NON-IDEAL FLUID BEHAVIOR

Homogeneous fluids are normally divided into two classes, liquids and gases (vapors). Gas: A phase that can be condensed by a reduction of temperature at constant pressure. Liquid: A phase that can be vaporized by a reduction of pressure at constant temperature. The distinction cannot always be made unambiguously, and the two phases become indistinguishable at the critical point.

THE CRITICAL POINT Critical point: The maximum pressure and temperature where a pure material can exist in vapor-liquid equilibrium. Beyond Tc and Pc, the designation of gas vs. liquid is arbitrary. At the critical point, the meniscus between phases slowly fades and dissappears. If one moves around the critical point, it is possible to get from the liquid to the vapor field without crossing a phase boundary.

P-T phase diagram for a pure material. Supercritical C - critical point P-T phase diagram for a pure material.

P-V phase diagram for a pure material. C - critical point. At high T we expect the isotherms to conform to the ideal gas law, i.e., P is inversely proportional to V.

P-V phase diagram for pure H2O.

THE P-V DIAGRAM We can use the lever rule on a P-V diagram to determine the proportion of vapor vs. liquid at any given pressure. The bending of the isotherms in the vapor field from the ideal hyperbolic shape as the critical point is approached indicates non-ideality. The P-V diagram illustrates the difficulty in developing an equation of state for all regions for a pure substance. However, this can be done for the vapor phase.

Schematic isotherms in the two-phase field for a pure fluid. For fluid of density A, the proportion of vapor is Y/(X+Y) and the proportion of liquid is X/(X+Y). For fluid of density B, the proportion of vapor is P/(P+Q) and the proportion of liquid is Q/(P+Q). A B P Q Y X

MOST GENERAL EQUATION OF STATE Two special cases: a) Incompressible fluid  =  = 0 dV/V = 0 (no equation of state exists) V = constant b)  and  are temperature- and pressure-independent

VIRIAL EQUATION OF STATE The most generally applicable EOS PV = a + bP + cP2 + … a, b, and c are constants for a given temperature and substance. In principle, an infinite series is required, but in practice, a finite number of terms suffice. At low P, PV  a + bP. The number of terms necessary to accurately describe the PVT properties of gases increases with increasing pressure.

The limit of PV as P  0 is independent of the gas. T = 273.16 K = triple point of water lim (PV)T, P  0 = (PV)T* = 22.414 (cm3 atm g-mol-1) = a So a is the same for all gases. It is in fact, RT!

I. THE COMPRESSIBILITY FACTOR

THE COMPRESSIBILITY FACTOR Z  PV/RT = 1 + B’P + C’P2 + D’P3 + … or Z = 1 + B/V + C/V2 + D/V3 + … The virial equation of state is the only one which has a firm basis in theory. It follows from statistical mechanics. It can be used to represent both liquids and gases. The term B/V arises due to pairwise interactions of molecules. The term C/V2 arises due to interactions among three molecules, etc.

The constants for the two versions of the virial equation are related by the equations: Disadvantages of the virial equation of state: 1) Cumbersome, many variables 2) Not much predictive value 3) Difficult to use for mixtures 4) Only really useful when convergence is rapid, i.e., at low to moderate pressures.

SOME APPROXIMATIONS Low pressure (0 - 15 bars at T < Tc) Becomes valid over greater pressure ranges as temperature increases. Easily solved for volume. Moderate pressure (0 - 50 bars) Only B and C are generally well known. At higher pressures, other EOS’s are required.

Compressibility factor diagram for methane. Note two things: 1) All isotherms originate at Z = 1where P  0. 2) The isotherms are nearly straight lines at low pressure, in accordance with the truncated virial equation:

The compressibility factor as a function of pressure for various gases The compressibility factor as a function of pressure for various gases. Z measures the deviation from the ideal gas law.

II. EQUATIONS OF STATE

THE OBJECTIVE IN THE SEARCH FOR AN EOS The objective is to find a single equation of state: 1) whose form is appropriate for all gases 2) that has relatively few parameters 3) that can be readily extrapolated 4) that can be adapted for mixtures This objective has only been partially fulfilled.

