Figure 1
Two reversible cycles demonstrating the fact that entropy is a property of a substance. P is pressure and V is volume.
By applying the inequality of Clausius and Figure 1, it can be demonstrated that the second law of thermodynamics leads to a property of a system that is called entropy. Let a system (control mass) undergo a reversible process from state 1 to state 2 along a path A, and let the cycle be completed along path B, which is also reversible.
Because this is a reversible cycle, it can be written:
Subtracting the second equation from the first, the following is:
∫
2
1
(
δQ
T
)
A
=
∫
2
1
(
δQ
T
)
C
Since the ∮δQ/T is the same for all reversible paths between states 1 and 2, it is concluded that this quantity is independent of the path and it is a function of the end states only; it is therefore a property. This property is called entropy and is designated S. It follows that entropy may be defined as a property of a substance in accordance with the relation:
(Eq1)
dS ≡
(
δQ
T
)
rev
Entropy is an extensive property, and the entropy per unit mass is designated s. It is important to note that entropy is defined here in terms of a reversible process.
The change in entropy of a system as it undergoes a change of state may be found by integrating Eq1 and getting:
(Eq2)
S2 − S1 =
∫
2
1
(
δQ
T
)
rev
To perform this integration, the relation between T and Q must be known. The important idea is that since entropy is a property, the change in entropy of a substance in going from one state to another is the same for all processes, both reversible and irreversible, between these two states. Eq2 enables one to find the change in entropy only along a reversible path. However, once the change has been evaluated, this value is the magnitude of the entropy change for all processes between these two states.
Eq2 enables one to calculate changes of entropy, but it tells nothing about absolute values of entropy. From the third law of thermodynamics, which is based on observations of low-temperature chemical reactions, it is concluded that the entropy of all pure substances (in the appropriate structural form) can be assigned the absolute value of zero at the absolute zero of temperature. It also follows from the subject of statistical thermodynamics that all pure substances in the (hypothetical) ideal-gas state at absolute zero temperature have zero entropy.
However, when there is no change of composition, as would occur in a chemical reaction, for example, it is quite adequate to give values of entropy relative to some arbitrarily selected reference state, such as was done earlier when tabulating values of iternal energy and enthalpy. In each case, whatever reference value is chosen, it will cancel out when the change of property is calculated between any two states. This is the procedure followed with the thermodynamic tables that is discussed in the lesson Entropy of a Pure Substance.
A word should be added here regarding the role of T as an integrating factor. It is noted in the lesson Heat that Q is a path function, and therefore δQ is an inexact differential. However, since (δQ/T)rev is a thermodynamic property, it is an exact differential. From a mathematical perspective, it is noted that an inexact differential may be converted to an exact differential by the introduction of an integrating factor. Therefore, 1/T serves as the integrating factor in converting the inexact differential δQ to the exact differential δQ/T for a reversible process.
