In the previous chapter we have introduced the concept of ideal gas and the corresponding equation of state (EOS). However, such a state obtains for a gas only at pressures around and below atmospheric (and at high temperatures), and, therefore, constitutes a limiting case. As we know, substances exist also in other forms: solids, liquids, etc.  Also more often than not, in practice (as in process plants) gases (as well as other phases) may exist at substantially higher pressures (up to several thousands of atmospheres). This necessitates the development of other EOSs not only for gases, but also for relating P-V-T behavior of liquids and solids. Such mathematical relations, if expressed in suitably generalized forms, provide the added advantage of being able to quantify P-V-T behaviour for a large number of individual substances.  

Such volumetric properties form a group of macroscopic thermodynamic state variables which are most easily measured. As we will see in later chapters, all other intensive thermodynamic properties can be represented in terms of mathematical expressions which denote functions of volumetric and a number of other directly measurable state variables. In the last chapter we have already introduced the ideas of thermodynamic work which clearly can be calculated if the relation between P and V is known. In chapters 3-5 we will demonstrate that heat transfer occurring under reversible conditions, as well as a host of other intensive, state variables (internal energy, enthalpy, entropy, Gibbs free energy, etc) can also be calculated using volumetric properties of a substance in question. Since work and heat are two principal modes of energy transfer in most thermodynamic systems of practical interest, it follows that the knowledge of volumetric properties is fundamental to all such calculations. Finally, as part of process plant design one needs volumetric properties for the purpose of sizing of process pipelines and all major process equipments such as reactors, heat exchangers, distillation columns, and so on. These considerations underscore the precise significance of P-V-T behavior of substances in all plant design activity. 

In the following sections we first describe the general nature of the P-V-T behavior of pure substances: in gaseous, liquid and solid forms. The various EOSs available to quantify such real fluid behaviour are then considered. Lastly, generalized correlations to relate gas and liquid behaviour are presented. Such analytical EOSs and generalized correlations allow prediction of P-V-T values of real fluids and are, therefore, of great value as they can obviate the need for detailed experimental data.

P-V Diagrams
Fig. 2.1 represents the general pure component, real fluid phase behavior that typically obtains from experimental measurements. Consider first the fig. 2.1a. Let us take a substance at some temperature T1and certain pressure P1 such that it is in a gaseous state A1. Keeping the temperature fixed at T1 if one pressurizes the gas (say in a piston-cylinder assembly as in fig. 1.2) its molar volume will decrease along the curve A-B. At point B, any further pressurization leads to commencement of condensation of the gas into a liquid from. Point ‘B’ is thus said to correspond to a state where the substance is in a saturated vapour state. Once condensation begins any attempt at reducing the volume by further 
Fig. 2.1 General P-V plots for real fluids

pressurization more of the saturated vapour present at B progressively liquefies until a point is reached where all the original gas (or saturated vapour) is fully converted to liquid state. Point is described as a saturated liquid state. It follows that at all point between and X the substance exists partitioned into two phases, i.e., part vapour and part liquid. As one transits from to X, pressure and temperature both remain constant; the only change that occurs is that the fraction of the original gas at point (or B) that is liquefied increases, until it is 1.0 at point X. The line B-X connecting the saturated vapour and liquid phases is called the tie-line. For a given and P, the relative amounts of the phases determine the effective molar (or specific) volume at any point within the two-phase region. Any further attempt to pressurize the saturated liquid results in relatively very little compression, and this is captured by the steep slope of the curve X-Y, which signifies that the liquid state is far less compressible, compared to the gas state (i.e., points over A-B). Essentially points between X-Y(including Y itself) represent compressed liquid states. 

