Thermodynamic Systems Basic Concepts

1.6 Zeroth Law of Thermodynamics and Absolute Temperature
Thermometers with liquid working fluids are usually used for measurement of temperature. When such a device is brought in contact with a body whose temperature is to be measured, the liquid column inside the thermometer expands due to heat conducted from the body. The expanded length can be said to represent the degree of hotnessin a somewhat quantitative manner.  
The Zeroth Law of Thermodynamics states that if two bodies are in thermal equilibrium with a third body, then the two given bodies will be in thermal equilibrium with each other. The zeroth law of thermodynamics is used for measurement of temperature. In the Celsius temperature scale, two fixed points – ice point and steam point – are used to devise the scale. Thus, the freezing point of water (at standard atmospheric pressure) is assigned a value of zero, while the boiling point of pure water (at standard atmospheric pressure) denoted as 100.  However for introducing detail, the distance between the two end points of the liquid column marks is arbitrarily divided into 100 equal spaces called degrees. This exercise can be extended both below zero and above 100 to expand the range of the thermometer.
The entire exercise can be carried out with any other substance as the thermometric fluid. However, for any specific measured temperature the extent of expansion of the liquid column will vary with the thermometric fluid as each fluid would expand to different extent under the influence of temperature. To overcome this problem, the ideal gas (see next section) has been arbitrarily chosen as the thermometric fluid. Accordingly, the temperature scale of the SI system is then described by the Kelvin unit (T0K). Its relation to the Celsius (t0C) scale is given by: 
Thus the lower limit of temperature, called absolute zero on the Kelvin scale, occurs at –273.150C.

1.7 The Ideal Gas
In the foregoing discussions we have pointed out that a thermodynamic system typically encloses a fluid (pure gas, liquid or solid or a mixture) within its boundary. The simplest of the intensive variables that can be used to define its state are temperature, pressure and molar volume (or density), and composition (in case of mixtures). Let us consider for example a pure gas in a vessel. As mentioned above, by phase rule the system has two degrees of freedom. It is an experimentally observed phenomenon that in an equilibrium state the intensive variables such as pressure, temperature and volume obey a definitive inter-relationship, which in its simplest form is expressed mathematically by the Boyle’s and Charles’s laws. These laws are compositely expressed in the form of the following equationthat is said to represent a behaviour termed as Ideal Gas Law
Where, P = system pressure (say, Pa = N/m2), T = system temperature (in 0K), V = gas molar volume (mol/m3) and, R = universal gas constant. The above relation is said to represent an equation of state, and may alternately be written as:
Where, Vt = total system volume; = total moles of gas in the system. Units of typical thermodynamic variables and that of the gas constant are provided in  Appendix I.
The equations (1.10) and (1.11) are also termed Equations of State (EOS) as they relate the variables that represent the thermodynamic state of a system in the simplest possible manner. It is obvious that the EOS indicates that if one fixes temperature and pressure the molar volume is automatically fixed as well, i.e., the latter is not an independent property in such a case. 
The ideal gas law is a limiting law in the sense that it is valid primarily for gaseous systems at low pressure, strictly speaking at pressure far below the atmospheric. However, for practical purposes it is observed to remain valid at atmospheric pressures as well. As we shall see later, the ideal gas law serves as a very useful approximation as well as a datum for estimation of both the volumetric (chapter 2) as well as all other real fluid thermodynamic properties of practical interest.

