The Second Law

The second law of thermodynamics concerns entropy and the spontaneity of processes. This chapter discusses theoretical aspects and practical applications. 

The Second Law establishes the directionality of the processes taking place in the system by defining a new function, entropy S. 

A more intuitive definition of entropy is in terms of probability a more random system has higher probability and therefore higher entropy. 

For any process to take place spontaneously, the Gibbs free energy must decrease. 

the aforementioned directionality arises from the combination of (A. 11) and (A. 13). All closed systems tend to the state of maximum entropy (minimum free energy), that is, the equilibrium. 

The state of the system at which there is no change of the Gibbs free energy (dG = 0) or of the entropy (dS = 0), is called the equilibrium. For an isothermal process (AT = 0) and two finite states, (A. 13) can be restated as 

The second law of thermodynamics introduces a new function of state, the entropy, S, in order to quantify the spontaneity and direction of change for natural systems 

There exists a thermodynamic function S, called the entropy, such that the change in entropy AS in going from an initial state i to a final state f has the following property 

The equal sign in Equation 4.6 only applies if the process is carried out reversibly. Note that Equation 4.6 contains the recipe for obtaining AS exactly by carrying out the change of state reversibly. 

It follows from Equation 4.6 for a thermally isolated system (hence q = 0) that 

In Chapter 9, we stressed the importance of thermodynamics in terms of the way human society uses energy. At that time, we noticed that whenever there is an attempt to convert energy from one form into another, some energy is lost or wasted. In other words, not all of the energy potentially available is directed into the desired process. How does this fact arise from thermodynamics Entropy provides the key to understanding that the loss of useful energy is inevitable. 

There are several equivalent ways to state the second law, but all lead to the same interpretations. 

In considering the energy economy, we alluded to the second law in conjunction with the notion that it is impossible to convert heat completely to work That is one way to express the second law of thermodynamics. Now let s try to understand why this is true. First consider heat. Heat flows due to random collisions of molecules, and an increase in temperature increases the random motions of molecules. Work, by contrast, requires moving a mass some distance. To yield a net movement, there must be a direction associated with a motion, and that direction implies that there is an order to the motion. Converting heat into work, therefore, is a process that moves from random motions toward more ordered ones. We have just seen how this type of change goes against nature s tendency to favor a more probable state (the more random one). How can we connect these ideas with entropy  

To make this connection, we must be careful to realize that changes in the universe involve both the system and its surroundings. If we focus on the system alone, we cannot understand how order is created at all. Yet, the synthesis of polymers shows that it does happen, as do everyday situations such as the growth of plants, animals, and people. To express the second law of thermodynamics in terms of entropy, we must focus on the total change in entropy for the universe, AS,. 

Because nature always tends to proceed toward a more probable state, we can assert an equivalent form of the second law of thermodynamics In any spontaneous process, the total entropy change of the universe is positive, (ASu 0). That this statement of the second law is equivalent to our original version is not at all obvious. But remember, energy that is not converted into work (a process that would decrease entropy) is transferred to the surroundings as heat. Thus the entropy of the surroundings increases, and the total entropy change in the system and surroundings is positive. 

The question of whether or not a process can occur is answered by the second law of thermodynamics. The mathematical statement of the 

The inequality in Equation (1.19) pertains to a process that will tend to occur irreversibly (spontaneously), whereas the equality pertains to a reversible process, i.e. one in which the system is never displaced from equilibrium by a finite amount. An equivalent expression for the second law, which does not require the constraint of an isolated system, is 

The surroundings are presumed to behave reversibly, so that Equation (1.21) may be written as 

Note that, for the special case of constant pressure and 8w = 0, Equation (1.20) may be written as 

although depends on path, dividing by P gives a change in a state function. 

That is, the reciprocal pressure is an integrating factor that converts into an 

However, it is not immediately obvious how to convert 8Q into an exact differential. We might tentatively guess that a form analogous to (2.3.2) can be found that is, perhaps there is another integrating factor X such that 

Here S stands for a new state function and the identity of the integrating factor X is yet to be discovered. The objectives of this section are to develop (2.3.3) and identify X. 

Our presentation of the second law is based on the rigorous development by Car-ath odory [2]. Caratheodory s approach has been described in detail by Chandrasekhar [3] and Kestin [4], so we need only outline the arguments here. 

