CHEE 311J.S. Parent1 5. Equations of State SVNA Chapter 3 Efforts to understand and control phase equilibrium rely on accurate knowledge of the relationship

CHEE 311 J.S. Parent 1

5. Equations of State SVNA Chapter 3

Efforts to understand and control phase equilibrium rely on accurate knowledge of the relationship between pressure, temperature and volume for pure substances and mixtures.

This PT diagramdetails the phaseboundaries of a pure substance. It provides no informationregarding molarvolume.


P-V-T Behaviour of a Pure Substance

The pure component PV-diagram shown here describes the relationship between pressure and molar volume for the various phases assumed by the the substance.


PV Diagram for Oxygen


Equations of State

Experimental data exists for a great many substances and mixtures over a wide range of conditions.

Tabulated P-V-T data is cumbersome to catalogue and use Mathematical equations (Equations of State) describing

P-V-T behaviour are more commonly used to represent segments of the phase diagram, usually gas-phase behaviour

Ideal Gas Equation of StateApplicable to non-polar gases at low pressure:

where V is the molar volume (m3/mole) of the substance.

In terms of compressibility, Z=PV/RT, the ideal gas EOS gives:

PV RT

.Z 1


Equations of State: Non-ideal Fluids

The ideal gas equation appliesunder conditions where molecular interactions are negligible and molecular volume need not be considered.

At higher pressures, the compressibility factor, Z, is not unity, but takes on a value that is different for each substance and various mixtures.

A more complex approach isneeded to describe PVT behaviour of non-ideal fluids


Virial Equation of State for Gases

If our goal to calculate the properties of a gas (not a liquid or solid), the PVT behaviour we need to examine is relatively simple.

The product of pressure and molar volume is relatively constant, and can be approximated by a power series expansion:

from which the compressibility is readily determined:

Eq 3.10

The coefficients B’,C’,D’ are called the first, second and third virial coefficients, respectively, and are specific to a given substance at a given temperature.

These coefficients have a basis in thermodynamic theory, but are usually empirical parameters in engineering applications.

.

2 3PV RT(1 B'P C 'P D 'P ...)

.

2 3PVZ 1 B'P C 'P D 'P ...

RT


Cubic Equations of State: Gases and Liquids

A need to describe PVT behaviour for both gases and liquids over a wide range of conditions using an equation of minimal computational complexity led to the development of cubic equations of state.

Peng-Robinson (PR): Sauve-Redlich-Kwong (SRK):

in terms of compressibility, Z:

PR-EOS:

SRK-EOS:

where a and b (or A and B) are positive constants that are tabulated for the substance of interest, or generalized functions of P and T.

These polynomial equations are cubic in molar volume, and are the simplest relationships that are capable of representing both liquid and gas phase properties.

.2 2

RT aP

V b V 2bV b

.1/ 2

RT aP

V b T V(V b)

.

3 2 2Z Z (A B B )Z AB 0 .

3 2 2 2 3Z (1 B)Z (A 3B 2B)Z (AB B B ) 0


Cubic Equations of State: Gases and Liquids

Given the required equation parameters (a and b in the previous cases), the system pressure can be calculated for a given temperature and molar volume.

At T > Tc, the cubic EOS has just one real, positive root for V.

At T<Tc there exists only one real, positive root at high pressure (molar volume of the liquid phase). However, at low pressures the cubic EOS can yield three real, positive roots; the minimum representing the liquid-phase molar volume, and the maximum the vapour-phase molar volume.


Theorem of Corresponding States

The virial and cubic equations of state require parameters (B’, C’, a, b, for example) that are specific to the substance of interest. In fact, the PVT relationships for most non-polar fluids is remarkably similar when compared on the basis of reduced pressure and temperature.

Simple fluids aside (argon, xenon, etc), some empiricism is required to achieve the required degree of accuracy. The three-parameter theorem of corresponding states is:

All fluids having the same value of acentric factor, , when compared at the same Tr and Pr, have the same value of Z.

The advantage of the corresponding states, or generalized, approach is that fluid properties can be estimated using very little knowledge (Tc, Pc and ) of the substance(s).

rc

PP

P r

c

TT

T


Theorem of Corresponding States


Pitzer Correlations: Gases and Liquids

Pitzer developed and introduced a general correlation for the fluid compressibility factor.

Eq 3.46

where Zo and Z1 are tabulated functions of reduced pressure and temperature.

This approach is equally suitable for gases and liquid, giving it a distinct advantage over the simple virial equation of state and most of the cubic equations.

Values of , Pc and Tc for a variety of substances can be found in Table B.1 of SVNA.

