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Chapter 4 FUGACITY

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**Fundamental equations for closed system consisting of n moles:**

(1) (2) (3) (4)

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**HOMOGENEOUS OPEN SYSTEM**

An open system can exchange matter as well as energy with its surroundings. For a closed homogeneous system, we consider U to be a function only of S and V: U = U(S, V) (5) In an open system, there are additional independent variables, i.e., the mole numbers of the various components present. nU = nU(S,V, n1, n2, ....., nm) (6) where m is the number of components.

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**The total differential of eq. (6) is**

(7) Where subscript ni refers to all mole numbers and subscript nj to all mole numbers other than the ith. Chemical potential is defined as: (8)

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We may rewrite eq. (7) as (9) For a system comprising of 1 mole, n = 1 and ni = xi (10) Eqs. (9) and (10) are the fundamental equations for an open system corresponding to eq. (1) for a closed system.

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**Using similar derivations, we can get the following relations:**

(11) (12) (13) It follows that: (14)

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(9) (15) (16) (17)

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(18)

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(19) (20) (21)

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**Equations (19 – 21) can be written as**

(22) (23) (24)

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**Substituting eqs. (22 – 24) into eq. (18) gives:**

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**All variations d(nS)(2), d(nV)(2), dn1(2), dn2(2), etc**

All variations d(nS)(2), d(nV)(2), dn1(2), dn2(2), etc., are truly independent. Therefore, at equilibrium in the closed system where d(nU) = 0, it follows that

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Thus, at equilibrium (25) (26) (27) (28) (29)

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(30)

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Eq. (14): (31) Eq. (30): Important relations for partial molar properties are: (32) and Gibbs-Duhem equation: (33)

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(34) (35)

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(36)

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GIBB’S THEOREM (37)

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(38) (32): (39)

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**Equation (20) of Chapter 3:**

(3.20) For ideal gas: (40)

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**For a constant T process**

According to eq. (37):

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whence (41) By the summability relation, eq. (32): Or:

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**This equation is rearranged as**

(42) the left side is the entropy change of mixing for ideal gases. Since 1/yi >1, this quantity is always positive, in agree- ment with the second law. The mixing process is inherently irreversible, and for ideal gases mixing at constant T and P is not accompanied by heat transfer.

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**Gibbs energy for an ideal gas mixture:**

Partial Gibbs energy : In combination with eqs. (38) and (41) this becomes or: (43)

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**An alternative expression for the chemical potential can be derived from eq. (2.4):**

At constant temperature: (constant T) Integration gives: (44) Combining eqs. (43) and (44) results in: (45)

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**Fugacity for Pure Species**

The origin of the fugacity concept resides in eq. (44), valid only for pure species i in the ideal-gas state. For a real fluid, we write an analogous equation: (46) where fi is fugacity of pure species i. Subtraction of eq. (44) from Eq. (46), both written for the same T and P, gives: (47)

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**Combining eqs. (3.41) with (47) gives:**

(48) The dimensionless ratio fi/P is another new property, the fugacity coefficient, given the symbol i: (49) Equation (48) can be written as (50) The definition of fugacity is completed by setting the ideal-gas-state fugacity of pure species i equal to its pressure: (51)

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Equation (3.50): (constant T) (3.50) Combining eqs. (50) and (3.50) results in: (constant T) (51) Fugacity coefficients (and therefore fugacities) for pure gases are evaluated by this equation from PVT data or from a volume-explicit equation of state.

