second law of thermodynamics is developed for systems undergoing cycles
while exchanging heat with two reservoirs. One corollary of
the second law is known as the Clausius inequality, which states that
the cyclic integral of is
always less than or equal to zero. That is,
delta;Q = differential heat transfer at the system
boundary during a cycle
T = absolute temperature at the
integration over the entire cycle
The Clausius inequality is valid for all cycles, reversible or irreversible.
Demonstration of the
To demonstrate the Clausius inequality, consider two
heat engines that work between the same two reservoirs. Heat
engine A is an reversible heat engine and heat engine B is an irreversible
one. Both of these two engines absorb
heat QH from a heat source, which has a temperature of TH and
reject heat to a heat sink at temperature TL. Applying the
first law to both of the engines gives,
WA,rev = QH -
WB,irr = QH - QB,L
Since a reversible engine has the highest efficiency of any heat engines
working between the same source and sink, a reversible
heat engine produces more work than an irreversible
heat engine for the
same heat input QH.
WA,rev = QH -
QA,L > WB,irr =
QH - QB,L
QA,L < QB,L
Since the temperature TH is constant during the heat transfer
QH, and the temperature TL is constant during the
heat transfer QA,L and QB,L, the cyclic integral
both heat engines is
Comparing the above two equations with QA,L < QB,L yields,
A reversible heat engine holds the following relation:
Therefore, the cyclic integral
a reversible engine is zero.
Combining the above equation with equation (1) gives,
To summarize the above analysis, the Clausius inequality states
internally reversible cycles
From the discussion above, ,
the Clausius inequality states, for an internally reversible cycle
The cyclic integral of is
zero. Therefore, is
a property. Clausius in 1865 named this new thermodynamic property entropy. The definition is
The entropy change for process1-2 can be determined as
The Entropy Change between
two Specified States are the
Same Regardless the
Process Is Reversible or Not
Note that the above definition actually defines the change of entropy, instead of entropy itself.
Because entropy is a property, the entropy change between two specified states is the same
whether the process is reversible or not.
The previous discussion only means that entropy is a useful property in the second law analysis of an engineering device. But what is entropy? Entropy can be viewed as a measure of molecular disorder. In other words, entropy is the amount of disorder in a system. A pure substance at absolute zero temperature is in perfect order, and its entropy is zero. This is the third law of thermodynamics.
|The entropy of a pure crystallite
substance at absolute zero is zero.
If the substance is not a pure crystal like substance, its entropy at absolute zero is not zero.
When the molecular disorder is produced, the ability of doing work is reduced.
Consider a cycle as described in the diagram at the left. Peocess1-2 is an arbitrary (reversible or irreversible) process, Process 2-1 is an internally reversible process. From the Clausius inequality,
where the equality holds for internally reversible process and inequality holds for irreversible process.
Negative Entropy Change of a System
Entropy is an extensive property, and thus the total entropy of a
system is equal to the sum of the entropies of all parts of the system.
The combination of a system and its surroundings is an isolated system.
For this case, the increase of entropy principle is
ΔSisolated = ΔSsys + ΔSsurr 0
The entropy change of a system can be negative. But the entropy change
of an isolated system formed by this system and its surroundings
can not be negative.
The second law of thermodynamics states that processes can occur only
in a certain direction. But which direction?
The increase of entropy principle answers this question: a process must
proceed in the direction that complies with the increase of entropy principle.
A process that violates this principle is impossible.