Chemical potential and fugacity

Thermodynamics of Liquid—Liquid Equilibrium. Phase splitting of a Hquid mixture into two Hquid phases (I and II) occurs when a single hquid phase is thermodynamically unstable. The equiUbrium condition of equal fugacities (and chemical potentials) for each component in the two phases allows the fugacitiesy andy in phases I and II to be equated and expressed as ... 

Recall the fugadties depend on the unknown mole fractions, but the standard state fugacities f° are constants whose values are obtained via judicious choices for standard states ( 10.4.1). Note that the standard states must be applied consistently to both fugacities and chemical potentials. Once those choices have been made, the six equations (10.3.43)-(10.3.45) and (10.3.49)-(10.3.51) can be solved for the three equilibrium mole fractions, the total number of moles (relative to the selected basis), and the two Lagrange multipliers. The calculation will be finished in 10.4.6. 

The key problem is to calculate the phase diagram of ideal solution band for this we first need an expression for fugacity. We start with the basic relationship between fugacity and chemical potential in eq. fio.iS). Taking state A to be component i in solution, and state B the pure component at the same... 

Calculations of chemical equilibrium, which will be the topic of the next section, are facilitated through the introduction of the activity, a property closely related to fugacity and chemical potential. The activity of a component i in mixture is defined as the ratio of its fugacity over the fugacity of the same component at its standard state ... 

Activity is dimensionless fugacity, normalized by the fugacity at the standard state. An immediate consequence of the definition of activity is its relationship to the chemical potential. First, recall that the standard state, which was introduced in Section m.2. is at the same temperature as the state of interest. We now return to eg. (lO.iS). which relates fugacities and chemical potentials between two states at the same temperature ... 

This relation allows us to calculate the fugacity coefficient of a gas from the state equation of a mixture and consequently its fugacity and chemical potential using expressions [8.15] and [8.13]. 

In general, equaHty of component fugacities, ie, chemical potentials, in the vapor and Hquid phases yields the foUowing relation for vapor—Hquid equiHbrium ... 

Substitution of vapor fugacities for chemical potentials (d/i, RT d ln /i) and dividing by RT gives... 

One method of calculating fugacity and hence y is based on the measured deviation of the volume of a real gas from that of an ideal gas. Considet the case of a pute gas The ftee energy F and chemical potential /i changes with pressure according to the equation... 

Thermodynamic Model. In the equilibrium state, the intensive properties -temperature, pressure and chemical potentials of each component- are constant in the overall system. Since the fugacities are functions of temperature, pressure and compositions, the equilibrium condition... 

The chemical potential difference of water molecule between empty and occupied hydrate, iy, — fi l, is calculated accoi ding to either equation (14) or (18) as a function of the pressure of propane. The second virial coefficient is taken into account in calculating both the density and fugacity (or chemical potential) of propane in the gas phase which is assumed to be in equilibrium with the hydrate. If ice Ic is in equilibrium with hydrate at the given temperature, then the chemical potential of ice, equals to fiy, and therefore H-wi assuming — //. is independent of the... 

The grand canonical partition function for a system of volume V, temperature T, and chemical potential g (the chemical potential and fugacity or activity z are related by z = exp ) is given by... 

Vapor sorption measurements yield equilibrium composition and fugacity or chemical potential the isopiestic version (19) is used to determine the uptake of a pure vapor by a nonvolatile material. This technique determines equilibrium composition of a phase which cannot be separated quantitatively from the liquid phase in equilibrium with it. In our application, a nonvolatile crystalline surfactant specimen S is equilibrated with vapor of V, which is, in turn, at equilibrium with a system of S and V consisting of two phases, one rich in S, and one rich in V. At equilibrium, the Gibbs-Duhem relation guarantees. that the initial specimen of S takes up enough V from the vapor phase that the chemical potential of S, as well as of V, is the same as in the biphasic system, and so the composition of the phase formed by vapor sorption is the same as that of the S-rich phase. This composition is easily determined by weight measurement. If the temperature were a triple point, i.e. three phases at... 

To decide among these possibilities we need a stability criterion for mixtures at fixed T, P, and fugacity Equivalently, we can develop the criterion in terms of T, P, and the chemical potential, then convert it to fugacities at the end. Imagine a one-phase binary mixture surrounded by a reservoir that imposes its temperature, pressure, and chemical potential on the system. The latter is accomplished by a semi-permeable membrane that separates the system from the reservoir. The membrane allows molecules of component 1 to pass, but it blocks passage of molecules of component 2. When diffusional equilibrium is established, the value of the chemical potential Gi is the same in the system and in the reservoir. The extensive state of the system is identified by giving values for the fixed quantities T, P, Gj, and N2. 

