Pressure Volume Tempertue

Pressure-Volume-Temperature Relationship Of Pure Gases

Scope:

The volumetric behavior of pure gases as functions of temperature and pressure is covered

in this assignment. The basis for the correlations is the thermodynamic fact that each substance

there exist a unique relation among Pressure, Temperature, and specific Volume for any single

phase.

General Considerations:

Methods of correlating the P-V-T behavior of pure gases can be conveniently, but somewhat

arbitrarily, divided in three groups.

First there are methods which are classified under the general term corresponding states and are

based on dimensional similitude considerations. The compressibility factor reduced property

correlation is a good example of this treatment.

Second there is the more theoretical approach employing a virial expansion of P-V-T equations

with the constants directly related to the intermolecular potential energy distance relationship

applicable to the substance in question.

Third and finally, many analytical equations of state have been proposed.

Introduction to volumetric properties: The volumetric properties of a pure fluid in a given state are commonly expressed with the

compressibility factor Z, which can be written as a function of T and P or of T and V.

Z ≡ = fp (T, P)

= fv (T, V) where V is the molar volume, P is the absolute pressure, T is the absolute temperature, and R is

called the universal gas constant. The value of R depends upon the units of the variables used. Z

is unity for ideal gases. Z approaches unity at low pressures for all gases and may be greater or

less than unity at higher pressures if the temperature is above the so-called Boyle temperature, Z

will always be greater than unity, but if the temperature is below the Boyle temperature, Z is less

than unity at low pressures and greater than unity at high pressures. Since the compressibility

factor is dimensionless, it is often represented by a function of dimensionless (reduced)

temperature, Tr = T/T*, and dimensionless (reduced) pressure, Pr = P/P*, where T*, and P* are

characteristic properties for the substance, such as the component’s vapor-liquid critical, Tc , and

Pc. It could also be given as a function of Tr and reduced volume, Vr = V/V*, where V* could be

chosen as Vc or RTc /Pc or another quantity with units of volume. Then Z is considered a

function of dimensionless variables:

Z = ƒPr (Tr , Pr)

=ƒVr (Tr , Vr )

The reasons why Z deviates from unity are:

1) Shape factors and Intermolecular Forces. Molecules are not point masses and have many diverse shapes. Even simple spherical atoms

attracts or repel other atoms, depending upon the separation distance, in addition, at high

pressures, the volume fractions occupied by the atoms themselves may be significant Molecules

behave in the similar manner. The fact that real gases have interacting forces and finite molecular

volumes is usually the most important reason that why Z deviates.

2) Electrical Forces: Electrical forces of permanent kind are usually important in polar molecules. Nonpermanent

electrical forces, such as electric moments arising during collisions are included in above and are

present in all real systems. The forces are included in the potential energy relationships between

molecules. So Z deviates from ideality.

3) Hydrogen bonding: Hydrogen bonding between molecules usually occurs in systems of polar nature. Suchj forces are

considered in same category as electrical forces.

4) Quantum Effects: Quantum effects are considered in those molecules where the translational energy modes must be

quantized.

These four effects must be considered when methods are proposed to describe the

variation of P-V-T behavior of a real gas from that of an ideal gas.

Corresponding State Correlations: The basis of the corresponding state correlations is the observation that the graphs of P vs V,

with isotherms, are quantitatively similar for all gases.

The variations in Z is small compared with possible variations in P, V, T. to show the P-V-T

behavior of a particular gas, Z is plotted against pressure, with curves representing isotherms and

isometrics. To show the behavior of Z in a general way i.e for all gases, recourse is usually had

to the principle of corresponding states. The principle has been stated in many ways, but it is best

visualized by applying dimensional analysis to the important system parameters. The set of

dimensionless groups which are obtained may then be utilized to define the system. The

equations of Z, graphs and the Equations of State are all formulations of a general principle of

dimensionless functions and dimensionless variables called the corresponding states principle

(CSP) or sometimes the “Law of Corresponding States.” It asserts that suitably dimensionless

properties of all substances will follow universal variations of suitably dimensionless variables of

state and other dimensionless quantities. Its general and specific forms can be derived from

molecular theory by Hakala.

For very high pressures and temperatures, Breedveld and Prausnitz have generated more accurate

extensions of graphs between these variables.

Many versions of these graphs have been published. All differ somewhat, as each reflects the

choice of experimental data and how they are smoothed. Those shown are as accurate as any

two-parameter plots published, and they have the added advantage that V can be found directly

from the lines of V =V/ (RTc /Pc).

Equations of state (EoS) are mathematical representations of graphical information such as

shown in these graphs.

