To do so, the
heat of vaporization and the specific volumes must be known
functions of temperature. This is an important problem in physical
chemistry but we shall not pursue it further here except to mention
that if
 variations in heat of vaporization can be neglected,
 the vapor phase is assumed to be an ideal gas, and
 the specific volume of the liquid is small compared to that of
the vapor phase,
the integration can be readily carried out^{8.1}. Making these approximations, the ClausiusClapeyron
equation becomes
Carrying out the integration, the resulting expression is
Note that the vapor pressure curves are straight lines if
is
plotted versus
and that the slope of the curves is
, directly related to the heat of vaporization.
Figures 8.9, 8.9, and
8.22 depict the vapor pressure curves for
various substances. The fact that all known substances in
the twophase region fulfill the ClausiusClapeyron equation
provides the general validity of the 1^{st} and
2^{nd} laws of thermodynamics!
Figure 8.9:
ClausiusClapeyron Experimental Proof
(1) [Gyarmathy, Thermodynamics I,
Eidgen
ssische Technische Hochschule
Z
rich, 1992]

Figure 8.10:
ClausiusClapeyron Experimental Proof
(2) [Mahan, Elementary Chemical Thermodynamics, 1963]

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