An important point to re-emphasize is that on the two-phase line B-X, the pressure of the system remains constant at a fixed value. This pressure is termed the saturation pressure () corresponding to the temperature T1. We recall your attention to the phase rule described in section 1.5, and eqn. 1.11.  By this eqn. the degrees of freedom is one, which is borne by the fact that if one fixes temperature the system pressure also becomes fixed. However, in both regions A-B and X-Y the degrees of freedom is two, as pressure becomes fixed only if one defines both temperature and volume.
In general, the same behviour as detailed above may repeat at another temperature (>T1). One can on the one hand connect all the saturated vapour phase points at different temperatures and on the other connect all the points representing saturated liquid phase, the locus of such points give rise to the dome-shaped portion X-C-B of the P-V diagram which essentially signifies that at any pressure and volume combination within this dome, the state of the system is biphasic (part gas and part liquid). The region right of the dome B-C represents saturated gas phase while to the left (X-C) the state is saturated liquid. If one continues to conduct the pressurization at increasingly higher temperatures, one eventually arrives at a temperature for which the tie-line is reduced to a point and the P-V curve turns into an inflexion point to the two-phase dome. The temperature which such a behavior obtains is called the critical temperature (TC), while the pressure at corresponding point of inflexion is termed the critical pressure (PC).  The molar volume at the point is termed the critical volume, and the state itself the criticalpoint. A fluid which is at a temperature and pressure above the critical point values is said to be in a supercritical state; this is indicated by the hatched region in fig. 2b. As has been shown for the P-Vcurves for a T > TC, there exists no liquid phase as the curve passes beyond the two-phase dome region. Thus, the critical temperature is a temperature above which a gas cannot be liquefied by compressing, as can be below it. Compilation ofvalues of critical properties and ω for a large number of substances are available readily from several sources see:srdata.nist.gov). Values of these parameters for some select substances are provided in Appendix II.              

In fig. 2.1b the phase behavior depicted in fig. 2.1a is extended and more generalized to include solid phase as well. Accordingly, not only vapour-liquid region, other two phase regions, i.e., solid-vapour and solid-liquid regions are also displayed. The same arguments as made above for explaining the nature of co-existence of vapour and liquid phases apply to the other two biphasic regions. 
P-T Diagrams
The phase behaviour described by fig. 2.1 can also be expressed in a more condensed manner by means of a pressure-temperature (P-T) diagram shown in fig 2.2. Just as P-V curves were depicted at constant temperature, the P-T diagram is obtained at a constant molar volume. The two phase regions which were areas in the P-V diagram are reduced to lines (or curves) in fig. 2.2. The P-T curves shown by lines X-Y, Y-Z, and Y-C resultfrom measurements of the vapour 
Fig. 2.2 Pressure-Temperature Diagram of a Pure Substance
pressure of a pure substance, both as a solid and as a liquid. X-Y corresponds to the solid-vapour (sublimation) line; X-Y represents the co-existence of solid and liquid phases or the fusion line, while the curve Y-C displays the vapour-liquid equilibrium region. The pressure at each temperature on the Y-C curve corresponds to the equilibrium vapour pressure. (Similar considerations apply for P-Trelation on the sublimation curve, X-Y). The terminal point C represents the critical point, while the hatched region corresponds to the supercritical region. It is of interest to note that the above three curves meet at the triple-point where all three phases, solid, liquid and vapour co-exist in equilibrium. By the phase rule (eqn. 1.11) the degrees of freedom at this state is zero. It may be noted that the triple point converts to a line in fig. 2.1b. As already noted, the two phases become indistinguishable at the critical point. Paths such as F to G lead from the liquid region to the gas region without crossing a phase boundary. In contrast, paths which cross phase boundary Z-Y include a vaporization step, where a sudden change from liquid to gas occurs.
A substance in the compressed liquid state is also often termed as sub-cooled, while gas at a pressure lower than its saturation vapour pressure for a given temperature is said to be “superheated”. These descriptions may be understood with reference to fig. 2.2. Let us consider a compressed liquid at some temperature (T) and pressure (P). The saturation temperature for the pressure P would be expected to be above the given T. Hence the liquid is said to be sub-cooled with respect to its saturation temperature. Consider next a pure vapour at some temperature (T) and pressure (P). Clearly for the given pressure P the saturation temperature for the pressure P would be expected to be below the given T.  Hence with respect to the saturation temperature the vapour is superheated.

The considerations for P-V and P-T diagrams may be extended to describe the complete P-V-T phase behaviour in the form of three dimensional diagrams as shown in fig. 2.3. Instead of two-dimensional plots in figs. 2.1 and 2.2 we obtain a P-V-T surface. The P-V plots are recovered
Fig. 2.3 Generalized Three-dimensional P-V-T Surface for a Pure Substance

if one takes a slice of the three dimensional surface for a given temperature, while the P-T curve obtains if one takes a cross-section at a fixed volume. As may be evident, depending on the volume at which the surface is cut the P-T diagram changes shape.

