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Equations of state
An equation of state (EOS) is a simplified mathematical model that calculates thermodynamic properties and the equilibrium state.
Developing EOS
To develop the EOS, we need equations that relate thermodynamic quantities in terms of pressure, molar volume, and temperature data (PVT data), and we want to eliminate any path dependence by eliminating all properties that are not state functions.
....................(1)
....................(2)
Substitution of Eq. 2 into Eq. 1 by elimination of dQ (a path dependent quantity) and selection of a reversible path (such that dSG = 0) gives
....................(3)
All of the properties in Eq. 3 are state functions; thus, Eq. 3 is independent of the path or process. After combining like terms, Eq. 3 becomes
....................(4)
where G ≡ H - TS is defined as the molar Gibbs energy. For a closed system (dn = 0), Eq. 4 becomes
....................(5)
Eqs. 4 and 5 are examples of fundamental property relations. Other fundamental property relations are possible. For example, differentiation of the definition for total Gibbs energy gives d(nG) = d(nH) - Td(nS) - (nS)dT. Similarly, differentiation of the total enthalpy gives d(nH) = d(nU) + pd(nV) + (nV)dp. Substitution of these relations into Eq. 4 (or Eq. 5) by elimination of the enthalpy term gives the fundamental property relation for the total Gibbs free energy of a closed system as
....................(6)
or, for an open system,
....................(7)
Equilibrium criteria for single-component liquid/vapor systems
Consider an isolated system of a pure fluid with two phases, vapor and liquid. Initially, the temperature, pressure, and other properties of the two phases are not in equilibrium. Fig. 3 illustrates the composite isolated system in which each phase is treated as a subsystem.
Fig. 3 – Isolated system with two open subsystems. The subsystems correspond to the vapor and liquid phases as shown by the dashed lines. The solid line is the boundary of the isolated system.
We begin by writing the differential entropy change from Eq. 4 for each open subsystem. The vapor phase equation is
....................(8)
and for the liquid phase,
....................(9)
The change in the total entropy of the isolated system can be written as the summation of Eqs. 8 and 9. The result is
....................(10)
For an isolated system, the change in the total internal energy is zero (see Eq. 11), as is the change in the total mass and volume. Thus, dnL = -dnV, d(nU)L = -d(nU)V, and d(nV)L = -d(nV)V. The differential entropy change for an isolated system at equilibrium must also be zero (see Eq. 12). Eq. 10 becomes
Because changes in internal energy, volume, and mass of the liquid phase can be arbitrarily set (i.e., are independent), we must have at equilibrium that 
. Thus, at equilibrium, TL = TV, pL = pV, and GL = GV. The first two equilibrium criteria are obvious. The equilibrium condition that the Gibbs free energy of the phases is equal is not as obvious.
Other systems lead to similar equilibrium conditions. For example, for a closed system at constant pressure and temperature (dp = 0, dT = 0) the fundamental property relation from Eq. 6 becomes d(nG) = 0. Thus, the equilibrium criterion here is that the Gibbs free energy must be a minimum. This criterion also leads to the equality of the Gibbs free energy of both phases at equilibrium, GL = GV.
Fugacity of a pure fluid
Fugacity criterion is often used as a substitute for the Gibbs free- energy criterion. The definition for fugacity comes from an analogue with ideal gases that is derived for a closed system under isothermal conditions. Eq. 6 for an isothermal process (dT = 0) is
....................(13)
For an ideal gas, 
, and Eq. 13 becomes
....................(14)
Fugacity is defined by analogy for a fluid that is not ideal. That is, we define the fugacity, f, based on a comparison with Eq. 14, which is written as
....................(15)
Eq. 15 shows that the value for fugacity is whatever is required to give the correct behavior of the real fluid. More exactly, fugacity measures how the Gibbs free energy of a real fluid deviates from that of an ideal gas. Fugacity has units of pressure, and for an ideal gas the fugacity is equal to the pressure (compare Eqs. 15 and 14).
We showed that at equilibrium for a pure fluid GL = GV. By integration of Eq. 15 under isothermal conditions, the Gibbs free-energy criterion implies that the fugacity of the liquid and vapor phases must also be equal at equilibrium. That is, at equilibrium for a pure fluid,
....................(16)
We would like an expression for fugacity in terms of our convenient quantities of pressure, molar volume, and temperature, so that an EOS can be used. Substitution of Eq. 13 into Eq. 15 gives
Subtraction of 
from both sides and some algebraic rearrangement gives
Finally, integration from a reference state of zero pressure (ideal gas state) to the actual pressure gives
From the definition of fugacity, 
(i.e., the fugacity is equal to the pressure for an ideal gas), we have
....................(17)
where 
is known as the fugacity coefficient, and 
is the compressibility factor. The fugacity coefficient is therefore equal to 1.0 for an ideal gas. Eq. 17 requires knowing the compressibility factor as a function of pressure.