1.8 State and Path Dependent Thermodynamic Variables 
Consider a gas at a certain temperature and a pressure within a piston-cylinder assembly (for example, fig. 1.2), which for arguments’ sake we may assume to be isolated. If the piston position is held fixed at this point the gas state is said to be characterized by the temperature and the pressure and its corresponding volume. In its simplest form the relationship between these intensive variables may be described by (say) eqn. 1.12. Consider next that the gas is compressed by application of an extra force on the piston so that it moves inwards into the cylinder. This motion will continue till it reaches a point when the internal gas pressure equals the externally applied pressure on the piston. If there is no further increase in the force applied to the piston, the gas will also attain a new equilibrium state wherein the pressure and temperature would attain a new set of values. If, on the other hand the extra applied pressure is removed and the gas reverts to the earlier state the original temperature and pressure (and, of course volume) is restored. Extending this argument, in general, if the gas is heated or cooled, compressed or expanded, and then returned to its initial temperature and pressure, its intensive properties are restored to their initial values. It is evident, therefore, that such properties do not depend on the past history of the fluid or on the path by which it reaches a given state. They depend only on present state, irrespective of how they are attained. Such quantities are thus defined as state variables. Mathematically, this idea may be expressed as follows:
The changes in the above intensive properties depend only on the initial and final states of the system. They constitute point functions and their differentials are exact.

Let us next consider the case of thermodynamic work as defined by eqn. 1.6. It may be readily evident that if one can depict the exact variation of pressure and volume during a change of state of a system on a two-dimensional P-V graph, the area under the curve between the initial and final volumes equal the work associated with process. This is illustrated in fig. 1.4.
Fig. 1.4: Depiction of thermodynamic work on P-V plot
As shown in the above figure the work associated with a thermodynamic process clearly in dependent on the path followed in terms of P and V. It follows that if one were to go from state ‘1’ to ‘2’ by path X and then return to ‘1’ by path Y the work in the two processes would differ and so one would not be giving and taking work out of the system in equal measure. An entity such as P-V work is, therefore, described as a path variable, and therefore is not directly dependent on the state of the system. This is obviously distinctive from the case of state variables such as P and V (and T). Thus, for quantifying work, one cannot write an equation of the same type as (1.12). The more appropriate relation for such variables may be written as: 
It may be pointed out that the notation is used to depict differential quantum of work in order to distinguish it from the differential quantity of a state variable as in eqn. 1.14. We demonstrate in chapter 3 that, like P-V work, heat transferred between a system and the surrounding is also a path variable and so one may also write:
Heat and work are therefore quantities, and not properties; they account for the energy changes that occur in the system and surroundings and appear only when changes occur in a system. Although time is not a thermodynamic coordinate, the passage of time is inevitable whenever heat is transferred or work is accomplished.

1.9 Reversible and Irreversible Thermodynamic Processes
We have seen above that in absence of any gradients (or motive forces) a thermodynamic system continues to remain in a state of equilibrium. Obviously, if a disturbance (i.e., mechanical, thermal or chemical potential gradient) is impressed upon such a system it will transit from its initial state of equilibrium. However, as it moves away from its initial state the originally applied gradients will diminish progressively in time, and ultimately when they are reduced to infinitesimal levels the system will attain a new equilibrium state. A question arises here as to the nature of the process of change: if the initially impressed disturbances are reversed in direction (not magnitude) can the system return to its first equilibrium state back through the same intermediate states as it went through during the first phase of change? If that happens we depict the process as reversible, if not, then the process is termed irreversible
It is necessary to understand the concept of reversibility of thermodynamic process more deeply as it is an idealized form of process of change and without that consideration it is not possible to represent or understand real thermodynamic processes, which are generally irreversible in nature.

What makes a thermodynamic process reversible? To answer the question let us again take the example of the simple gas-in-piston-and-cylinder system as shown in figure 1.5.
Fig. 1.5 Illustration of Reversibility of Thermodynamic Process
The system initially contains a pure gas whose pressure equals that exerted externally (due to piston weight), and its temperature is the same as that of the environment. Thus it is at equilibrium (say state ‘A’) as there are no mechanical, thermal or chemical concentration gradients in the system. Now a ball of a known weight is transferred on to the piston, whereupon the external pressure exceeds the gas pressure and the piston moves down to attain a new lower position at which point the gas has been compressed and its pressure once again equals that applied externally. At the same time if any differentials in temperature (within or across the system boundary) and internal concentration distribution of the gas molecules result due to the applied mechanical imbalance, heat and mass transfer will take place simultaneously until these gradients are also annulled and the system eventually comes to rest at a new equilibrium point (say, ‘B’). We say that the system has undergone a process due to which its state has changed from A to B. Note that this process can be continued as long as desires by sequentially transferring more and more balls individually onto the piston and impelling the system to change in steps till say the end point state ‘X’. The question that one may pose: is the process A-X reversible?  That is, if one reversed all the initial steps of sequentially moving each ball off the piston so as to reach from state ‘X’ back to ‘A’ would all the interim states of the system as defined by temperature, pressure and volume at any point be identical to those obtained during the process of going from A to X?