Fig. 2.4 illustrates the phase diagram for the specific case of water. The data that is pictorially depicted so, is also available in the form of tables popularly known as the “steam table”.  The steam table (www.steamtablesonline.com) provides values of the following thermodynamic properties of water and
Fig. 2.4 Three-dimensional P-V-T Plot for Water
vapour as a function of temperature and pressure starting from its normal freezing point to the critical point: molar volume, internal energy, enthalpy and entropy (the last three properties are introduced and discussed in detail in chapters 3 and 4). 

The steam tables are available for saturated (two-phase), the compressed liquid and superheated vapour state properties. The first table presents the properties of saturated gas and liquid as a function of temperature (and in addition provides the saturation pressure). For the other two states the property values are tabulated in individual tables in terms of temperature and pressure, as the degree of freedom is two for a pure component, single state. For fixing the values of internal energy, enthalpy and entropy at any temperature and pressure those for the saturated liquid state at the triple point are arbitrarily assigned zero value.  The steam tables comprise the most comprehensive collection of properties for a pure substance.
2.2 Origins of Deviation from Ideal Gas Behaviour
The ideal gas EOS is given by eqn. 1.12. While this is a relationship between the macroscopic intensive properties there are two assumptions about the microscopic behaviour of molecules in an ideal gas state:
  1. The molecules have no extension in space (i.e., they posses zero volume)
  2. The molecules do not interact with each other 
In particular, the second assumption is relatively more fundamental to explaining deviations from ideal gas behavior; and indeed for understanding thermodynamic behavior of real fluids (pure or mixtures) in general.  For this, one needs to understand the interaction forces that exist between molecules of any substance, typically at very short intermolecular separation distances (~ 5 – 20(where 1= 10-8m).

Uncharged molecules may either be polar or non-polar depending on both on their geometry as well as the electro-negativity of the constituent atoms. If the centre of total positive and negative charges in a molecule do not coincide (for example, for water), it results in a permanent dipole, which imparts a polarity to the molecule. Conversely, molecules for which the centres of positive and negative charge coincide (for example, methane) do not possess a permanent dipole and are termed non-polar. However, even a so-called non-polar molecule, may possess an instantaneous dipole for the following reason. At the atomic level as electrons oscillate about the positively charged central nucleus, at any point of time a dipole is set up. However, averaged over time, the net dipole moment is zero. 

When two polar molecules approach each other closely the electric fields of the dipoles overlap, resulting in their re-orientation in space such that there is a net attractive force between them. If on the other hand a polar molecule approaches a non-polar molecule, the former induces a dipole in the latter (due to displacement of the electrons from their normal position) resulting once again in a net attractive interaction between them. Lastly when two non-polar molecules are close enough their instantaneous dipoles interact resulting in an attractive force. Due to these three types of interactions (dipole-dipole, dipole-induced dipole, and induced dipole-induced dipole) molecules of any substance or a mixture are subjected to an attractive force as they approach each other to very short separation distances.


However, intermolecular interactions are not only attractive. When molecules approach to distances even less than ~ 5 or so, a repulsive interaction force comes into play due to overlap of the electron clouds of each molecule, which results in a repulsive force field between them. Thus if one combines both the attractive and repulsive intermolecular interactions the overall interaction potential Uresembles the schematic shown in fig. 2.5.  
Fig. 2.3 Schematic of Intermolecular potential energy for a pair of uncharged molecules

Many expressions have been proposed for the overall interaction potential  [see, J.M. Prausnitz, R.N Lichtenthaler and E.G. Azevedo, Molecular Thermodynamics of Fluid Phase Equilibria, (3rd ed.), 1999, Prentice Hall, NJ (USA)]. These are essentially empirical, although their functional forms often are based on fundamental molecular theory of matter. The most widely used equation in this genre is the Lennard-Jones (LJ) 12/6 pair-potential function which is given by eqn. 2.1
(2.1)   
Where, = intermolecular separation distance; = characteristic L-J parameters for a substance. The  term represents the repulsive interaction, whereas the  term corresponds to the attractive interaction potential. As already indicated, the domain of intermolecular interactions is limited to relatively low range of separation distances. In principle they are expected to be operative over  but for practical purposes they reduce to insignificant magnitudes for separations exceeding about 10 times the molecular diameter.

The L-J parameters are representative of the molecular interaction and size respectively. Typical values of the L-J equation parameters for various substances may be found elsewhere (G. Maitland, M. Rigby and W. Wakeham, 1981, Intermolecular Forces: Their Origin and Determination, Oxford, Oxford University Press.)
Since gases behave ideally at low pressures, intermolecular separation distances therein are typically much higher than the range over which intermolecular interactions are significant. This is the reason why such interactions are negligible in case of ideal gas, which essentially is one of the assumptions behind the definition of ideal gas state. Indeed while the ideal gas EOS is expressed in macroscopic terms in eqn. 1.12, the same equation may be derived from microscopic (thermodynamic) theory of matter.