It has been shown that models for DCFIs can lead to successful descriptions of solution properties and phase equilibria, especially for strongly nonideal systems such as dilute solutions of gases and solids. Because FST formulations are for composition derivatives of pressure and chemical potential or fugacity, the evaluation can appear complex and requires property values at certain reference states. This may be the reason such models have not been implemented into process simulators. However, the reliability and accuracy of models based on perturbations from hard spheres, such as Equation 9.4, is quite good for very many systems, and the results can at least be used to generate local parameterizations and to validate EOS models. Ultimately, results from molecular simulations, as described in Chapter 6, may lead to new relations for the thermodynamic models for use in process simulators. 

The simulations proceeds by first using a gas-phase equation of state to determine the pressure and the fugacity for the gas phase. The simulation then follows a series of trial moves which involve particle displacement, particle insertion and particle removal in order to establish equilibrium. The particles (molecules) in the simulation box are allowed to move, rotate or rearrange their configuration based upon the Boltzmann-weighted Metropolis sampling probability described earlier in Eq. (B12). In order to establish a constant volume, temperature and chemical potential, the number of molecules in the box can increase or decrease. In addition to the displacement moves described already, particle insertions and particle removals are also present. A new particle or molecule can be inserted into the system at a randomly chosen point based on the following probability ... 

Both convention and convenience suggest use of the fugacity in practical calculations in place of the chemical potential ]1. Equation 218 is then replaced by the equal fugacity criterion which follows directiy from equation 160 ... 

General Properties of Computerized Physical Property System. Flow-sheeting calculations tend to have voracious appetites for physical property estimations. To model a distillation column one may request estimates for chemical potential (or fugacity) and for enthalpies 10,000 or more times. Depending on the complexity of the property methods used, these calculations could represent 80% or more of the computer time requited to do a simulation. The design of the physical property estimation system must therefore be done with extreme care. 

Perhaps the most significant of the partial molar properties, because of its appHcation to equiHbrium thermodynamics, is the chemical potential, ]1. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equihbrium problems. The natural logarithm of the Hquid-phase activity coefficient, Iny, is also defined as a partial molar quantity. For Hquid mixtures, the activity coefficient, y, describes nonideal Hquid-phase behavior. 

The chemical potential pi plays a vital role in both phase and chemical-reaction equilibria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence pi, is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, pi approaches negative infinity when either P or Xi approaches zero. While these characteristics do not preclude the use of chemical potentials, the application of equilibrium criteria is facilitated by introduction of the fugacity, a quantity that takes the place of p. but which does not exhibit its less desirable characteristics. 

Driving Force Gas moves across a membrane in response to a difference in chemical potential. Partial pressure is sufficiently proportional to be used as the variable for design calculations for most gases of interest, but fugacity must be used for CO9 and usually for Hg... 

In Chapter 5, we considered systems in which composition becomes a variable, and defined and described the chemical potential. We showed that the chemical potential provides the condition for spontaneity or equilibrium. It is the potential that drives the flow of mass in a chemical process, A useful quantity related to the chemical potential is the fugacity. It can also be thought of as a measure of the flow of mass in a chemical process, and can be used to determine the point of equilibrium. It is often known as the escaping tendency since it can be used to describe the ease with which mass flows from one phase to another, particularly the flow from a solid or liquid phase to a gas phase. 

The isopiestic method is based upon the equality of the solvent chemical potentials and fugacities when solutions of different solutes, but the same solvent, are allowed to come to equilibrium together. A method in which a solute is allowed to establish an equilibrium distribution between two solvents has also been developed to determine activities of the solute, usually based on the Henry s law standard state. In this case, one brings together two immiscible solvents, A and B, adds a solute, and shakes the mixture to obtain two phases that are in equilibrium, a solution of the solute in A with composition. vA, and a solution of the solute in B with composition, a . 

From the equilibrium condition, we know that the chemical potentials /iA and fij of the solute in the two solutions are equal. Since the chemical potentials are equal, the fugacities are also equal,... 

We now have the foundation for applying thermodynamics to chemical processes. We have defined the potential that moves mass in a chemical process and have developed the criteria for spontaneity and for equilibrium in terms of this chemical potential. We have defined fugacity and activity in terms of the chemical potential and have derived the equations for determining the effect of pressure and temperature on the fugacity and activity. Finally, we have introduced the concept of a standard state, have described the usual choices of standard states for pure substances (solids, liquids, or gases) and for components in solution, and have seen how these choices of standard states reduce the activity to pressure in gaseous systems in the limits of low pressure, to concentration (mole fraction or molality) in solutions in the limit of low concentration of solute, and to a value near unity for pure solids or pure liquids at pressures near ambient. 

If we could prevent the mixture from separating into two phases at temperatures below Tc, we would expect the point of inflection to develop into curves similar to those shown in Figure 8.17 as the dotted line for /2, with a maximum and minimum in the fugacity curve. This behavior would require that the fugacity of a component decreases with increasing mole fraction. In reality, this does not happen, except for the possibility of a small amount of supersaturation that may persist briefly. Instead, the mixture separates into two phases. These phases are in equilibrium so that the chemical potential and vapor fugacity of each component is the same in both phases, That is, if we represent the phase equilibrium as... 

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