VAN DER WAALS EQUATION (1873) The a term accounts for forces of attraction between molecules (long-range forces). The b term accounts for the non-zero volume of molecules (short-range repulsion). At low pressure’s real gases are easier to compress than ideal gases; at higher pressures they are more difficult to compress (see Z plot). An alternate form is:

The van der Waals isotherms (labelled with values of T/Tc. The van der Waals equation predicts the shape of the isotherms fairly well in the one-phase region, but shows unrealistic oscillations in the two-phase region. The theory fails because it only considers two-body interactions. P/Pc V/Vc

THE VAN DER WAALS PARAMETERS We can determine how to calculate the a and b parameters by setting the 1st and 2nd derivatives of the van der Waals equation to zero at the critical point (an inflection point), i.e., Solving these equations we get:

CRITICAL CONSTANTS OF GASES - I

CRITICAL CONSTANTS OF GASES - II The van der Waals equation does better than the ideal gas law but is not great. No two-parameter EOS can predicts all these gases.

We can rearrange the previous equations to get the van der Waal parameters in terms of the critical parameters: where Note that, the actual measured value of Vc is not used to calculate a and b!

VAN DER WAALS CONSTANTS FOR GASES a - dm6 atm mol-2; b - 10-2 dm3 mol-1

SOME OTHER EOSs Bertholet (1899) The higher the temperature, the less likely particles will come close enough to attract one another significantly. a and b are different from VdW. Dieterici (1899) Keyes (1917) , , A and l are correction factors.

BEATTIE-BRIDGEMAN (1927) a, b, c, A0, B0 are constants We usually don’t know V, but we know P, so an iterative approach is required: calculate A, B and  with an assumed V value and compute P. If Pcalc  Pexp, then adjust V accordingly and recalculate P.

Rearrangement of the Beattie-Bridgeman equation gives: Where This shows the B-B equation to be simply a truncated form of the virial equation.

a, b, and c are the same as for the B-B EOS BEATTIE (1930) perfect gas volume a, b, and c are the same as for the B-B EOS One can use the Beattie equation to obtain a first guess for the Beattie-Bridgeman equation, which is more accurate because it allows for the variation of A, B and  with volume.

SOME MORE EOS’s Jaffé (1947) - a modification of the Dieterici EOS Wohl (1949)

McLeod (1949) where and a, A, B, and c are constants. Benedict-Webb-Rubin (1940) - specifically devised for hydrocarbons. Useful for both liquids and gases. Define

Martin-Hou (1955) Introduces the reduced temperature: Tr = T/Tc Like many others, this EOS is also a version of the virial EOS.

REDLICH-KWONG (1949) This has been one of the most useful to geology where The R-K EOS is quite accurate for many purposes, particularly if the a and b parameters are adjusted to fit experimental data. However, there have been a number of attempts at improvement.

MODIFICATIONS OF THE R-K EOS de Santis et al. (1974) but b is a constant and a(T) = a0 + a1T. Peng and Robinson (1976) where and  is a parameter for the fluid called the acentric factor.

Carnahan and Starling (1969) - “hard-sphere” model Kerrick and Jacobs (1981) - Hard-Sphere Modified Redlich-Kwong (HSMRK) a(P,T) = an empirically-derived polynomial. where z = c, d, or e

LEE AND KESLER (1975)

DUAN, MOLLER AND WEARE (1992) This is just a modified form the the Lee and Kesler EOS.

CALCULATING FUGACITY COEFFICIENTS BY INTEGRATING AN EOS Using the van der Waals equation:

Using the original Redlich-Kwong equation:

Using the HSMRK EOS of Kerrick and Jacobs (1981)

Activities in CO2-H2O mixtures predicted by a MRK EOS after Kerrick & Jacobs (1981).

Predicted H2O and CO2 activities in H2O-CO2-CH4 mixtures at 400°C and 25 kbar. Calculated for XCH4 = 0.0, 0.05 and 0.20. Dotted curves imply a miscibility gap of H2O-rich liquid and CO2-rich vapor. After Kerrick & Jacobs (1981).

CO2-H2O solvus at 1 kbar. The solid line is a fit of a MRK EOS to experimental data (solid dots). After Bowers & Helgeson (1983).

Effect of 12 wt. % NaCl on the CO2-H2O solvus at 1 kbar Effect of 12 wt. % NaCl on the CO2-H2O solvus at 1 kbar. After Bowers & Helgeson (1983).

