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For a closed system consists of n moles, eq. (1.14) becomes: (2.1) This equation may be applied to a single-phase fluid in a closed system wherein no chemical reactions occur. For such a system the composition is necessarily constant, and therefore: (2.2) (2.3)

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Kalau n dan T konstan maka dT = 0 (T dan n konstan)

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Kalau n dan P konstan maka dP = 0 (P dan n konstan)

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Consider now the more general case of a single-phase, open system that can interchange matter with its surroundings. The total Gibbs energy nG is still a function of T and P. Since material may be taken from or added to the system, nG is now also a function of the numbers of moles of the chemical species present. Thus, where n i is the number of moles of species i. (2.4)

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The total differential of nG is: 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: (2.5) (2.6)

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Using similar derivations, we can get the following relations: (2.7) (2.8) (2.9) It follows that: (2.10)

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With this definition and with the first two partial derivatives replaced by (nV) (eq. 2.2) and – (nS) (eq. 2.3), the preceding equation becomes: : Equation (2.11) is the fundamental property relation for single-phase fluid systems of constant or variable mass and constant or variable composition, and is the foundation equation upon which the structure of solution thermodynamics is built. For the special case of one mole of solution, n = 1 and n i = x i : (2.11) (2.12)

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Consider a closed system consisting of two phases in equilibrium. Within this closed system, each individual phase is an open system, free to transfer mass to the other.

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Equation (2.11) may be written for each phase: (2.13) (2.14) where superscripts and identify the phases. The presumption here is that at equilibrium T and P are uniform throughout the entire system.

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The change in the total Gibbs energy of the two-phase system is the sum of these equations. When each total- system property is expressed by an equation of the form, the sum is: (2.15) (+)

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Since the two-phase system is closed, eq. (2.1) is also valid. (2.16) Comparison of the two equations shows that at equilibrium: The changes dn i and dn i result from mass transfer between the phases, and mass conservation requires:

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Since the dn i are independent and arbitrary, the only way the left side of this equation can in general be zero is for each term in parentheses separately to be zero. Hence, (i = 1, 2,..., N) where N is the number of species present in the system. Although not given here, a similar but more compre- hensive derivation shows (as we have supposed) that for equilibrium the same T and P apply to both phases. Therefore:

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Thus, multiple phases at the same T and P are in equilibrium when the chemical potential of each species is the same in all phases. By successively considering pairs of phases, we may readily generalize to more than two phases the equality of chemical potentials; the result for phases is: (i = 1, 2,..., N)(2.17)

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The definition of the chemical potential by Eq. (2.10) as the mole-number derivative of nG suggests that other derivatives of this kind should prove useful in solution thermodynamics. Thus, (2.18) respresents It is a response function, representing the change of total property nM due to addition at constant T and P of a differential amount of species i to a finite amount of solution.

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Eq. (2.6): Eq. (2.18): (2.19)

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For thermodynamic property M: nM = M (T, P, n 1, n 2,..., n i,...) The total differential of nM is:

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Because the first two partial derivatives on the right are evaluated at constant n and because the partial derivative of the last term is given by Eq. (2.18), this equation has the simpler form: (2.19) where subscript x denotes differentiation at constant composition.

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Since and Equation (2.19) becomes: The terms containing n are collected and separated from those containing dn to yield:

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In application, one is free to choose the size of the system (n) and to choose any variation in its size (dn). Thus n and dn are independent and arbitrary. The only way that the left side of this equation can then, in general, be zero is for each term in brackets to be zero. Therefore, or (2.20) (2.21) Multiplication of Eq. (35) by n yields the alternative expression: (2.22)

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Equation (2.20) is in fact just a special case of Eq. (2.19), obtained by setting n = 1, which also makes n i = x i. Equations (2.21) and (2.22) on the other hand are new and vital. Known as summability relations, they allow calculation of mixture properties from partial properties, playing a role opposite to that of Eq. (2.18), which provides for the calculation of partial properties from mixture properties

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One further important equation follows directly from Eqs. (2.21) and (2.22). Since Eq. (2.22) is a general expression for M, differen- tiation yields a general expression for dM: Comparison of this equation with Eq. (2.21), another general equation for dM, yields the Gibbs/Duhem equation: (2.23)

