Fundamental equations for closed system consisting of n moles: (1) (2) (3) (4)
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, n 1, n 2,....., n m ) (6) where m is the number of components.
The total differential of eq. (6) is (7) Where subscript n i refers to all mole numbers and subscript n j to all mole numbers other than the i th. Chemical potential is defined as: (8)
We may rewrite eq. (7) as (9) For a system comprising of 1 mole, n = 1 and n i = x i (10) Eqs. (9) and (10) are the fundamental equations for an open system corresponding to eq. (1) for a closed system.
Using similar derivations, we can get the following relations: (11) (12) (13) It follows that: (14)
Equation (20) of Chapter 3: (3.20) For ideal gas: (40)
For a constant T process (constant T) According to eq. (37):
whence By the summability relation, eq. (32): Or: (41)
This equation is rearranged as 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. (42)
Gibbs energy for an ideal gas mixture: Partial Gibbs energy : In combination with eqs. (38) and (41) this becomes or: (43)
An alternative expression for the chemical potential can be derived from eq. (2.4): At constant temperature: (2.4) (constant T) Integration gives: (44) Combining eqs. (43) and (44) results in: (45)
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 f i is fugacity of pure species i. Subtraction of eq. (44) from Eq. (46), both written for the same T and P, gives: (47)
Combining eqs. (3.41) with (47) gives: The dimensionless ratio f i /P is another new property, the fugacity coefficient, given the symbol i : (48) (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)
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.
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)
FUGACITY COEFFICIENT DERIVED FROM VOLUME-EXPLICIT EQUATION OF STATE Use equation (3.63): Combining eqs. (3.63) and (50) gives: (53)
VAPOR/LIQUID EQULIBRIUM FOR PURE SPECIES Eq. (46) for species i as a saturated vapor: (54) For saturated liquid: (55) By difference:
Phase transition from vapor to liquid phase occurs at constant T dan P (P i sat ). According to eq. (4): d(nG) = 0 Since the number of moles n is constant, dG = 0, therefore : Therefore: (56) (57) For a pure species, coexisting liquid and vapor phases are in equilibrium when they have the same temperature, pressure, and fugacity
An alternative formulation is based on the corresponding fugacity coefficients whence: (58) (59)
FUGACITY OF PURE LIQUID The fugacity of pure species i as a compressed liquid is calculated in two steps: 1.The fugacity coefficient of saturated vapor is determined from Eq. (53), evaluated at P = P i sat and V i = V i sat. The fugacity is calculated using eq. (49). (53) (49)
2.the calculation of the fugacity change resulting from the pressure increase, P i sat 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)
Eq. (60) = Eq. (61): Since V i, the liquid-phase molar volume, is a very weak function of P at T << T c, an excellent approximation is often obtained when V i is assumed constant at the value for saturated liquid, V i L : (62)
Remembering that: The fugacity of a pure liquid is: (63)
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.
P1P1 P2P2 PP V V2V2 V1V1 (P 1, V 1 ) (P 2, V 2 ) W
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.
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.
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).
PURE HOMOGENEOUS FLUID T, P, V N = 1 = 1 F = 2 – + N = 2 – 1 + 1 = 2 The state of a pure homogeneous fluid is fixed whenever two intensive thermodynamic properties are set at definite values
N = 1 = 2 F = 2 – + N = 2 – 2 + 1 = 1 VAPOR/LIQUID EQULIBRIUM FOR PURE SPECIES T V, P V, V V T L, P L, V L 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 101.325 kPa can exist only at 373.15 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
Intensive variables: T V, P V, V V, V, T L, P L, V L, L = 8 Independent equations: T V = T L = T P V = P L = P V = L V V = f(T, P) ……. biggest root of the eos V = f(T, P, V V ) V L = f(T, P) ………. smallest root of the eos L = f(T, P, V L ) Here we have 7 equations with 8 unkowns. It means that we must define one intensive variable (T or P)
Algorithm : 1.Input: T 2.Assume P 3.Calculate Z V and Z L (cubic equation) 4.Calculate V V and Z L 5.Calculate V (eq. 53 with V = V V ) 6.Calculate L (eq. 53 with V = V L ) 7.Calculate Ratio = V / L 8.If Ratio 1, assume new value of P 9.Go to step no. 3