Two-Parameter CSP: Pitzer and Guggenheim describe the molecular conditions for this lowest level CSP when only

two characteristic properties, such as Tc and Pc, are used to make the state conditions

dimensionless, and the dimensionless function may be Z. Thus, these graphs and Equations are

examples of this two parameter form of CSP where the characteristics are Tc and Pc or Tc and Vc.

Only the monatomic substances Ar, Kr, Xe, or “simple fluids” follow this behavior accurately;

all others show some deviation. For example, two parameter CSP requires that all substances

have the same critical compressibility factor, Zc = PcVc / RTc. Simple transformations of EoS

models show that all those with only two substance-specific constant parameters such as the

original van der Waals EoS (1890), also predict the same Zc for all compounds. Analysis shows

that popular two-parameter cubic EoS models yield values greater than 0.3. This behavior

suggests that more than two dimensionless characteristics must be used both in concept and in

modeling.

Three-Parameter CSP: In general, successful EoS has included one or more dimensionless characteristic parameters

into the function expressed by Equations especially to obtain good agreement for liquid

properties. The first step in accomplishing this is to introduce a third parameter; usually it is

related to the vapor pressure, Pvp , or to a volumetric property at or near the critical point. This

improves accuracy for many substances. Molecular theory by Hakala, suggests that effects of

non polar non sphericity and of “globularity” (the range and sharpness of non polar repulsive

forces) should require separate CSP parameters, but in practice, only a single characteristic

accounts for both. Several different third parameters were introduced at about the same time but

the most popular have been Zc and the acentric factor, ,.Lydersen, and a later revision by

Hougan, tabulated Z at increments of T/Tc and P/Pc for different values of Zc.

For example, the compressibility factor was given as:

Z =Z(0) (

T/Tc , P/Pc) + Z(1)

(T/Tc , P/Pc )

where Z

(0) and Z

(1) are generalized functions of reduced temperature and pressure with Z

(0)

obtained from the monatomic species and Z(1)

by averaging (Z - Z(0)

) / for different substances.

It was expected that would describe only “normal fluids,” not strongly polar or

associating species such as via hydrogen-bonding, and limited to smaller values of . Pitzer

focused on the surface tension as being the most sensitive property to indicate when the

molecular forces were more complex than those for “normal” substances. These ideas can be

coalesced into a single equation for surface tension, , made dimensionless with critical

properties.

Pitzer states that if a substance deviates more than 5% from above Equation it appears to indicate

significant abnormality. Otherwise, the substance is expected to be normal and three-parameter

CSP should be reliable.

Higher Parameter and Alternative CSP Approaches: One method to extend CSP to substances more complex than normal fluids is to use more

terms in above equation with new characteristic parameters to add in the effects of polarity and

association on the properties.

Alternative expansions have also appeared. Rather than use simple fluids as a single reference,

multiple reference fluids can be used. For example, Lee and Kesler and Teja developed CSP

treatments for normal fluids that give excellent accuracy. The concept is to write Z in two terms

with two reference substances having acentric factors (R1)

and (R2)

, Z(R1)

(Tr , Pr, (R1)

)and

Z(R2)

(Tr, Pr, (R2)

). The expression is:

For “quantum fluids,” H2, He, Ne, Gunn, showed how to obtain useful values for CSP over wide

ranges of conditions. The equations are

where M is the molecular weight, T

cl, P

cl and V

cl are “classical” critical constants found

empirically. The “classical” acentric factor for these substances is defined as zero.

EQUATIONS OF STATE: An equation of state (EoS) is an algebraic relation between P, V, and T. This section discusses

what behavior of Nature must be described by EoS models. First, the virial equation, which can

be derived from molecular theory, but is limited in its range of applicability. It is a polynomial in

P or 1/V (or density) which, when truncated at the Second or Third Order term, can represent

modest deviations from ideal gas behavior, but not liquid properties.

Two Constant Equations Of State: There are four two constant equation of state which have been widely used,

Van der waals:

Berthelot:

Dieterici:

Redlich-Kwong:

The first three are discussed by Hirschfelder, Curtiss, and Bird, and the last is considered

to be the best of the four by Gambill. Only Dieterici’s equation gives a reasonable value

of Zc, and none predicts the second virial coefficient curvature as would be obtained from

theory by using the Lennard-Jones potential.

Three Constant Equations Of State:

Gambill discusses the generalized Wohl equation as a three constant equation of state.

Wohl Equation:

Where

a1= 6PcVc2

a2= 4PcVc2

b =Vc/4

This equation is not much, if any, more accurate than any of the two constant equations

and it is restricted to regions not near the saturation curve; nor should it be considered

valid at reduced densities greater than unity.

Four Constant Equations Of State:

The most useful four constant equation of state is McLeod equation.

McLeod equation:

(P + /V2)(V-b) = RT

Where

b = A – Bπ + Cπ2

π = P + /V2

The relationship of the four constants , A, B, C to critical properties has been studied

by Rush and Gamson and by Kobe and Murti.

