Abstract
The fugacity is a term that has Latin origins, referring to pace and speed. Early work somewhere in the 20th century, on fugacity, demonstrated that chemical equilibrium is established by the free energies of reacting material. Since then, various approaches and concepts on fugacity have been provided. The gas fugacity of noble gasses has been studied, and models have been proposed for calculating the solubility of noble gasses. The development of various models has considered pressure effects, differential equilibrium between components and temperature changes.
The paper discusses the evolution of various multimedia fugacity models and the fugacity of pure substances. A literature review is performed to discuss the earlier work done on fugacity approaches and various correlations and equations. Application models are discussed that include all four models which are Models Level I-IV.
Introduction
When multiple phases exist in the system, a material and energy transfer takes place until a stage reaches where all the phases are in equilibrium with each other. In other words, the system inclines to a state where the all mechanical, thermal and chemical potential gradients inside all phases discontinue existing. This paper focuses the concept of the property critical to the account of real mixtures, as well as the idea of the chemical perspective essential for obtaining the principle of chemical or phase reaction equilibrium. Such properties assist the purpose of the first and second law ideology to describe the alterations in internal energy, entropy and enthalpy of multiphase and multicomponent systems, quantitatively. A different category of properties of multi-component and multi-phase systems include the partial molar property and the chemical potential. The partial molar property describes behaviour of the homogeneous multi-component systems; on the other hand, the chemical potential provides a fundamental description of equilibrium in multi-phase and reactive systems.1
In the case of pure gases, the ideal gas mixture acts as a facilitator for property estimation of real gas mixtures. A comparative evaluation of the properties of the real and ideal gas mixtures escorts the introduction of the perception of fugacity, a property associated with the chemical potential. This word was firstly coined by Gilbert N. Lewis in 1901, which is often construed as ‘a propensity to flee or escape.’2 The functional similarity of fugacity with the chemical potential offers an expedient pathway to link the pressure, temperature and phase composition of the system under the state of equilibrium.1, 3, 4
Definition of Fugacity in terms of Gibbs energy
The particular Gibbs function for a compressible substance is:
dG=VdP-SdT (1)
Gibbs function for a pure substance is equivalent to the chemical potential, so for an isothermal process it can be written as:
dµT= vdP (2)
Substituting by an ideal gas EOS it obtains:
dµT,ideal = RTdP/P=RTd lnP (3)
Through applying equation (3) the chemical potential of a pure substance can be calculated that acts as an ideal gas. For a real gas, an EOS can be implemented, and the chemical potential is calculated by integration. This concept is not much appreciated. Instead, implementing fugacity (f) still holds good for a real gas. 4
dµT,real = RTd ln f , (4)
Additionally the ideal gas behaves similarly of real gas, at very low pressure, so it is clear that:
lim1→ 0 f/ P = 1 (5)
Thus, equation (4) with the reference value of f =0 at pressure=0 the fugacity is completely defined. 4, 5
Literature review and purpose of research
The term fugacity is a Latin word which means fleetness that was first coined by an American physical chemist Gilbert N. Lewis in early 20th century. Lewis’ work on fugacity completed the concept of chemical equilibria introduced 24 years back by Willard Gibbs, which demonstrated that chemical equilibrium was resolved established by the free energies of the reacting material. On the other hand, a vast data collection was available on the enthalpies of formation and the chemical reactions of substances. Apart from latest approaches, a series of empirical laws of behavior of dilute solutions and ideal gases transformed the face of physical chemistry. Such amount of data made it necessary to either directly measure the missing free-energy standards or supplement the current enthalpy data with entropy standards that can facilitate their calculation. The inclusion of the behavior of real gases and concentrated solutions to the empirical laws became a significant requirement. To bridge this gap, Lewis instigated a dynamic experimental program considered to calculate the missing entropy and free-energy standards. Secondly he sequentially pioneered the concepts of activity (1907), fugacity (1901) and ionic strength (1921). 6
Various approaches and concepts have provided on the gaseous fugacity time to time. In a study, for the calculation of helium concentration in the melt stage, chemist computed the noble gas fugacity through a modified Redlich-Kwong state equation for the H2O-CO2-noble gas.7 Another proposed model for calculating the solubility of other noble gases theoretically is the Paonita model where parameters can be assumed in the similar way as for dry melts. To describe the temperature and pressure effects, differentiation of equilibrium between vapor and silicate melt was described using the below thermodynamic relation. 8
a and f stand for the activity of noble gas in melts, P° and T° illustrate the standard state and ∆H0 P°,T denotes the enthalpy of dissolution of the noble gas at P° and T. 4,9
A wide literature review was accomplished on the mutual solubilities of CO2−water systems and CO2 hydrate-forming conditions. 10
Klauda proposed a conventional thermodynamic model for the equilibrium pressures of various structures of gas hydrates is that eliminates the requirement of energy parameters utilized in Van der Waals and Platteeuw (vdWP) type of models. This model employs published Kihara cell latent parameters resolved by viscosity and second virial coefficient data.11
