Gibbs Duhem Equation

8 Key excerpts on "Gibbs Duhem Equation"

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  Physical Chemistry

  • Robert J. Silbey, Robert A. Alberty, George A. Papadantonakis, Moungi G. Bawendi(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  In a multiphase system there is a separate Gibbs–Duhem equation for each phase. For a system with two species at constant temperature and pressure that contains 1 mol of material, x 1 d 1 + x 2 d  2 = 0 (4.104) x 1 d 1 + (1 − x 1 ) d  2 = 0 (4.105) where x 1 is the mole fraction of component 1 and (1 – x 1 ) is the mole fraction of component 2. Thus, when the composition is changed at constant T and P, the change in the chemical potential of species 2 is not independent of the change in the chemical potential of species 1. Later, in Section 6.6, we will use this form of the Gibbs–Duhem equation to show that if Henry’s law holds for the solute (species 2), Raoult’s law holds for the solvent (species 1). Comment: We can generalize on what we have here by pointing out that the Gibbs–Duhem equation is the complete Legendre transform for a system with a certain set of intensive variables. You may have thought of the intensive variables of a system as being independent, as the extensive variables are, but they are not. Suppose that the intensive variables for a system are T, P,  1 ,  2 and  3 . According to the Gibbs-Duhem equation, if you specify the values of T, P,  1 and  2 , then  3 can have only a particular value. This will be even more evident when we discuss equilibrium constants in the next chapter. 4.9 Special Topic: Additional Applications of Maxwell Relations 123 4.9 SPECIAL TOPIC: ADDITIONAL APPLICATIONS OF MAXWELL RELATIONS A number of applications have already been made of Maxwell relations, but some others are of special interest. For the purpose of making calculations about a mole of a substance, the Maxwell relations from the fundamental equations for U, H, A, and G can be written ( T  V ) S = − ( P  S ) V (4.106) ( T P ) S = (  V  S ) P (4.107) (  S  V ) T = ( P T ) V (4.108) − (  S P ) T = − (  V T ) P (4.109) In Section 2.6, we found that ( U∕ V ) T = 0 for an ideal gas.

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  Characterization of Metamorphism through Mineral Equilibria

  • John M. Ferry(Author)
  • 2018(Publication Date)
  • De Gruyter

    (Publisher)

  It is useful to point out that the equations of thermodynamics used in the Gibbs method describe relations among surfaces, slopes, tangent planes, and intercepts; as such, these relations must obey all of the rules of dif-ferential calculus. It turns out that the different variables, functions, slopes, and intercepts are all quantities with thermodynamic significance such as P, T, X , G, S and V. The Gibbs method, therefore, can be thought of as a mathematical description of the relations among surfaces, planes, slopes, and intercepts. Of course, this is not to imply that thermodynamics is not fundamentally rooted in experiment and observation, but only that the mathematical relations among variables, which will be described in this chap-ter, also obey the rules of calculus. In the next four sections we derive the necessary equations for the analytical formulation of phase equilibria. Some techniques for solution are given in the appendices. This will be followed by some examples of applica-tions. The last section is a review of the literature pertaining to this type of analysis and the information that can be gained from rocks through its application. [A discussion of the Gibbs method and some applications can also be found in Rumble (1974, 1976a).] CONDITIONS OF HOMOGENEOUS EQUILIBRIUM: THE GIBBS-DUHEM EQUATION The Gibbs-Duhem equation 0 = SdT - VdP + EX.dp. (1) 1 1 is a relationship among the intensive variables of a single phase in internal, homogeneous equilibrum. Homogeneous equilibrium refers to equilibrium within a specific phase, and the conditions of homogeneous equilibrium are the equations that specify that each phase is in internal equilibrium with respect to processes such as internal diffusion, cation ordering, and chemical speci-ation. If a phase is in homogeneous equilibrium, the Gibbs-Duhem equations 110 must hold true. A rigorous derivation of the equation is given by Tunell (1979).