Five Constant Equations Of State:

By far the most accurate and best known of the five constant equations is Beattie-

Bridgman.

Beattie Bridgman equation:

Where

Multi Constant Equations of State:

A number of very complex but highly accurate equations of state have been proposed.

The eight constant Benedict-Webb-Rubin equation is accurate for light hydrocarbons.

There are two quite accurate but somewhat cumbersome generalized equations of state

which have been widely used when machine computation was possible. They are given

below.

1) Hirschfelder, Buehler, McGee, and Sutton Equation Of State.

2) Martin and Hou Equation Of State.

For values of Vr = 1 / r > 1, the Martin and Hou equation appears somewhat more accurate, and

errors in estimating the pressure should not exceed 1 per cent.

For values of Vr < 1, the Hirschfelder equations were considered more accurate than the original

Martin Hou equation.

Virial Equation Of State:

The virial equation of state is a polynomial series in pressure or in inverse volume whose

coefficients are functions only of T for a pure fluid. The consistent forms for the initial terms are

Z = 1 + B( ) + (C – B2) ( )

2 + . . . . . .

= 1 + + + . . . . . . . . . . . . . . . ..

where the coefficients B, C, . . . are called the second, third, . . . virial coefficients. Except at high

temperatures, B is negative and, except at very low T where they are of little importance, C and

higher coefficients are positive.

Because 1) the virial expansion is not rigorous at higher pressures, 2) higher order molecular

force relations are intractable, and 3) alternative EoS forms are more accurate for dense fluids

and liquids, the virial equation is usually truncated at the second or third term and applied only to

single-phase gas systems.

The general ranges of state for applying these equations are given in Table 4-4; they were

obtained by comparing very accurately correlated Z values of Setzmann and Wagner with those

computed with their highly accurate virial coefficients over the entire range of conditions that

methane is described. When only B is used, the equations are equivalent at the lowest densities.

2nd

Equation in density is more accurate to somewhat higher densities but if it is used at higher

pressures, it can yield negative Z values. Thus, it is common to use 1st equation in pressure if

only the second virial, B, is known. If the term in C is included, 2nd

equation in density is much

more accurate than 1st equation.

Estimation of Second Virial Coefficients:

Unlike for empirical EoS, there is direct theoretical justification for extending simple CSP for B

to complex substances by merely adding terms to those for simple substances. Thus, essentially

all of the methods referenced above can be written in the form:

where V* is a characteristic volume, such as Vc or Pc /RTc, the ai are strength parameters for

various intermolecular forces, and the ƒ(i)

are sets of universal functions of reduced temperature,

T/T*, with T* typically being Tc. Then, ƒ(0)

is for simple substances with a0 being unity, ƒ(1)

corrects for non spherical shape and globularity of normal substances with a1 commonly being,

, ƒ(2)

takes account of polarity with a2 being a function of the dipole moment, and ƒ(3)

takes

account of association with a3 an empirical parameter.

Estimation of Third Virial Coefficients:

The trio intermolecular potential includes significant contributions that cannot be determined

from the pair potentials that describe second virial coefficients. Thus, CSP is also used for C,

though the range of substances considered has been much more limited.

The principal techniques for C are the CSP methods of Chueh and Prausnitz, De Santis and

Grande and Orbey and Vera. All use T* = Tc in the equation:

Analytical Equations Of States:

An EoS used to describe both gases and liquids requires the form Z in terms of V and it must be

at least cubic in V. The term analytical equation of state implies that the function ƒV (T, V) has

powers of V no higher than quartic. Then, when T and P are specified, V can be found

analytically rather than only numerically.

It is possible to formulate all possible cubic EoS in a single general form with a total of five

parameters. If one incorporates the incompressibility of liquids to have P go to infinity as V

approaches a particular parameter b, the general cubic form for P is:

where, depending upon the model, the parameters , b , , and η may be constants, including

zero, or they may vary with T and/or composition.

Above equation in terms of Z is:

When it is rewritten as the form to be solved when T and P are specified and Z is to be found

analytically, it is :

where the dimensionless parameters are defined as:

When a value of Z is found by solving from above equation for given T, P and parameter values,

V is found from V = ZRT/P. V must always be greater than b.

Non Analytic Equations Of State:

The complexity of property behavior cannot be described with high accuracy with the cubic or

quartic EoS that can be solved analytically for the volume, given T and P. So five approaches

that are available for pure components. Two are strictly empirical: BWR/MBWR models and

Wagner formulations. Two are semi empirical formulations based on theory: perturbation

methods and chemical association models. The last method attempts to account for the

fundamentally different behavior of the near-critical region by using Crossover expressions.