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**An example of volume-explicit equation of state is the 2-term virial equation:**

(constant T) Because the second virial coefficient Bi is a function of temperature only for a pure species, (constant T) (52)

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**FUGACITY COEFFICIENT DERIVED FROM VOLUME-EXPLICIT EQUATION OF STATE**

Use equation (3.63): Combining eqs. (3.63) and (50) gives: (53)

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**VAPOR/LIQUID EQULIBRIUM FOR PURE SPECIES**

Eq. (46) for species i as a saturated vapor: (54) For saturated liquid: (55) By difference:

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Phase transition from vapor to liquid phase occurs at constant T dan P (Pisat). According to eq. (4): d(nG) = 0 Since the number of moles n is constant, dG = 0, therefore : (56) Therefore: (57) For a pure species, coexisting liquid and vapor phases are in equilibrium when they have the same temperature, pressure, and fugacity

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**An alternative formulation is based on the corresponding fugacity coefficients**

(58) whence: (59)

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**FUGACITY OF PURE LIQUID**

The fugacity of pure species i as a compressed liquid is calculated in two steps: The fugacity coefficient of saturated vapor is determined from Eq. (53), evaluated at P = Pisat and Vi = Visat. The fugacity is calculated using eq. (49). (53) (49)

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the calculation of the fugacity change resulting from the pressure increase, Pisat to P, that changes the state from saturated liquid to compressed liquid. An isothermal change of pressure, eq. (3.4) is integrated to give: (60) According to eq. (46): ( – ) (61)

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Eq. (60) = Eq. (61): Since Vi, the liquid-phase molar volume, is a very weak function of P at T << Tc, an excellent approximation is often obtained when Vi is assumed constant at the value for saturated liquid, ViL: (62)

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Remembering that: The fugacity of a pure liquid is: (63)

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**Phase Rule First law of thermodynamics:**

thermodynamic properties (U, T, P, V) reflect the internal state or the thermodynamic state of the system. Heat and work quantities, are not properties; they account for the energy changes that occur in the surroundings and appear only when changes occur in a system. They depend on the nature of the process causing the change, and are associated with areas rather than points on a graph.

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(P1, V1) P1 W P (P2, V2) P2 V1 V V2

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The intensive state of a PVT system containing N chemical species and phases in equilibrium is characterized by the intensive variables, temperature T, pressure P, and N – 1 mole fractions for each phase. These are the phase-rule variables, and their number is 2 + (N – 1)(). The masses of the phases are not phase-rule variables, because they have no influence on the intensive state of the system.

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An independent phase-equilibrium equation may be written connecting intensive variables for each of the N species for each pair of phases present. Thus, the number of independent phase-equilibrium equations is ( – 1)(N). The difference between the number of phase-rule variables and the number of independent equations connecting them is the number of variables that may be independently fixed.

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**Called the degrees of freedom of the system F:**

F = 2 + (N – 1)() – ( – 1)(N) F = 2 – + N (60) The intensive state of a system at equilibrium is established when its temperature, pressure, and the compositions of all phases are fixed. These are therefore phase-rule variables, but they are not all independent. The phase rule gives the number of variables from this set which must be arbitrarily specified to fix all remaining phase-rule variables number of degrees of freedom (F).

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**PURE HOMOGENEOUS FLUID**

= 1 F = 2 – + N = 2 – = 2 T, P, V The state of a pure homogeneous fluid is fixed whenever two intensive thermodynamic properties are set at definite values

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**VAPOR/LIQUID EQULIBRIUM FOR PURE SPECIES**

N = 1 = 2 F = 2 – + N = 2 – = 1 TV, PV, VV TL, PL, VL The state of the system is fixed when only a single property is specified. For example, a mixture of steam and liquid water in equilibrium at kPa can exist only at K (100°C). It is impossible to change the temperature without also changing the pressure if vapor and liquid are to continue to exist in equilibrium

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**Intensive variables: TV, PV, VV, V, TL, PL, VL, L = 8**

Independent equations: TV = TL = T PV = PL = P V = L VV = f(T, P) ……. biggest root of the eos V = f(T, P, VV) VL = f(T, P) ………. smallest root of the eos L = f(T, P, VL) Here we have 7 equations with 8 unkowns. It means that we must define one intensive variable (T or P)

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**Algorithm: Input: T Assume P Calculate ZV and ZL (cubic equation)**

Calculate VV and ZL Calculate V (eq. 53 with V = VV) Calculate L (eq. 53 with V = VL) Calculate Ratio = V/ L If Ratio 1, assume new value of P Go to step no. 3

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