The root of non-ideal gas behavior, which typically obtains at higher pressure, thus is due to the fact that at elevated pressures, the intermolecular separations tend to lie within the interactive range and hence the ideal gas assumption is no longer valid. Thus, the ideal gas EOS is insufficient to describe the phase behavior of gases under such conditions.

Intermolecular interactions also help explain the behavior of fluids in other states. Gases can condense when compressed, as molecules are then brought within the separations where the attractive forces constrain the molecules to remain within distances typical of liquid phase. It follows that a pure component liquid phase cannot be ideal in the same sense as a gas phase can be. Further, the fact that liquids are far less compressible also is due to the repulsive forces that operate at close intermolecular distances. Obviously these phenomena would not be observed unless there were interactions between molecules. Thus, it follows that while properties of the ideal gas depend only on those of isolated, non-interacting moleclues, those of real fluids depends additionally on the intermolecular potential. Properties which are determined by the intermolecular interaction are known as configurational properties, an example of which is the energy required for vapourization; this is because during the process of vapourization energy has to be provided so as to overcome the intermolecular attractive force between molecules in the liquid phase and achieve the gas state where the seprations are relatively larger.


2.3 Equations of State for Real Fluids
The generic form of an equation of state (EOS) is: 
However, as we have already seen by the phase rule, for a single phase pure component the degrees of freedom are two. This may be expressed in the form of an EOS equation as follows: 
It follows that:
(2.2)
Defining (2.3)
And Isothermal compressibility as (2.4)
The generic EOS (2.2) may then be written as:


2.3.1 EOS for Liquids
For liquids, which are relatively incompressible, the factors β and κ are generally show an weak dependence on T and and hence averaged values of these parameters may be used for estimating the liquid volume at any temperature using the following integrated form of the equation (2.5): 
(2.6)
For liquids the usual datum volume (i.e., V1 in eqn.2.6) can be the saturated volume at a given temperature, which may be obtained from the Rackett equation (H. G. Rackett, J. Chem. Eng. Data, 1970, vol. 15, pp. 514-517); i.e.,: 
 ; where Tr=reduced temperature = T/Tc(2.7)
Where ZC is the critical compressibility factor (see below).
2.3.2 EOS for Gases
In contrast to liquids, gases are relatively far more compressible, and so volume is strongly dependant on temperature and pressure. Consequently eqn. 2.6 cannot be used easily to estimate volume at a given T & P, as both β and κ are strong functions of T and P. Thus, various EOSs have been proposed to describe gas phase volumetric properties. The next section presents select EOSs that are typically used for the gas phase, ranging from those applicable to moderate pressure to others which are more accurate at high pressures. 
Virial EOS:
Generally applicable to moderate deviations from ideal gas behavior, the virial EOS is given by two alternate forms:
(2.8)
Where B, C = second and third virial coefficients 
Or: (2.9)
It may be shown that the following relations hold further:
(2.10)
As may be evident, the second and third terms on the right side of eqns. 2.8 and 2.9 constitute corrections for the non-ideal behavior of a gas. The virial coefficients are essentially dependent of temperature. The more the number of virial coefficients used in the equation the better is the prediction of gas molar volume. While the estimation of the second virial coefficient is relatively straightforward, that of the third virial coefficient is generally far more complex, and there is scant data in the literature on its value for different substances. Because of this the virial EOS is most commonly used in the truncated form, and applies to moderate deviations from ideal gas behavior: 
(2.11) 
Alternately,                                                                     
(2.12)
Where,  are reduced pressure and temperature respectively, given by: 


A set of generalized correlations have been proposed by Pitzer and co-workers (K. S. Pitzer, Thermodynamics, 3d ed., App. 3, McGraw-Hill, New York, 1995) for computing the second virial coefficients whereby:
(2.13) 
The terms  are given by:
                                  
(2.14)
               
