This example of Carnot cycle demonstrates how the introduction of entropy by the
second law facilitates idealized power-cycle analyses. Various degrees of removal of
idealization to representing real cycles can be considered. In those considerations, the
departure of real power-cycle steps from idealized isentropic power-cycle steps is
associated with entropy increase. Thus, universal entropy increase as predicted by the
entropy law is usually interpreted to be the central meaning of the second law as the
law of dissipation. But, there is a more important meaning of the second law as the law
of driving force—as discussion in the two sections below suggests.
5.9 Mixtures of Ideal Gases and Their Properties
The specific internal energy and the enthalpy of ideal gases are functions of temperature alone. The total values of internal energy and enthalpy for a mixture can be
expressed as the sum of the products of the mole fraction and the specific internal
energy or enthalpy of each component in the mixture
U T
ð Þ ¼
X n
i¼1
N i u i ¼ N
X n
i¼1
x i u i T
ð Þ
ð76Þ
H T
ð Þ ¼
X n
i¼1
N i h i ¼ N
X n
i¼1
x i h i T
ð Þ
ð77Þ
The entropy of each ideal gas depends on temperature and volume or temperature and pressure. The determination of the total value for a mixture requires the
application of Gibbs’ theorem.
Gibbs’ theorem is demonstrated by a thought experiment: Consider an isolated
composite system made of one stationary cylinder- chamber of volume V and its
extension chamber of equal volume, and one sliding cylinder (on the right in the figure
on next page) of equal volume (see Figs. 5.5 and 5.6). The stationary cylinder chamber
with its extension chamber is made of three walls: two end walls and a middle partition, which separates gas A in the stationary cylinder chamber from the extension
chamber. Initially, the sliding cylinder is occupied with gas B at the same temperature
of gas A. Of the three fixed walls, the middle partition is a semi-permeable partition
permeable to gas B but impermeable to gas A, while both end walls are impermeable.
Of the two walls of the sliding cylinder, the left wall is a semi-permeable membrane
permeable to A, but impermeable to B, while the right wall is an impermeable wall.
Initially, the sliding cylinder and the extension chamber are coincident in space
with gas B inside the cylinder (as shown in the top view of Fig. 5.5). Reversible
mixing can take place by sliding the sliding cylinder into the stationary cylinder.
Imagine an intermediate position of the sliding cylinder in the middle of the process
as shown in Fig. 5.6. And imagine that there are four spaces (designated as a, b, c,
5.8 Isentropic Processes and Carnot Cycles
119
second law facilitates idealized power-cycle analyses. Various degrees of removal of
idealization to representing real cycles can be considered. In those considerations, the
departure of real power-cycle steps from idealized isentropic power-cycle steps is
associated with entropy increase. Thus, universal entropy increase as predicted by the
entropy law is usually interpreted to be the central meaning of the second law as the
law of dissipation. But, there is a more important meaning of the second law as the law
of driving force—as discussion in the two sections below suggests.
5.9 Mixtures of Ideal Gases and Their Properties
The specific internal energy and the enthalpy of ideal gases are functions of temperature alone. The total values of internal energy and enthalpy for a mixture can be
expressed as the sum of the products of the mole fraction and the specific internal
energy or enthalpy of each component in the mixture
U T
ð Þ ¼
X n
i¼1
N i u i ¼ N
X n
i¼1
x i u i T
ð Þ
ð76Þ
H T
ð Þ ¼
X n
i¼1
N i h i ¼ N
X n
i¼1
x i h i T
ð Þ
ð77Þ
The entropy of each ideal gas depends on temperature and volume or temperature and pressure. The determination of the total value for a mixture requires the
application of Gibbs’ theorem.
Gibbs’ theorem is demonstrated by a thought experiment: Consider an isolated
composite system made of one stationary cylinder- chamber of volume V and its
extension chamber of equal volume, and one sliding cylinder (on the right in the figure
on next page) of equal volume (see Figs. 5.5 and 5.6). The stationary cylinder chamber
with its extension chamber is made of three walls: two end walls and a middle partition, which separates gas A in the stationary cylinder chamber from the extension
chamber. Initially, the sliding cylinder is occupied with gas B at the same temperature
of gas A. Of the three fixed walls, the middle partition is a semi-permeable partition
permeable to gas B but impermeable to gas A, while both end walls are impermeable.
Of the two walls of the sliding cylinder, the left wall is a semi-permeable membrane
permeable to A, but impermeable to B, while the right wall is an impermeable wall.
Initially, the sliding cylinder and the extension chamber are coincident in space
with gas B inside the cylinder (as shown in the top view of Fig. 5.5). Reversible
mixing can take place by sliding the sliding cylinder into the stationary cylinder.
Imagine an intermediate position of the sliding cylinder in the middle of the process
as shown in Fig. 5.6. And imagine that there are four spaces (designated as a, b, c,
5.8 Isentropic Processes and Carnot Cycles
119