To answer this question we need to understand the process occurring in the system a little more deeply. Consider first that a mass mo is suddenly moved onto the piston from a shelf (at the same level). The piston assembly accelerates downwards, reaching its maximum velocity at the point where the downward force on the piston just balanced by the pressure exerted by the gas in the cylinder. However, the initial momentum of the plunging piston would carry it to a somewhat lower level, at which point it reverses direction. If the piston were held in this position of maximum depression brought about by transfer of the mass m0, the decrease in its potential-energy would very nearly equal the work done on the gas during the downward movement. However, if unrestrained, the piston assembly would oscillate, with progressively decreasing amplitude, and would eventually come to rest at a new equilibrium position at a level below its initial position.
The oscillation of the piston assembly cease because it is opposed by the viscosity of the gas, leading to a gradual conversion of the work initially done by the piston into heat, which in turn is converted to internal energy of the gas.
All processes carried out in finite time with real substances are accompanied in some degree by dissipative effects of one kind or another. However, one may conceive ofprocesses that are free of dissipative effects. For the compression process depicted in Fig. 1.4, such effects issue from sudden addition of a finite mass to the piston. The resulting imbalance of forces acting on the piston causes its acceleration, and leads to its subsequent oscillation. The sudden addition of smaller mass increments may reduce but does not eliminate this dissipative effect. Even the addition of an infinitesimal mass leads to piston oscillations of infinitesimal amplitude and a consequent dissipative effect. However, one may conceive ofan ideal process in which small mass increments are added one after another at a rate such that the piston movement downwards is continuous, with minute oscillation only at the end of the entire process.

This idealized case derives if one imagines of the masses added to the piston as being infinitesimally small. In such a situation the piston moves down at a uniform but infinitesimally slow rate. Since the disturbance each time is infinitesimal, the system is always infinitesimally displaced from the equilibrium state both internally as well with respect to external surroundings. Such a process which occurs very slowly and with infinitesimal driving forces is called a quasi-static process. To freeze ideas let us assume that the gas in the system follows the ideal gas law. Thus the pressure, temperature and volume at any point during the process are related by eqn. 1.12 (or 1.13). Now imagine that the process of gradual compression is reversed by removing each infinitesimal mass from the piston just as they were added during the forward process. Since during the expansion process also the system will always be differentially removed from equilibrium state at each point, the pressure, temperature and volume will also be governed by the relation 1.12. Since the latter is an equilibrium relationship and hence a unique one, each interim state of the system would exactly converge during both forward and backward progress of system states.  Under such a condition the process of compression is said to be thermodynamically reversible.  Both the system and its surroundings are ultimately restored to their initial conditions. In summary, therefore, if both the system and its surroundings can be restored to their respective initial states by reversing the direction of the process, then the process is said to be reversible. If a process does not fulfill this criterion it is called an irreversible process.

It need be emphasized that a reversible process need be a quasi-static process, and that the origin of irreversibility lie in the existence of dissipative forces in real systems, such as viscosity, mechanical friction. These forces degrade useful work irreversibly into heat which is not re-convertible by simply reversing the direction of the process, since during a reverse process a fraction of useful work will again be lost in the form heat in overcoming the dissipative forces. Thus, in the above example system if there were no viscous or frictional forces opposing the motion of the piston the processes of compression and expansion would be reversible, provided of course all changes occur under infinitesimal gradients of force. The argument in the last sentence may be extended to state that if changes are brought about by finite gradients (in this case finite difference in force across the piston, associated with addition of finite mass to the piston), the process would necessarily be irreversible. This is because finite gradients will force the system to traverse through non-equlibrium interim states, during which the pressure, temperature and volume will not be constrained by a unique relationship such as eqn. 1.12, which holds for equilibrium states. Indeed, it would not be possible to define the non-equilibrium states in terms of a single temperature or pressure, as there would be internal gradients of these variables during processes induced by finite force imbalances across the system boundary. These very same considerations would apply for the reverse process of expansion as well, if it occurs under finite mechanical gradients. So in general during such processes it would not be possible to ascribe unique intensive properties to interim states during a change, and hence the forward and reverse “paths” would not coincide as they would if the process occurs under quasi-static conditions.         