(2.15)
The parameter  discussed in a later section, is the acentric factor, which is a fundamental thermodynamic property of a substance, and is a measure of asphericity of molecular shape (see Appendix-II for values of acentric factor for select substances). The value of acentric factor may be obtained from the following expression from experimental vapour pressure data of a pure substance (K. S. Pitzer, Thermodynamics, 3d ed., App. 3, McGraw-Hill, New York, 1995): 
(2.16)
 = reduced saturation vapour pressure at a reduced temperature equalling 0.7.
Since the virial EOS applies to only moderate deviations from ideal behaviour several complex extensions of it have been proposed for increasing accuracy of prediction at higher pressures; an example of such an EOS is the so-called Benedict-Webb-Rubin (BWR) EOS (G.B.Benedict, G. B. Webb, and L. C. Rubin, J. ChemPhys., vol. 8, pp. 334-345, 1940) used in the petroleum and natural-gas industries:
(2.17) 
where A0, B0 , Co, a, b, c, and  are all constant for a given fluid. 
Example 2.2

Calculate the molar volume for butane at 2.5bar and 298 K using the truncated virial EOS using the following data:
(Click for Solution) 
2.3.3 Cubic EOS for Gases
While the truncated virial EOS applies to moderate pressures, it is often necessary to obtain volumetric properties of gases at relatively much higher pressures. It has generally proved to be computationally unwieldy to use extensions of the virial EOS at higher pressure. To obviate this problem an entire range of alternate EOSs, termed cubic EOS, have been proposed by various workers. The term “cubic equation of state” implies an equation which, if expanded, would contain the volume terms raised to the first, second, or third power. Such an EOS provides two distinct advantages over many other more complex EOS found in the literature; they allow:
  • Prediction of both gas and liquid (saturated) phase molar volumes
  • Provide a trade-off between complexity and accuracy of prediction
The first of such cubic EOS was proposed by the Dutch physicist Johannes Diderik van der Waals in 1873, and has the following form:
(2.18) 
Where, a and b are characteristic constants for a pure substance. For example, for nitrogen the values of a and b are  Values of these parameters for other substances may be found in the literature (see, for example: S.I. Sandler, Chemical, Biochemical and Engineering Thermodynamics, 4th edition, Wiley India, 2006).

Once can take advantage of the phase P-V behaviour at the critical point (see fig. 2.1) to determine the expressions for a and b. Since the P-V curve offers a point of inflection at the critical point one may write:  
                                                                                                                                                                                                                 
(2.19)

(2.20)
On applying eqns. (2.19) and (2.20) on 2.18 two independent equations obtain, which may solved simultaneously to show that for the vdW EOS:
 




It follows that the critical compressibility factor is then given by:
The vdW EOS suggests that for all pure substances the critical compressibility factor is a constant (=0.375). This, however, is contrary to experience. Indeed, although similar in magnitude, the value of ZC is specific to a pure substance. For example, for methane it is 0.286, while for helium 0.302. This indicates that although an improvement over ideal gas EOS, the vdW EOS does not reflect the specificity needed to predict the behaviour of all substances at high pressures.

To bridge this gap several other cubic EOS of higher accuracy have been advanced, the three most widely used of which are:
  • Redlich-Kwong (RK) EOS (Otto Redlich and J. N. S. Kwong, Chem. Rev.,vol. 44, pp. 233-244, 1949)
  • Soave-Redlich-Kwong (SRK) EOS (G. Soave, Chem. Eng. Sci., vol. 27, pp. 1197-1203, 1972.) 
  • Peng-Robinson (PR) EOS (D. Y. Peng and D. B. Robinson, Ind. Eng. Chem. Fundam., vol. 15, pp. 59-64, 1976.) 
All the above EOSs including that due to van der Waals may be expressed by a single equation of the following form: 
(2.21)
As part of a generalized approach to representing non-ideal gas behavior parameter Z, termed the compressibility factor, may be defined. This is discussed further in the following section. However, here we provide the expression for it:
Z = PV/RT      (2.22)
Using eqn. 2.20, eqn. 2.19 may be rewritten as: 
(2.23)
Where:             

Table 2.1 Values of 
EOSuwab
vdW00– 1 – BA– AB
RK10– 1A – B – B2– AB
SRK10– 1A – B – B2– AB
PR2– 1– 1 + BA – 2B – 3B2– AB + B2 +B3
The parameters  given by: 
As may be evident from table 2.1 all the parameters are determinable from the critical properties,  and the temperature at which the molar volume or pressure needs to be calculated. Once these are computed the eqn. 2.21 may be solved to yield the value of Z, which in turn may be used to obtain the molar volume. Eqn. 2.21 may be solved using a suitable iterative algorithm or more readily by the general analytical solution for cubic algebraic equations, which is outlined below.



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