Lebowitz, proved the presence of phase transitions in numerous two‐component lattice gases. Out of which some were isomorphic to spin organizations or to fluids with asymmetrical molecules of diverse orientations. 12
An extended model for calculating the gas phase fugacities involves the use of Soave-Redlich-Kwong (SRK) equation of state that employs a quadratic mixing rule for the parameter “a” while a linear mixing approach for the parameter “b”. 13
Fugacity Capacity
The fugacity of a component determines the chemical potential in a phase. Unequal fugacities of a component across two different phases will be redistributed the component between those phases. The transfer of component occurs from higher fugacity phase to lower fugacity phase. With equal fugacities, equilibrium exists, and the net transfer is zero. The expected relationship between fugacity [f (Pa)], and concentration [C (mol/m3 )], is
C = Zf (1)
Z is a proportionality constant also known as “fugacity capacity".4, 14
The fugacity capacity Z I determined by the chemical nature, temperature, and the medium. The probable Z values for chemicals in the water and air phases are 1/H and 1/R*Ta respectively. Here, H is Henry's law constant (Pa m3/mol), R is constant of gas (8.314 Pa m^3/mol K) and Ta is the absolute temperature (K). H can also be determined as PS/CS, where CS is water solubility (mol/m3), and PS is the saturated pressure of the vapor (Pa). For soil, Z is described as rs*Kp/H. Here rs is the soil density (kg/l) and Kp (l/kg) is a coefficient that describes the soil and water phase partitioning. 14
At equilibrium, the fugacity for each compartment is equal to the other compartment. The Fugacities are determined for every one of the compartments in the ecosystem. Based on the fugacity the prediction for a given chemical distribution patterns is done. The aim is to determine the expected behavior rather than the expected concentrations. 14
There are various computer simulated models that are used to determining the common features of an existing or a new chemical's behavior. A general Fugacity Model can predict the media that the chemical will partition into. The model can also predict the intermediate transport tendency and primary loss mechanisms. The model is available for download and run using QBASIC. The program requires input data that are listed on the download page and also requires registration. The program is easy to run but requires understanding of the values. It also creates a lot of data that requires time to decipher. 14
Fugacity in chemical thermodynamics
In the chemical thermodynamics, the fugacity (f) of a real gas is an effectual pressure that substitutes the factual mechanical pressure in precise chemical equilibrium calculations. It is equivalent to an ideal gas pressure that owns the similar chemical potential as a real gas. For instance, at 0°C and a pressure P= 100 atm, nitrogen gas (N2) shows the fugacity (f) of 97.03 atm. It indicates that the chemical potential of real N2 at 100 atm pressure is not as much as of an ideal nitrogen gas. It also suggests that the value of the chemical potential would be that which id for the ideal nitrogen gas at a pressure of 97.03 atm. Fugacities are calculated experimentally or evaluated through various models such as Vander Waals gas which is more close to real gas than an ideal gas. The fugacity is connected with ideal gas pressure through the dimensionless fugacity coefficient ϕ. 1, 3, 10, 14
Φ= f/P
The fugacity is strongly associated with the thermodynamic activity. The activity in gas can be described as the deviation of fugacity from the standard state fugacity in a liquid phase. 13 Calculations involving real gases chemical equilibrium use fugacity instead of pressure. The thermodynamic condition for chemical equilibrium indicates that the value of total chemical potential of reactants is equivalent to the products. If the chemical potential of gas is articulated through the function of fugacity, the equilibrium state might be altered into the recognizable reaction quotient form; only the pressures will be replaced by fugacities. 13,14
For a condensed liquid or solid phase in equilibrium along with their vapor phase, the chemical potential is equal to the chemical potential of vapour phase. Thus, this fugacity is mostly similar to the vapor pressure when the vapor pressure is low. 13
Fugacity of pure substances in terms of chemical potential
The chemical potential presents a basic description of phase equilibrium. It also facilitates as an effective tool in illustrating chemical reaction equilibrium. Nevertheless, it is not easy to directly connect the chemical potential with thermodynamic properties acquiescent to simple experimental determination, for instance, the volumetric properties. To fill this gap, the function of fugacity relates the chemical potential with thermodynamic properties. 14, 15
The concept of fugacity is highly developed on the basis of the following thermodynamic relation for an ideal gas. For a single component closed system that contains an ideal gas the specific Gibbs function for a plain compressible substance is:
dG=VdP-SdT
For a pure ideal gas i, at constant temperature the above equation reduces to:
(6)
Γi (T) stands for the constant of integration
Through the essential simplicity of equation 6 it is applied to a real fluid but by replacing the pressure with the fugacity we found equation 7:
dGi = RTd ln f
Hence equation will transform as Gi= Γi (T) + RTd ln fi (7)
fi is the units of pressure and often described as fictitious pressure. It is noticeable that equation 8 is generalized to all phases and can be integrated with solids and liquids as well. However, the fugacity calculation for solids and liquids varies from that for gases. This equation offered a partial definition to fi, the fugacity for pure species ‘i’. Subtracting equation 6 form 7 produces: 1, 2, 4, 12, 14
(8)
And the dimension ratio of fi/P will be termed as Fugacity coefficient (ϕ). Therefore:
For an ideal gas the following relations hold and ϕi =1.
