The Second Law of Thermodynamics
The Second Law
In our qualitative description of processes we have already emphasized the trend of any isolated system towards an unique and stable equilibrium state. The Second Law of Thermodynamics is the quantitative formulation of this observation. Its importance goes well beyond the computation of the unique equilibrium states for isolated systems. In particular, as will be seen, it gives strong restrictions for the efficiency of energy conversion systems, and thus is of enormous importance for engineering applications.
The original derivation of the second law through Rudolf Clausius (1822–1888) was based on the argument that the direction of heat transfer is restricted and then relied heavily on statements on thermodynamic cycles. The following derivation postulates an inequality to describe the trend to equilibrium, and uses arguments on process direction for simple equilibration processes to identify terms in the postulated equation. This approach allows us to introduce the second law quite early, before any thermodynamic processes and cycles are discussed. With this, entropy and the second law will be available for the evaluation of processes and cycles from the start. All equations and conclusions agree to the classical approach, as presented in most textbooks on engineering thermodynamics, just the order of arguments is different.
Entropy and the Trend to Equilibrium
To set the stage, we briefly summarize our earlier statements on processes in closed systems: a closed system can be manipulated by exchange of work and heat with its surroundings only. In non-equilibrium—i.e., irreversible— processes, when all manipulation stops, the system will undergo further changes until it reaches a final equilibrium state. This equilibrium state is stable, that is the system will not leave the equilibrium state spontaneously.
It requires new action—exchange of work or heat with the surroundings—to change the state of the system.
The following non-equilibrium processes are well-known from experience, and will be used in the considerations below: (a) Heat goes from hot to cold. When two bodies at different temperatures are brought into thermal contact, heat will flow from the hotter to the colder body until both reach their common equilibrium temperature. (b) Work can be transferred without restriction, by means of gears and levers. However, in transfer some work might be lost to friction.
The process from an initial non-equilibrium state to the final equilibrium state requires some time. However, if the actions on the system (only work and heat!) are sufficiently slow, the system has enough time to adapt and will be in equilibrium states at all times. We speak of quasi-static—or, reversible— processes. When the slow manipulation is stopped at any time, no further changes occur.
The behavior of isolated systems described above—a change occurs until a stable state is reached—can be described mathematically by an inequality.
The final stable state must be a maximum (alternatively, a minimum) of a suitable extensive property describing the system. We call that extensive property entropy, denoted S, and write an inequality for the isolated system,
S˙gen is called the entropy generation rate. The entropy generation rate is pos- itive in non-equilibrium (S˙gen > 0), and vanishes in equilibrium (S˙gen = 0). The new equation (4.1) states that in an isolated system the entropy will grow in time ( dS > 0) until the stable equilibrium state is reached ( dS = 0).
Non-zero entropy generation describes the irreversible process towards equilibrium, e.g., through internal heat transfer and friction. There is no entropy generation in equilibrium. Since entropy only grows before the equilibrium state is reached, the latter is a maximum of entropy.
The above postulation of an inequality is based on phenomenological arguments. The discussion of irreversible processes has shown that all isolated systems will in time evolve to a unique equilibrium state. The first law alone does not suffice to describe this behavior. We have seen this in the description of temperature equilibration in Sec. 3.12, where the first law has infinitely many solutions for the final temperatures TA, TB , and additional input is needed to state that TA = TB in equilibrium. Above, we relied on experience as additional input, the second law is a formalization of that experience. Non- equilibrium processes aim to reach equilibrium, and the inequality is required to describe the clear direction in time.
In the next sections we will extend the second law to non-isolated system, and identify entropy as a measurable property.