An additional point that obtains from the above considerations is that only under reversible conditions can one calculate the thermodynamic work by integrating eqn. 1.6, since at all points during the process the variables P, V and T are always uniquely related by the eqn. 1.12. Clearly if the process were occurring under irreversible conditions no such relation would hold and hence the calculation of the thermodynamic work would not be possible through a simple integration of eqn. 1.6.

The foregoing discussion has used the example of a single-phase closed-system, where compression and expansion processes are induced by gradients of mechanical force across the system boundary.  There are, however, many processes which are occur due to potential gradients other than mechanical forces. For example, heat flow is induced by temperature differences, electromotive force gradients lead to flow of electricity, and chemical reactions take place as there is a difference between the chemical potential of reactants and products. In general, it may be shown that all such processes brought about by potential gradients of various kinds would tend to reversibility if the gradients are themselves infinitesimal. For example, heat transfer across the boundary of a thermodynamic system would be reversible if the difference across it is of a differential amount ‘dT’, and so on. 

1.10 Significance of Chemical Engineering Thermodynamics: Process Plant Schema
Before we conclude the present chapter it would be appropriate to obtain a brief preview of the scope and utility of the principles of thermodynamics insofar as application to real world processes is concerned. Although based on relatively abstract principles, the laws of thermodynamics provide the fundamental constraints under which all real world process take place. The ultimate application of the knowledge of the core principles of chemical engineering is in the design of a chemical process plant. Engineering thermodynamics constitutes one of the principal elements of such knowledge. Typically such a plant converts a set of raw materials to a desired product through a variety of steps that are schematically represented by Fig. 1.6.      
Fig. 1.6 Chemical Process Plant Schema
The raw materials most often are mixtures which need to be purified to obtain the right composition required for conversion to products A wide variety of separation processes are available for carrying out such purification; examples include distillation, liquid-liquid extraction, precipitation from solutions, crystallization, etc. Practically all such separation processes involve generation of two or more phases, in one of which the desired raw material components are preferentially concentrated, which is then used recover the substances in a relatively purer form. For a typical large scale chemical plant the separation process equipments may constitute more than half of the total capital investment.  

The chemical reactor forms the “heart” of a chemical plant. It is here that once the feed materials are available in the right proportions (and compositions) they are reacted to yield the product. Obtaining the desired product requires an optimal choice of conditions under which the reactor may be operated. However, the product formed is very rarely obtained in a pure form. This is because typically the feed is never fully converted to product molecules and therefore the stream exiting the reactor is not a pure substance. In addition it is usually a common phenomenon that the intended chemical reaction is accompanied by often more than a single side reaction. The latter leads to the formation of side products, which results in “contamination” of the final product. Therefore, it is usually required to subject the reactor exit stream to another round of purification to obtain a product with the desired specifications of the product.
With regards to all such processes of purification and reaction, the laws of thermodynamics play a very fundamental role: they allow the calculation of the principal entities that form the basis of design and operation of process plants: 
  1. The maximum degree of purification that is possible under a given set of processing conditions
  2. The maximum degree of conversion possible under the reaction conditions
  3. The optimal operating conditions for separation and reaction processes
  4. The total energy required to achieve the intended degree of separation and reaction, and therefore the plant energy load 
The calculation of the above parameters tends to constitute 50-70% of the computational load encountered during the stage of basic process plant design. Thus, the principles of chemical engineering thermodynamics is one of the mainstays of knowledge needed to realize the goal of plant design and operation.