Using an equation with residual property equation of real gases
ME=M-Mid
(Where M denotes molar mass and ME is the difference between the actual property value of a solution and the value it would obtain as an ideal solution at similar pressure, temperature pressure, and composition.), we got at constant T: 15
(9)
Vapour phase fugacities, ϕ approach
The ϕ approach points towards the fugacities’ modelling through using fugacity coefficients as a replacement for activities. Instead of using an activity coefficient and describing various standard states, only one standard state of realistic symmetrical ideal gas is characteristically used denoted by Dig and y on mol fraction scale. This is the standard state applied on gas phases and acknowledged as the ideal gas standard state. This state extremely differs from the liquid phase standard states because it is reliant on composition.1, 5, 15
The fugacity coefficient (T, P, n ) defines component i’s divergence in the mixture from the ideal gas law at (T,P,n):
(10) denotes fugacity for the non-ideal gas phase and denotes standard state fugacity. The standard state is a function of composition. is illustrated as:
(11)
Which, is inserted with equation 10:
(12)
(Fosbøl, P; Stenby, E. H.; Thomsen, K)
Fugacity expressions for pure gases
Fugacity coefficient for pure gases can be easily evaluated by applying below equation to a volume-explicit equation.
(13)
The condensed virial EOS provides an exemplary of this type, where compressibility factor of pure species (i) is illustrated by:
Thus using the above equation (13): at constant T
Hence :
General Correlations for Fugacity Coefficient
A general correlation approach presented for the calculation of compressibility factor Z and the remaining enthalpy and entropy of gases can be implemented to compute the fugacity coefficients for pure gases, as well as gaseous mixtures. The generalized form can be expressed as follows:
Where
For acquiring the values from above equations the tabulated data for Z0 and Z1 for the values of Tr and Pr can be applied for numerical or graphical computations as well.
Fugacity and Fugacity Coefficient of Species in Mixture
The definition of fugacity for mixture species is identical to the meaning of the wholesome species fugacity. The chemical potential of an ideal gas mixture is:
Therefore:
pi is the partial pressure of ith species.1,2,3,15
Evolution of Multimedia Fugacity Models
Multimedia fugacity model summarizes the progression controlling chemical behavior in the environmental standards through producing and implementing of mathematical statements or models of chemical providence. Generally most of the chemicals have capability to shift from one medium to another. Multimedia Fugacity Models are specifically designed and used to analyse and predict the chemicals’ behaviour in diverse environmental section. Based on the thermodynamic properties of materials these models are highly valuable n environmental studies. The models are invented using the conception of fugacity, as a standard of balance of equilibrium and expedient means of calculating multimedia equilibrium separation.16 A mathematical expression describing the rate of diffusion of chemicals or the transportation between phases is known as the fugacity of chemicals. The difference in fugacity between source and destination phases determines the transfer rate. A first step to model building is to set up a mass balance equation phase-wise including concentrations, fugacities, amounts and fluxes. ‘Fugacity capacity’ is proportionality constant for various media and is expressed as Z-values (SI Unit: mol/m3 Pa). Transport parameters are articulated using D-values (SI unit: mol/Pa h) and used for processes like reaction, advection and intermediatef transport. Henry’s law constant, equilibrium partitioning coefficients and other similar physical-chemical properties are used for calculating the Z-values. It is evident that the mass balance models are used for determining the long term behaviour of spilled materials. These models can be the basis of guidance and prioritizing the emergency response measures. The Level I, II, III models based on fugacity are easy to use, transparent and easy to understand. These models can be used for predictions of a chemical spill. These models are well-tested and widely accepted and standard models that can be used for chemical evaluations.17