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In Section XVII-16C there is mention of S-shaped isotherms being obtained. That is, as pressure increased, the amount adsorbed increased, then decreased, then increased again. If this is equilibrium behavior, explain whether a violation of the second law of thermodynamics is implied. A sketch of such an isotherm is shown for nitrogen adsorbed on a microporous carbon (see Ref. 226). 

As we shall see, because of the limitations that the second law of thennodynamics imposes, it may be impossible to find any adiabatic paths from a particular state A to another state B because In this... 

Obviously die first law is not all there is to the structure of themiodynamics, since some adiabatic changes occur spontaneously while the reverse process never occurs. An aspect of the second law is that a state fimction, the entropy S, is found that increases in a spontaneous adiabatic process and remains unchanged in a reversible adiabatic process it caimot decrease in any adiabatic process. 

The next few sections deal with the way these experimental results can be developed into a mathematical system. A reader prepared to accept the second law on faith, and who is interested primarily in applications, may skip section A2.1.4.2 and section A2.1.4.6 and perhaps even A2.1.4.7. and go to the final statement in section A2.1.4.8. 

The surfaces in which the paths satisfying the condition = 0 must lie are, thus, surfaces of constant entropy they do not intersect and can be arranged in an order of increasing or decreasmg numerical value of the constant. S. One half of the second law of thennodynamics, namely that for reversible changes, is now established. 

The total change d.S can be detennined, as has been seen, by driving the subsystem a back to its initial state, but the separation into dj.S and dj S is sometimes ambiguous. Any statistical mechanical interpretation of the second law requires that, at least for any volume element of macroscopic size, dj.S > 0. However, the total... 

There are many equivalent statements of the second law, some of which involve statements about heat engines and perpetual motion machines of the second kind that appear superficially quite different from equation (A2.T21). They will not be dealt with here, but two variant fonns of equation (A2.T21) may be noted in... 

One may note, in concluding this discussion of the second law, that in a sense the zeroth law (thennal equilibrium) presupposes the second. Were there no irreversible processes, no tendency to move toward equilibrium rather than away from it, the concepts of thennal equilibrium and of temperature would be meaningless. 

Snch a generalization is consistent with the Second Law of Thennodynamics, since the //theorem and the generalized definition of entropy together lead to the conchision that the entropy of an isolated non-eqnilibrium system increases monotonically, as it approaches equilibrium. 

In 1872, Boltzmaim introduced the basic equation of transport theory for dilute gases. His equation detemiines the time-dependent position and velocity distribution fiinction for the molecules in a dilute gas, which we have denoted by /(r,v,0- Here we present his derivation and some of its major consequences, particularly the so-called //-tlieorem, which shows the consistency of the Boltzmann equation with the irreversible fomi of the second law of themiodynamics. We also briefly discuss some of the famous debates surrounding the mechanical foundations of this equation. 

This completes the heuristic derivation of the Boltzmann transport equation. Now we trim to Boltzmaim s argument that his equation implies the Clausius fonn of the second law of thennodynamics, namely, that the entropy of an isolated system will increase as the result of any irreversible process taking place in the system. This result is referred to as Boltzmann s H-theorem. 

We are going to carry out some spatial integrations here. We suppose that tire distribution function vanishes at the surface of the container and that there is no flow of energy or momentum into or out of the container. (We mention in passing that it is possible to relax this latter condition and thereby obtain a more general fonn of the second law than we discuss here. This requires a carefiil analysis of the wall-collision temi The interested reader is referred to the article by Dorfman and van Beijeren [14]. Here, we will drop the wall operator since for the purposes of this discussion it merely ensures tliat the distribution fiinction vanishes at the surface of the container.) The first temi can be written as... 

The are many ways to define the rate of a chemical reaction. The most general definition uses the rate of change of a themiodynamic state function. Following the second law of themiodynamics, for example, the change of entropy S with time t would be an appropriate definition under reaction conditions at constant energy U and volume V ... 

The Boltzmann distribution is fundamental to statistical mechanics. The Boltzmann distribution is derived by maximising the entropy of the system (in accordance with the second law of thermodynamics) subject to the constraints on the system. Let us consider a system containing N particles (atoms or molecules) such that the energy levels of the... 

The Carnot cycle is formulated directly from the second law of thermodynamics. It is a perfectly reversible, adiabatic cycle consisting of two constant entropy processes and two constant temperature processes. It defines the ultimate efficiency for any process operating between two temperatures. The coefficient of performance (COP) of the reverse Carnot cycle (refrigerator) is expressed as... 