Models for compressibility factor, such as a cubic EOS, however, are typically not explicit functions of pressure. A more convenient form would be to transform the integral with respect to pressure to one with respect to volume. Eq. 17 can be transformed to
....................(18)
The importance of Eq. 18 is that the fugacity can be calculated if the molar volume, temperature, and pressure are known over the full range of molar volumes from V to ∞ . Typically, sufficient laboratory data (p, V, T) is not available, and mathematical models, such as cubic EOS, are used. Eqs. 16 and 18 are used on the volumetric properties of pure fluids page to calculate the intensive and extensive state of a pure fluid at equilibrium using a cubic EOS.
Equilibrium criteria for multicomponent liquid/vapor systems
The procedure to determine the equilibrium criterion for multicomponent systems is similar to that used for pure fluids. We consider a closed system with a multicomponent mixture of n moles as illustrated in Fig. 1. Transfer of mass from one phase to the other is allowed, but the overall system is closed, such that the overall composition of the system is constant. Given the overall compositions (zi) , pressure, and temperature, we seek to determine the amount of liquid and vapor present at equilibrium, as well as the component mole fractions for the phases (xi and yi).
As before, the closed system consists of two subsystems, the liquid and vapor phases (see Fig. 3). The primary difference between the derivation for pure fluids and the derivation for multiple components is that the fundamental property relations for the open system must be modified to include mass transfer of different components. That is, we must compute the change in the total Gibbs energy of the liquid phase as small amounts of each component (dni) are transferred from the vapor phase to the liquid phase (or vice versa for the vapor phase). For example, Eq. 7 becomes
....................(19)
where 
is the molar Gibbs free energy added to the liquid phase when dni moles are added to it. The partial molar Gibbs energy 
is also named the chemical potential, 
. The chemical potential measures how much Gibbs energy is added to a mixture when dni is added to it.[1] Thus, Eq. 9 is commonly written as
for the liquid phase, and
for the vapor phase. As for pure fluids, these two equations are added to obtain the differential total Gibbs energy of the entire closed system. Because the differential total Gibbs free energy of the closed system must be zero when pressure and temperature are constant, we obtain
....................(20)
Conservation of mass requires that any component that enters a phase must have come from the other phase so that dniL=-dniV , and upon substitution into Eq. 20,
Because the dni are independent and arbitrary, we must have at equilibrium
....................(21)
Eq. 21 says that at equilibrium the chemical potential of a component in the liquid phase must be equal to the chemical potential of the same component in the vapor phase. This equilibrium criterion reduces to GL = GV for the case of a pure fluid.
Fugacity of a component in a mixture
The equilibrium criterion expressed as component fugacities is often used instead of chemical potentials. The reason for this is primarily one of convenience because component fugacity has units of pressure. Just as for pure fluids, the fugacity of a component is defined as an analogue to an ideal gas mixture.
Consider an ideal gas mixture at a temperature T. The pressure for n moles is 
. In this mixture, each component has ni moles. If ni moles of each component in this mixture occupy the same total volume alone at the same temperature, the pressure would be 
. Division of this result by the pressure gives the partial pressure of a component in an ideal gas mixture. That is, pi = yip, where 
is the vapor molar fraction of each component. The sum of the partial pressures equals the pressure 
.
For an ideal pure gas at constant temperature, we had 
(see Eq. 14). It follows, therefore, that the partial molar Gibbs energy of a component should be evaluated at the partial pressure, or
....................(22)
For a real mixture (not an ideal gas or solution), the component fugacity is defined by analogue with Eq. 22.
....................(23)
where 
is the fugacity of component i in a mixture. The component fugacity for real fluids is sometimes referred to as a corrected partial pressure. Comparison of Eqs. 22 and 23 show that for ideal mixtures, 
. From the integration of Eq. 23 and the use of the equilibrium criteria of Eq. 21, we obtain the equilibrium criteria for component fugacities as
....................(24)
Eq. 24 is often used instead of the equality of chemical potentials to determine equilibrium.