III. CORRESPONDING STATES

PRINCIPLE OF CORRESPONDING STATES Reduced variables of a gas are defined as: Pr = P/Pc Tr = T/Tc Vr = V/Vc Principle of corresponding states - real gases in the same state of reduced volume and temperature exert approximately the same pressure. Another way to say this is, real gases in the same reduced state of temperature and pressure have the same reduced compressibility factor. This fact can be used to calculate PVT properties of gases for which no EOS is available.

The reduced compressibility factor vs. the reduced pressure

Generalized compressibility chart. Medium- and high-pressure range. Reduced pressure Generalized compressibility chart. Medium- and high-pressure range. 3.0 2.8 2.6 Z 2.4 Z 2.2 2.0 1.8 1.6 1.4 1.2 1.0 Reduced pressure

EXAMPLE: Calculate the specific volume of NH3 at 500°C and 2 kbar using the reduced Z chart and compare to the ideal gas law prediction. Ideal gas law Corresponding states: Tr = (773 K)/(405.5 K) = 1.91; Pr = (2000 atm)/(111 atm) = 18.02.

Measured compressibility factors for H2O vs Measured compressibility factors for H2O vs. those obtained from corresponding state theory

Measured compressibility factors for CO2 vs Measured compressibility factors for CO2 vs. those obtained from corresponding state theory

Generalized density correlation for liquids. r = /c

PITZER’S ACENTRIC FACTOR The acentric factor of a material is defined with reference to its vapor pressure. The vapor pressure of a subtance may be expressed as: but the L-V curve terminates at the critical point where Tr = Pr. So a = b and If the principle of corresponding states were exact, all materials would have the same reduced-vapor pressure curve, and the slope a would be the same for all materials. However, the value of a varies.

The linear relation is only approximate; a is not defined with enough precision to be used as a third parameter in generalized correlations. Pitzer noted that Ar, Kr and Xe all lie on the same reduced-vapor pressure curve and this passes through log Prsat = -1 at Tr = 0.7. We can then characterize the location of curves for other gases in terms of their position relative to that for Ar, Kr and Xe. The acentric factor is:  can be determined from Tc, Pc and a single vapor pressure measurement at Tr = 0.7.

Approximate temperature-dependence of reduced vapor pressure Slope  -3.2 (n-octane) Approximate temperature-dependence of reduced vapor pressure

ACENTRIC FACTORS FOR GASES

PRINCIPLE OF CORRESPONDING STATES - REVISITED Restatement of principle of corresponding states: All fluids having the same value of  have the same value of Z when compared at the same Tr and Pr. The simplest correlation is for the second virial coefficients: The quantity in brackets is the reduced 2nd virial coefficient.

The range where this correlation can be used safely is shown on the chart on the next slide. For the range where the generalized 2nd virial coefficient cannot be used, the generalized Z charts may be used: Z = Z0 + Z1 These correlations provide reliable results for non-polar or only slightly polar gases. The accuracy is ~3%. For highly polar gases, the accuracy is ~5-10%. For gases that associate, even larger errors are possible. The generalized correlations are not intended to be substitutes for reliable experimental data!

Generalized correlation for Z0 Generalized correlation for Z0. Based on data for Ar, Kr and Xe from Pitzer’s correlation.

Generalized correlation for Z1 based on Pitzer’s correlation.

EXAMPLE -1 What is the volume of SO2 at P = 500 atm and T = 500°C? According to the ideal gas law: Using the acentric factor:  =0.273; Tr = 773/430.8 = 1.79; Pr = 500/77.8 = 6.43. From the charts: Z0 = 0.97; Z1 = 0.31. Z = 0.97 + 0.273(0.31) = 1.055

saturation Line defining the region where generalized second virial coefficients may be used. The line is based on Vr  2.

EXAMPLE - 2 What is the volume of SO2 at P = 150 atm and T = 500°C? According to the ideal gas law: Using the acentric factor:  =0.273; Tr = 773/430.8 = 1.79; Pr = 150/77.8 = 1.93.

Vc = 0.122 L mol-1 Vr = 0.392/0.122 = 3.25

CORRESPONDENCE PRINCIPLE FOR FUGACITY Correspondence principles and generalized charts exist for fugacity and other thermodynamic properties. For fugacity, both two- and three-parameter generalized charts have been developed. Again, these are to be used only in the absence of reliable experimental data.