The Lee/Kesler generalized correlation (found in Tables E.1-E.4 of the SVNA) is accurate for non-polar, or only slightly polar, gases and liquids to about 3 percent.

o 1Z Z Z


Generalized Virial-Coefficient Correlation: Gases

The tabulated compressibility information that is the basis of the generalized Pitzer-type approach can be cumbersome (especially in an exam)

the complex PVT relationship of non-ideal fluids is difficult to represent by a simple equation, necessitating the use of tables if the corresponding states approach is to be accurate.

SVNA provides a generalized virial EOS correlation that allows you to apply the virial EOS with coefficients that are based on a corresponding states approach (Page 89 SVNA, 4thed).

where

and

PV BPZ 1

RT RT

.

o 1c

c

RTB (B B )

P

o1.6r

0.422B 0.083

T

14.2r

0.172B 0.139

T


PVT Behaviour of Mixtures

Most equations of state prescribe mixing rules that allow you to calculate EOS parameters and describe the PVT behaviour of mixtures.

The Virial EOS,

the composition dependence of the virial coefficient B is:

where y represents the mole fractions in the mixture and the indices i and j identify the species. Values of Bij are determined using generalized correlations and/or formulae specifically developed for the mixture of interest.

Mixture behaviour will be examined in greater detail later in the course

PVZ 1 BP

RT

i j iji j

B y y B


5. Non-Ideality in 1-component Systems

Pure, Non-ideal GasesThe ideal gas assumption:

PV = RTwhere V = molar volume holdsonly for low pressures, wheremolecular interactions are negligible and molecular volumeneed not be considered.

At higher pressures, we have usedthe compressibility factor, Z, tocharacterize gas behaviour.

Z = PV / RT = 1 for ideal gases


Gibbs Energy of Pure Gases

For any pure gas, ideal or non-ideal, the fundamental equation applies:

dG = VdP - SdT

At constant T, changes in the Gibbs energy of a pure gas arise only from changes in pressure, and:

dG = VdP (constant T)

We can integrate between two pressures, Pref and P to obtain:

For an ideal gas, we can substitute for the molar volume, V=RT/P

P

Pref

ref

VdP)P,T(G)P,T(G

ref

P

Pref

igig

PP

lnRT

dPP

RT)P,T(G)P,T(G

ref


Gibbs Energy of Pure, Ideal Gases

For the ideal gas case, we have

If we consistently select unit pressure (1 bar, 1 psi, etc) as our reference state, we can simplify the expression:

10.27

where i(T) is only a function of temperature.

This expression provides the Gibbs energy per mole of a pure, ideal gas at a given P and T

We would like to develop an analogous expression for non-ideal systems, for which V RT/P

Like all non-ideal systems, we can’t predict how V,T and P relate, but we can perform experiments and correlate our data

refref

igig

PPlnRT)P,T(G)P,T(G

PlnRT)T()P,T(G iig


Gibbs Energy for Pure, Non-ideal Gases

The utility of Equation 10.27 leads us to define a direct analogue

10.30where

i(T) the same function of temperaturefi is a defined intensive variable called the fugacity (units of pressure)

Fugacity is used to describe the Gibbs energy of non-ideal gases. In these cases, Gibbs energy does not vary with lnP, so we define a new “chemical pressure” such that the Gibbs energy varies directly with ln fi.

Equation 10.30 is the first part of the definition of fugacity. The second part specifies that as the pressure approaches zero (and the pure gas becomes more ideal) the fugacity approaches the pressure.

ii flnRT)T()P,T(G

Pf:0PAs i


Pure Gases: Fugacity and Fugacity Coefficient

In summary, the fugacity of a pure, non-ideal gas is defined as:

with the specification that:

Together, these definitions allow us to quantify the Gibbs energy of non-ideal gases.

A closely related parameter is the fugacity coefficient, defined by:

such that

Note that a gas behaving ideally is defined as having i = 1, in which case the expression reduces to equation 10.27.

Pfi

i

PlnRT)T()P,T(G ii

ii flnRT)T()P,T(G

Pf:0PAs i


Calculating the Fugacity of a Pure Gas

The simplest means of calculating the fugacity of a pure gas is to compare its behaviour to an ideal system. We will do this frequently in our treatment of non-ideality.

For the non-ideal gas:

For the ideal gas:

Taking the difference of these equations:

ref

ref,iiP

P

ig

P

f

Pf

lnRTdP)VV(ref

)f/fln(RTVdP)P,T(G)P,T(G ref,ii

P

Pref

ref

)P/Pln(RTdPV)P,T(G)P,T(G ref

P

P

igref

igig

ref



We can simplify this relation by an appropriate choice of Pref. As pressure goes to zero, a real gas approaches ideality. Therefore,

With Pref = 0, we have:

or

Substituting V = ZRT/P and Vig = RT/P, we arrive at:

10.34

1P/f:0PAs refref,iref

)P/f(lnRTdP)VV( i

P

0

ig

P

0

igi dP)VV(

RT1

)P/f(ln

P

0i dP

P)1Z(

)P/f(ln



Equation 10.34 is commonly written in terms of the fugacity coefficient:

at a given T.