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For the important special case of changes at constant T and P, it simplifies to: (constant T and P) (2.24) In summary, the three kinds of properties used in solution thermodynamics are distinguished by the following symbolism: Solution propertiesM, for example: U, H, S, G Partial properties, for example: Pure-species propertiesMi, for example: U i, H i, S i, G i

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If n moles of an ideal-gas mixture occupy a total volume V t at temperature T, the pressure is: If the n i moles of species i in this mixture occupy the same total volume alone at the same temperature, the pressure is:

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Dividing the latter equation by the former gives: The partial molar volume of species i in an ideal-gas mixture : (i = 1, 2,... )or (2.25)

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An ideal gas is a model gas comprised of imaginary molecules of zero volume that do not interact. Thus, properties for each chemical species are independent of the presence of other species, and each species has its own set of private properties. This is the basis for the following statement of Gibbs's theorem: A partial molar property (other than volume) of a constituent species in an ideal -gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature but at a pressure equal to its partial pressure in the mixture. A partial molar property (other than volume) of a constituent species in an ideal -gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature but at a pressure equal to its partial pressure in the mixture. Mathematical expression for Gibb’s theorem: for(2.26)

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Enthalpy of an ideal gas is independent of pressure. Hence It follows that where H ig is the pure-species value at the mixture T and P. Application of the summability relation yields: (2.27) (2.28) Analogous equations apply for U ig and other properties that are independent of pressure.

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(constant T) From eq. (1.33) At constant temperature Integration from p i to P gives:

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According to eq. (2.26): eq. (2.29) becomes: (2.29) (2.30) where S i ig is the pure-species value at the mixture T and P

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By the summability relation (eq. 2.21): (2.31) When this equation is rearranged as: the left side is the entropy change of mixing for ideal gases. Since 1/y i > 1, this quantity is always positive, in agreement 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 [Eq. (2.28)]. (2.32)

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For the Gibbs energy of an ideal-gas mixture, The parallel relation for partial properties is: In combination with Eqs. (2.28) and (2.30) this becomes: or (2.33)

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An alternative expression for the chemical potential is obtained when G ig is eliminated from Eq. (2.33) by Eq. (1.14). (1.14) At constant T Eq. (1.14) for an ideal gas becomes: (constant T) Integration gives: where i (T), the integration constant at constant T, is a function of temperature only. (2.34)

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Equation (2.33) therefore becomes: (2.35)

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As evident from Eq. (2.17), the chemical potential i provides the fundamental criterion for phase equilibria. This is true as well for chemical-reaction equilibria. However, it exhibits characteristics which discourage its use. The Gibbs energy, and hence i, is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. As a result, absolute values for i do not exist.

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Moreover, Eq. (2.35) shows that for an ideal-gas mixture i approaches negative infinity when either P or y i approaches zero. This is in fact true for any gas. 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 i but which does not exhibit its less desirable characteristics.

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The origin of the fugacity concept resides in Eq. (2.34), valid only for pure species i in the ideal-gas state. For a real fluid, we write an analogous equation: (2.36) in which pressure P is replaced by a new property f i with units of pressure. This equation provides a partial definition of f i, the fugacity of pure species i.

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Subtraction of Eq. (2.34) from Eq. (2.36), both written for the same T and P, gives: According to the definition of Eq. (1.37), G i – 1 G i ig is the residual Gibbs energy, G R. The dimensionless ratio f i /P is another new property, the fugacity coefficient, given the symbol i. Thus, where (2.37) (2.38)

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The definition of fugacity is completed by setting the ideal-gas- state fugacity of pure species I equal to its pressure: Thus for the special case of an ideal gas, G i R = 0 and i = 1. Eq. (2.37) can be written as: (2.39) (constant T)(2.40) Combining the above equation with e. (1.44): 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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The definition of the fugacity of a species in solution is parallel to the definition of the pure-species fugacity. For species i in a mixture of real gases or in a solution of liquids, the equation analogous to Eq. (2.35), the ideal-gas expression, is: Where is the fugacity of species i in solution, replacing the partial pressure y i P. This definition of does not make it a partial molar property, and it is therefore identified by a circumflex rather than by an overbar. (2.41)

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Equation (2.17) is the fundamental criterion for phase equilibrium. Since all phases in equilibrium are at the same temperature, an alternative and equally general criterion follows immediately from Eq. (2.41): (i = 1, 2,..., N)(2.42) Thus, multiple phases at the same T and P are in equilibrium when the fugacity of each constituent species is the same in all phases.

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