To calculate component fugacities of a real mixture, we subtract the chemical potential for component i in an ideal gas (Eq. 22) from both sides of Eq. 23. The result is
....................(25)
where 
is the component fugacity coefficient. Eq. 25 is used to calculate the deviation of the component fugacity from ideal behavior (this is also known as the residual partial Gibbs energy of component i). Integration of Eq. 25 from zero pressure to the actual pressure gives 
, where the chemical potential is zero and the component fugacity coefficient is 1.0 at zero pressure (the mixture is ideal at zero pressure). From Eqs. 13 and 14 and the definition of fugacity, we obtain using calculus:
....................(26)
where 
is the partial molar compressibility factor.
Eq. 26 is similar in form to Eq. 27 for a pure fluid. Table 1 compares the fundamental equilibrium equations for pure and multicomponent fluids. Because cubic EOS represent Z as an explicit function of V and not Z as a function of p, Eq. 26 is often rearranged to
....................(27)
Eq. 27 shows that the fugacity of a component in a mixture can be calculated when the molar volume, temperature, pressure, and compositions are known over the full range of molar volumes from V to ∞ . Sufficient laboratory data is typically not available for the integration and an EOS must be used. Eqs. 24 and 27 will be used on the volumetric properties and phase behavior of mixtures page to calculate the intensive and extensive state of a multicomponent mixture at equilibrium using a cubic EOS.
Table 1
Nomenclature
| A | = | area normal to specified direction, m2 |
| B | = | formation volume factor of the fluid, volume/volume |
| c | = | isothermal compressibility of a fluid, 1/pressure, 1/Pa |
| cp | = | isobaric compressibility of a fluid, 1/pressure, 1/Pa |
| f | = | fugacity of a pure fluid, pressure, Pa |
|
= | fugacity of a component in a mixture, mole2-pressure/mole2, Pa |
| F | = | external force on one side of system, energy/length, J/m |
|
= | external force vector of surroundings on system, energy/length, J/m |
| G | = | molar Gibbs free energy, energy/mole, J/mole |
| h | = | heat transfer coefficient, energy/temperature/time, J/(Kelvin-sec) |
| H | = | molar enthalpy of fluid, energy/mole, J/mole |
| k | = | binary interaction parameter, dimensionless |
| Ki | = | K-value of ith component, yi/xi, dimensionless |
|
= | displacement vector of system, length, m |
| L | = | liquid mole fraction, moles liquid/total moles, dimensionless |
| M | = | net mass transferred, mass, moles |
| n | = | total moles of all components, moles |
| nc | = | number of components |
| np | = | number of phases |
| p | = | pressure, force/area, Pa |
| Q | = | net heat transferred, energy, J |
| R | = | gas constant, pressure-volume/temperature/mole, Pa-m3/(Kelvin-mole) |
| S | = | molar entropy of fluid, entropy/mole, J/(Kelvin-mole) |
| t | = | time, seconds |
| T | = | temperature, Kelvin |
| u | = | velocity of fluid, length/time, m/sec |
| U | = | molar internal energy, energy/mole, J/mole |
| V | = | vapor mole fraction, moles vapor/total moles, dimensionless or molar volume of fluid, volume/mole, m3/mole |
| W | = | net work performed, energy, J |
| x | = | x-coordinate, length, m |
| xi | = | mole fraction of ith component in liquid, moles ith component in liquid/total moles liquid, dimensionless |
| y | = | y-coordinate, length, m |
| yi | = | mole fraction of ith component in vapor, moles ith component in vapor/total moles vapor, dimensionless |
| z | = | z-coordinate, length, m |
| zi | = | overall mole fraction of ith component, moles ith component/total moles, dimensionless |
| Z | = | compressibility factor of a fluid, dimensionless |
| β | = | temperature dependence function in cubic EOS, typically set to 1.0, dimensionless |
| μi | = | chemical potential of ith component, energy/mole, J/mole |
| ρ | = | molar density of fluid, moles/volume, mole/m3 |
| φ | = | fugacity coefficient for a pure fluid, pressure/pressure, dimensionless |
|
= | fugacity coefficient for a component in a mixture, mole2-pressure/mole2-pressure, dimensionless |
| ω | = | acentric factor, dimensionless |
References
- ↑ Gibbs, J.W. 1961. The Scientific Papers of J. Willard Gibbs. H.A. Bumstead and R.G. Van Name, eds. New York City: Dover.
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See also
Thermodynamics and phase behavior