BWR and MBWR Models:

The BWR expressions are based on the pioneering work of Benedict, Webb and Rubin who

combined polynomials in temperature with power series and exponentials of density into an

eight-parameter form. Additional terms and parameters were later introduced by others to

formulate modified Benedict-Webb-Rubin (MBWR) EoS.

The general form of BWR/MBWR correlations is:

this equation form was standard for IUPAC and NIST compilations of pure component fluid

volumetric and thermodynamic properties. Kedge and Trebble have investigated an expression

similar to above equation with 16 parameters that provides high accuracy. However, other

formulations described below have become more prevalent in use.

Wagner Models:

Setzmann and Wagner describe a computer-intensive optimization strategy for establishing

highly accurate EoS models by a formulation for the residual Helmholtz energy.

where Ao(T, V) is the ideal gas Helmholtz energy at T and V. The compressibility factor

equation is found as using a thermodynamic partial derivative:

There are actually a few additional terms in the expression of Setzmann and Wagner. Equations

in this form can describe all measured properties of a pure substance with an accuracy that

probably exceeds that of the measurements. It gives excellent agreement with the second virial

coefficient where B/Vc is all terms in the sums of above equation when = 0. It can predict the

properties of fluids at hyper pressures and hyper temperatures.

Perturbation Models:

The technique of perturbation modeling uses reference values for systems that are similar

enough to the system of interest that good estimates of desired values can be made with small

corrections to the reference values. For EoS models, this means that the residual Helmholtz

energy of above equation is written as:

where the form of the perturbation terms [Ar (T, V) /RT](i)

can be obtained from a rigorous or

approximate theory, from a Taylor’s expansion or from intuition.

Chemical Theory EoS:

The interactions between the molecules are quite strong due to charge-transfer and hydrogen

bonding. This occurs in pure components such as alcohols, carboxylic acids, water and HF and

leads to quite different behavior of vapors of these substances. For example, Twu, show that Z

for the saturated vapor of acetic acid increases with temperature up to more than 450 K as

increased numbers of molecules appear due to a shift in the dimerization equilibrium. However,

the liquid Z behaves like most other polar substances. Also, the apparent second virial

coefficients of such components species are much more negative than suggested by

corresponding states and other correlations based on intermolecular forces and the temperature

dependence is much stronger.

Instead of using parameters of a model from only non polar and polar forces, one approach has

been to consider the interactions so strong that new “chemical species” are formed. Then the

thermodynamic treatment assumes that the properties deviate from an ideal gas mainly due to the

“speciation” plus some physical effects. It is assumed that all of the species are in reaction

equilibrium. Thus, their concentrations can be determined from equilibrium constants having

parameters such as enthalpies and entropies of reaction in addition to the usual parameters for

their physical interactions.

An example is the formation of dimers (D) from two monomers (M)

2M = D

The equilibrium constant for this reaction can be exactly related to the equation:

KD = yD / y2M P = - B / RT

The model of Hayden and O’Connell explicitly includes such contributions so that it can also

predict the properties of strongly interacting substances. Anderko notes that there are two general

methods for analyzing systems with speciation. The first, exemplified by the work of Gmehling

is to postulate the species to be found, such as dimers, and then obtain characteristic parameters

of each species such as critical properties from those of the monomers along with the enthalpy

and Gibbs energy of each reaction. The alternative approach was first developed by Heidemann

and Prausnitz and extended by Ikonomu and Donohue Anderko, Twu, and Visco and Kofke.

This approach builds the species from linear polymers whose characteristics and reaction

equilibrium constants can be predicted for all degrees of association from very few parameters.

EoS Models for the Near-Critical Region: Considerable work has been done to develop EoS models that will suitably bridge the two

regimes. The first is to use a “switching function” that decreases the contribution to the pressure

of the classical EoS and increases that from a non classical term. The advantage of this method is

that no iterative calculations are needed. Another approach is to “renormalize” Tc and Pc from

the erroneous values that a suitable EoS for the classical region gives to the correct ones.

Examples of this method include Fox, Pitzer and Schreiber, Chou and Prausnitz, Vine and

Wormald, Solimando, Lue and Prausnitz and Fornasiero.

The final approach to including non classical behavior has been the more rigorous approach via

crossover functions of Sengers and coworkers. The original method was to develop an EoS

model that was accurate from the critical point to well into the classical region, but did not cover

all conditions. Anisimov, and Tang show results for several substances.

Recent efforts with this method have led to EoS models applicable to all ranges. Though not

applied extensively yet, indications are that it should be broadly applicable with accuracies

similar to the scaling methods. In addition, theoretical analyses of this group have considered the

differences among approaches to the critical point of different kinds of systems such as

electrolytes, micelles and other aggregating substances, and polymers where the range of the non

classical region is smaller than molecular fluids and the transition from classical to non classical

can be sharper and even non universal.