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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.

Substitution of **Eq. 2** into **Eq. 1** by elimination of *dQ* (a path dependent quantity) and selection of a reversible path (such that *dS*_{G} = 0) gives

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

where *G* ≡ *H* - *TS* is defined as the molar Gibbs energy. For a closed system (*dn* = 0), **Eq. 4** becomes

**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

or, for an open system,

## 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.

We begin by writing the differential entropy change from **Eq. 4** for each open subsystem. The vapor phase equation is

and for the liquid phase,

The change in the total entropy of the isolated system can be written as the summation of Eqs. 8 and 9. The result is

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, *dn*_{L} = -*dn*_{V}, *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, *T*_{L} = *T*_{V}, *p*_{L} = *p*_{V}, and *G*_{L} = *G*_{V}. 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, *G*_{L} = *G*_{V}.

### 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

For an ideal gas, , and **Eq. 13** becomes

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

**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 *G*_{L} = *G*_{V}. 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,

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

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

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 (*z*_{i}) , 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 (*x*_{i} and *y*_{i}).

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 (*dn*_{i}) are transferred from the vapor phase to the liquid phase (or vice versa for the vapor phase). For example, **Eq. 7** becomes

where is the molar Gibbs free energy added to the liquid phase when *dn*_{i} 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 *dn*_{i} 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

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 *dn*_{i} are independent and arbitrary, we must have at equilibrium

**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 *G*_{L} = *G*_{V} 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 *n*_{i} moles. If *n*_{i} 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, *p*_{i} = *y*_{i}*p*, 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

For a real mixture (not an ideal gas or solution), the component fugacity is defined by analogue with **Eq. 22**.

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

**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

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:

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

**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.

## Nomenclature

## 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