Thermodynamics is a deductive science built on the foundation of two fundamental laws that circumscribe the behavior of macroscopic systems the first law of thermodynamics affirms the principle of energy conservation the second law states the principle of entropy increase. In-depth treatments of thermodynamics may be found in References 1—7. 

Under isothermal conditions where energy is not added or removed from the system, the second law of thermodynamics obtains, and... 

On purely kinetic grounds, however, the term random must be used carefully in describing a MaxweUian gas. The probabUity of a MaxweUian gas entering a duct is not a random function. This probabUity is proportional to the cosine of the angle between the molecular trajectory and the normal to the entrance plane of the duct. The latter assumption is consistent with the second law of thermodynamics, whereas assuming a random distribution entry is not. 

If FCCU operations are not changed to accommodate changes ia feed or catalyst quaUty, then the amount of heat required to satisfy the heat balance essentially does not change. Thus the amount of coke burned ia the regenerator expressed as a percent of feed does not change. The consistency of the coke yield, arising from its dependence on the FCCU heat balance, has been classified as the second law of catalytic cracking (7). 

The second law of thermodynamics focuses on the quaUty, or value, of energy. The measure of quaUty is the fraction of a given quantity of energy that can be converted to work. What is valued in energy purchased is the abiUty to do work. Electricity, for example, can be totally converted to work, whereas only a small fraction of the heat rejected to a cooling tower can make this transition. As a result, electricity is a much more valuable and more costly commodity. 

Unlike the conservation guaranteed by the first law, the second law states that every operation involves some loss of work potential, or exergy. The second law is a very powerful tool for process analysis, because this law tells what is theoretically possible, and pinpoints the quantitative loss in work potential at different points in a process. 

The second law can also suggest appropriate corrective action. Eor example, in combustion, preheating the air or firing at high pressure in a gas turbine, as is done for an ethylene (qv) cracking furnace, improves energy efficiency by reducing the lost work of combustion (Eig. 4). 

In the same way that the first law of thermodynamics cannot be formulated without the prior recognition of internal energy as a property, so also the second law can have no complete and quantitative expression without a prior assertion of the existence of entropy as a property. 

The second law reqmres that the entropy of an isolated system either increase or, in the limit, where the system has reached an equilibrium state, remain constant. For a closed (but not isolated) system it requires that any entropy decrease in either the system or its surroundings be more than compensated by an entropy increase in the other part or that in the Emit, where the process is reversible, the total entropy of the system plus its surroundings be constant. 

Since heat transfer with respec t to the surroundings and with respect to the system are equal but of opposite sign, = —Q. Moreover, the second law requires for a reversible process that the entropy changes of system and surroundings be equalbut of opposite sign AS = —AS Equation (4-356) can therefore be written Q = TcAS In terms of rates this becomes... 

Thermodynamic Analyses of Cycles The thermodynamic quahty measure of either a piece of equipment or an entire process is its reversibility. The second law, or more precisely the entropy increase, is an effective guide to this degree of irreversibility. However, to obtain a clearer picture of what these entropy increases mean, it has become convenient to relate such an analysis to the additional work that is required to overcome these irreversibihties. The fundamental equation for such an analysis is... 

Hence, the second-law efficiency of the expander-heat-exchanger-compressor system is p p... 

As pointed out in Section 2.4, shock waves are such rapid processes that there is no time for heat to flow into the system from the surroundings they are considered to be adiabatic. By the second law of thermodynamics, the quantity (S — Sg) must be positive for any thermodynamic process in an isolated system. According to (2.54), this quantity can only be positive if the P-V isentrope is concave upward. Thus, the thermodynamic stability condition for a shock wave is... 

A thermodynamic change can take place in two ways - either reversibly, or irreversibly. In a reversible change, all the processes take place as efficiently as the second law of thermodynamics will allow them to. In this case the second law tells us that... 

The second law of thermodynamics was actually postulated by Carnot prior to the development of the first law. The original statements made concerning the second law were negative—they said what would not happen. The second law states that heat will not flow, in itself, from cold to hot. While no mathematical relationships come directly from the second law, a set of equations can be developed by adding a few assumptions for use in compressor analysis. For a reversible process, entropy, s, can be defined in differential form as... 

It is recognized that a truly reversible process does not exist in the real world. II it is further recognized that real processes result in an increase in enl ropy, the second law can be stated. 

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