The early forms of model like Level-I replicate the chemical equilibrium situations that arise between different phases of varied compositions and volume. The existing fugacity can be shown by M=ViZi where, M is the quantity of chemical (mol), Vi is the volume per m3, and Zi is the phase Z value. Although very elementary a Level-I solution is mostly useful for indicating the chemical partition. Level-II models determine the chemical reaction or degradation rate, even though maintaining the interphase equilibrium. Level-III models are more advanced and establish intercompartmental transfer rates where the equilibrium does not apply. Level-III models require to be informed about the mode-of-entry of the chemical into the environment that is water, air, soil or some other media. Each of these models provides precious insights into the chemical persistence and the potential LRT (Long Range Transport) in water or air. Level-IV models consist of engaging the differential mass balance equations that are used for determining the dynamic or time-dependent behavior of chemicals.17, 18
Application of Models
The multimedia fugacity models have four levels that are applied for prediction and determination of transport and outcome of the organic chemicals in a multi-compartmental environment. The complexity and the number of the phases determine the level of the model to be applied. Some of the models apply to steady-state conditions while others apply to time-dependent conditions utilizing differential equations. The models are utilized for assessing the inclination of the chemicals to change state in different temperature zones. The multi-compartmental approach is found to be beneficial in quantifying the water-air sediment interactions, also beneficial in ‘QWASI’ models designed for chemical outcomes in lakes. The models are also applied in POPCYCLING-BALTIC models that described the outcomes of organic pollutants in the Baltic region. 16
Conclusion
There are numerous approaches proposed to evaluate the gas fugacity. A number of problems come up during the determination of the fugacities of mixture of gasses. The problems are mainly due to experimental data that is significantly different. There may be the availability of only critical pressure and temperature values, or P-V-T data of components though sometimes data of the mixtures is available completely. In each case complete use of the available data must be done. For moderate pressure, short methods are advantageous, thereby avoiding unnecessary waste of time.
References
- Mackay, D.; Multimedia environmental models: the fugacity approach; CRC press;
2001; pp. 185
- Mackay, D. Finding fugacity feasible. Environmental Science & Technology.
1979, 13, 1218-1223.
- Holloway, John R.; Fugacity and activity of molecular species in supercritical fluids;
Springer Netherlands; 1977; pp. 161-181
- Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular thermodynamics
of fluid-phase equilibria; Pearson Education; 1998.
- Koretsky, M. D. Engineering and chemical thermodynamics; Hoboken, NJ:
Wiley; 2004.
- Jensen, M. R.; Ungerer, P.; De Weert, B.; Behar, E. Crystallisation of heavy
hydrocarbons from three synthetic condensate gases at high pressure.
Fluid phase equilibria. 2003, 208, 247-260.
- de Santis, R.; Breedveld, G. J. F.; Prausnitz, J. M. Industrial & Engineering
Chemistry Process Design and Development. 1974, 13, 374-377.
- Paonita, A. Noble gas solubility in silicate melts: a review of experimentation and
theory, and implications regarding magma degassing processes.
Annals of Geophysics. 2005
- Flowers, G. C. Correction of Holloway's (1977) adaptation of the modified
Redlich-Kwong equation of state for calculation of the fugacities of molecular
species in supercritical fluids of geologic interest. Contributions to Mineralogy
and Petrology. 1979, 69, 315-318.
- Chapoy, A.; Mohammadi, A. H.; Chareton, A.; Tohidi, B.; & Richon, D.
Measurement and modeling of gas solubility and literature review of the
properties for the carbon dioxide-water system.
Industrial & engineering chemistry research. 2004, 43, 1794-1802.
- Klauda, J. B.; Sandler, S. I. A fugacity model for gas hydrate phase
equilibria. Industrial & engineering chemistry research. 2000, 39, 3377-3386.
- Lebowitz, J. L.; Gallavotti, G. Phase transitions in binary lattice gases.
- Fosbøl, P. L.; Stenby, E. H.; Thomsen, K. Carbon Dioxide Corrosion: Modelling and
Experimental Work Applied to Natural Gas Pipelines. Technical University of
DenmarkDanmarks Tekniske Universitet, CenterCenters, Center for Energy
Resources EngineeringCenter for Energy Resources Engineering.
2007; pp 5-11
- Redlich, O.; Kwong, J. N. S. On the Thermodynamics of Solutions. V.
An Equation of State. Fugacities of Gaseous Solutions.
Chemical Reviews. 1949, 44, 233-244.
- Hildebrand, J. H. Gilbert Newton Lewis. 1875-1946. Obituary Notices of Fellows of
the Royal Society. 1947, 5, 491-506.
- Maddalena, R. L.; McKone, T. E.; Layton, D. W.; Hsieh, D. P. Comparison of
multi-media transport and transformation models: regional fugacity
model vs. CalTOX. Chemosphere. 1995, 30, 869-889.
- Mackay, D.; Arnot, J. A.; Webster, E.; Reid, L. The evolution and future
of environmental fugacity models. In Ecotoxicology Modeling; Springer US.
2009, 355-375.
- Mackay, D.; Paterson, S. Calculating fugacity. Environmental Science & Technology.
1981, 15, 1006-1014.