To calculate the fugacity of a pure, non-ideal gas, all we need is information on the relationship of Z as a function of P at T.

Experimental data Equations of State (van Der Waals, Virial) (Sections 3.1-3.5) Generalized correlations (Sections 3.6, 6.6 of text)

P

0i dP

P)1Z(

ln


5. Calculating Fugacity of Pure Gases

To calculate the fugacity of a pure gas requires a knowledge of the P,V,T behaviour of the substance. This can take many forms, and our choice is often governed by the required precision, and the availability of data/correlations.

In all cases, we can apply the following relation:

Section 10.7 of the text presents a generalized method of calculating fi for pure gases that are non-polar or slightly polar.

Lee-Kesler Correlation:

= (o)(1) (10.63)

where o and 1 are tabulated functions of reduced P and T and

is the acentric factor of the substance

P

0i dP

P)1Z(

ln


Calculating Fugacity of Pure Gases

Virial Equation: We have already used another correlation in an example. In cases where the simplest form of the virial equation of state applies, we can calculate fugacity from:

(10.64)

where(3.50)

(3.51)

See the previously worked out example for a demonstration of this approach.

)BB(TP

ln 1o

r

r

2.4r

1

6.1r

o

T

172.0139.0B

T

422.0083.0B


Applicability of Simple Correlations

It is very important to understand under what conditions the simple correlations apply.


6.2 Pure Component VLE in Terms of Fugacity

Consider a pure component at its vapour pressure: Phase rule tells us, F=2-2+1 = 1 degree of freedom Therefore, at a given T, there can only be a single pressure,

Psat for which a vapour and a liquid are stable

Along the phase boundary, the chemical potentials are equal How do the fugacities of the liquid and gas relate?

P liquid

gas

T


Pure Component VLE in Terms of Fugacity

For the non-ideal, pure gas we can write:(10.36)

For a non-ideal liquid, we can define an analogous expression:(10.37)

At equilibrium, we apply the criterion on the basis of chemical potential to give us:

or(10.38)

In terms of fugacity coefficients:(10.40)

All of these equations apply to a pure substance at Pisat

liqii

liqi

vapii

vapi flnRT)T(flnRT)T(

vapii

vapi

vapi flnRT)T(G

sati

liqi

vapi fff

sati

liqi

vapi

liqii

liqi

liqi flnRT)T(G


Review of Chemical Equilibrium Criteria

We now have several different forms of the criterion for chemical equilibrium. While they stem from the same theory, they differ in practical applicability.

A system at equilibrium has the following properties: the total Gibbs energy of the system is minimized, meaning

that no change in the number of phases or their composition could lower the Gibbs energy further

the chemical potential of each component, i, is the same in every phase within the system

in phases

the fugacity of each component, i, is equal in every phase of the system

in phases iii f...ff

iii ...

0)nG(dP,T


Calculating the Fugacity of Pure Liquids

The derivation of the fugacity of a pure liquid at a given T, P is comprised of four steps:

Step 1. Calculate the fugacity of a vapour at Pisat

Step 2. Calculate the change in Gibbs energy between Pisat and the

given pressure P using the fundamental equation:

dG = VdP - SdT (constant T)which after integration yields:

Given that liquids are nearly incompressible (Viliq is not a strong

function of P) the integral is easily equated to:(A)

satiP

0

sati dP

P)1Z(

)P/f(ln

dPV)P,T(G)P,T(GP

P

liqi

sati

liqliq

sati

ii

)PP(V)P,T(G)P,T(G sati

liqi

sati

liqliq

ii



3. Using the definitions of fugacity:

we can take the difference:(B)

4. Substituting A into B:

or

or

(10.41)

satii

sati

liqi

liqii

liqi

flnRT)T()P,T(G

flnRT)T()P,T(G

)f/fln(RT)P,T(G)P,T(G sati

liqi

sati

liqi

liqi

RT

)PP(VexpPf

RT

)PP(Vexpff

)PP(V)f/fln(RT

sati

liqisat

isati

liqi

sati

liqisat

iliqi

sati

liqi

sati

liqi



We can now calculate the fugacity of any pure liquid using two equations:

(10.41)

and

(10.34)

The exponential within Equation 10.41accounts for the change in Gibbs energy as we compress the liquid from Pi

sat to the specified pressure, P. This is known as the Poynting factor.

This contribution to fugacity is slight at all pressures near Pisat, and

is often assumed to be unity.

RT

)PP(VexpPf

sati

liqisat

isati

liqi

satiP

0

sati dP

P)1Z(

exp

RT

)PP(VexpfactorPoynting

sati

liqi

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CHEE 311J.S. Parent1 5. Equations of State SVNA Chapter 3 Efforts to understand and control phase equilibrium rely on accurate knowledge of the relationship