I. We can use this equation together with the generalized Z charts. 1) Look up Pc and Tc of gas 2) Calculate Pr and Tr values for desired T’s and P’s 3) Make a Table of Z from the generalized charts at various values of Tr and Pr. Of course, we must have Pr values from 0 to the pressure of interest at each temperature. 4) Graph (Z-1)/Pr vs. Pr for each Tr. 5) Determine the area under the the graph from Pr = 0 to Pr = Pr to get ln . II. Used generalized fugacity charts.

USE OF TWO-PARAMETER GENERALIZED FUGACITY CHARTS EXAMPLE 1: Calculate the fugacity of CO2 at 600°C (873 K) and 1200 atm. Tc = 304.2 K; Pc = 72.8 atm Tr = 2.87; Pr = 16.48 From the chart   1.12 so f = (1.12)(1200) = 1344 bars

EXAMPLE 2: What is the fugacity of liquid Cl2 at 25°C and 100 atm EXAMPLE 2: What is the fugacity of liquid Cl2 at 25°C and 100 atm? The vapor pressure of Cl2 at 25°C is 7.63 atm. For the vapor coexisting with liquid: Tc = 417 K; Pc = 76 atm Tr = 0.71; Pr = 0.10 from the chart   0.9 f = (0.9)(7.63) = 6.87 atm Now we must correct this to 100 atm. V = 51 cm3 mol-1; assume to be constant. f2 = 8.36 atm

THREE-PARAMETER CORRELATIONS FOR FUGACITY ETC. Enthalpy: Entropy: Density: Tables for these correlations can be found in Pitzer (1995) Thermodynamics. McGraw-Hill.

IV. GASEOUS MIXTURES

IDEAL GAS MIXTURES Mixture as a whole obeys: Two such mixtures are in equilibrium with each other through a semi-permeable membrane when the partial of each component is the same on each side of the membrane. There is no heat of mixing. The gas mixture must therefore consist of freely moving particles with negligible volumes and having negligible forces of interaction.

DALTON’S LAW VS. AMAGAT’S LAW Dalton’s Law: Pi = XiPT Amagat’s Law: Vi = XiVT These two laws are mutually exclusive at a given pressure and temperature. At constant VT and T At constant PT and T

THERMODYNAMICS OF IDEAL MIXING - REVISITED We have previously shown that: using Dalton’s Law we can derive: and for entropy we have:

NON-IDEAL MIXTURES OF NON-IDEAL GASES For a perfect gas mixture: For an ideal mixture of real gases: For a real mixture of real gases: Lewis Fugacity Rule Correction for non-ideal mixing Correction for non-ideal gas

DALTON’S LAW AND GENERALIZED CHARTS Calculate reduced pressure according to:

AMAGAT’S LAW AND GENERALIZED CHARTS Calculate reduced pressure according to:

PSEUDOCRITICAL CONSTANTS

KAY’S METHOD Assumes a linear critical curve between the critical points for A and B. When answers are near the critical point for the mixture, we cannot be certain that we are not dealing with a liquid-vapor mixture.

JAFFÉ’S METHOD For binary mixtures only.

MIXING CONSTANTS IN EQUATIONS OF STATE Van der Waals and simple Redlich-Kwong EOS: Use if no mixture data are available.

Beattie-Bridgeman EOS:

Benedict-Webb-Rubin EOS:

Virial Equation of State: Z = 1 + B/V + C/V2 + D/V3 + …

PREDICTION OF CRITICAL CONSTANTS Critical Temperature: I. All compounds with Tboil (1 atm) < 235 K and all elements: Tc = 1.70Tb - 2.00. II. All compounds with Tboil (1 atm) > 235 K. A. Containing halogens or sulfur Tc = 1.41Tb + 66 - 11F F = No. of fluorine atoms B. Aromatics and napthenes Tc = 1.41Tb + 66 - r(0.388Tb - 93) r = ratio of non-cyclic carbon atoms to total carbon atoms.

C. All other compounds Tc = 1.027 Tb + 159 Critical Pressure: where Tc is in K and Vc is in cm3 g-1. Critical Volume: where is a parameter called the Sugten Parachor.

SUGTEN PARACHOR VALUES FOR ATOMS AND STRUCTURAL UNITS