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  Classical and Geometrical Theory of Chemical and Phase Thermodynamics

  • Frank Weinhold(Author)
  • 2009(Publication Date)
  • Wiley-Interscience

    (Publisher)

  Thus, as stated in Inductive Law 1 (Table 2.1), only two degrees of freedom (independently variable intensive properties) suffice to describe the thermodynamic variability of a simple c ¼ 1 system. This confirms (as expected) that chemical potential m only becomes an informative thermodynamic variable when chemical change is possible, that is, for c  2 chemical components. We may also note the special form of the Gibbs – Duhem equation (6.34) under labora- tory conditions of constant T and P, namely, X c i¼1 n i dm i ¼ 0 (at constant T , P) (6:36a) or, if we divide by the total number of moles to express the relationship in mole fraction terms, X c i¼1 x i dm i ¼ 0 (at constant T , P) (6:36b) 6.3 THE GIBBS – DUHEM EQUATION 203 For a binary (two-component) solution of solute B and solvent A, for example, this establishes that the solute variations dm B are always calculable from the solvent variations dm A by the equation dm B ¼ 1  x B x B dm A (c ¼ 2; constant T , P) (6:37) Because variations in solvent chemical potential are generally much easier to determine experimentally (e.g., by osmotic pressure measurements, as described in Section 7.3.6), (6.37) gives the recipe for determining the more difficult dm solute from its Gibbs–Duhem dependence on other easily measured thermodynamic intensities. Equations such as (6.35) – (6.37) are sometimes referred to as “Gibbs – Duhem equation(s),” but they are really only special cases of (and thus less general than) “the” Gibbs – Duhem equation (6.34). Finally, let us note some interesting identities for other thermodynamic potentials that follow from Equation (6.31).

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  Phase Equilibrium in Mixtures

  International Series of Monographs in Chemical Engineering

  • M. B. King, P. V. Danckwerts(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Application of equa-tions (1.21) or (1.22) gives rise to the result (p. 17) that the chemical potential of any 7 See for example PLANCK, Treatise on Thermodynamics, pp. 118-119, Longmans 1926, Dover 1960; and also The Scientific papers of Willard Gibbs, p. 61, Longmans 1901, Dover 1961. 8 GIBBS, ref. 7. 9 Chemical Thermodynamics, McGraw-Hill, 1961. INTRODUCTION AND SOME THERMODYNAMIC CONSIDERATIONS 13 component should be the same in all phases at equilibrium. The more general equa-tions (1.20a) and (1.20b) show that, in the hypothetical case where a component is totally absent from one phase, the chemical potential of this component suitably defined, may be greater in this phase than in the others. 10 SOME DIFFERENTIAL RELATIONSHIPS BETWEEN THE EQUILIBRIUM THERMODYNAMIC PROPERTIES OF A CLOSED SYSTEM Rearranging (1.17) it follows that, for an infinitesimal change in a closed system, dU = TdS -P d V ] where U, S and V are equilibrium values of the internal energy, entropy and volume of the system at the temperature Tand pressure P. Here, as elsewhere in this section, the closed bracket serves as a reminder that the system re-mains closed throughout. Now G = H - FS, so dG = dH ■-TdS -SdT = dU + Pd + VdP -TdS -SdT] Whence, substituting for dU from (1.17), dG = V d P -S d r ] (1.23) Similarly by differentiating (1.6) and substituting for dU from (1.17), dA = -P d V -S d 7 1 (1.24) It follows from equations (1.23) and (1.24) that Y (1.25) dG = - S (1.26) dT ]p 'See also GIBBS, ref. 7, p. 66. i-(1.28)

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  Chemical Thermodynamics: Principles and Applications

  Principles and Applications

  • J. Bevan Ott, Juliana Boerio-Goates(Authors)
  • 2000(Publication Date)
  • Academic Press

    (Publisher)

  Chapter 3

  Thermodynamic Relationships and Applications

  In Chapter 1 we gave the defining equations for enthalpy, Helmholtz free energy, and Gibbs free energy:

  (3.1)

  (3.2)

  (3.3)

  In Chapter 2 we used the laws of thermodynamics to write equations that relate internal energy and entropy to heat and work.

  (3.4)

  with

  (3.5)

  (3.6)

  where the inequality applies to the spontaneous process and the equality to the reversible process.

  These equations can be used to derive the four fundamental equations of Gibbs and then the 50,000,000 equations alluded to in Chapter 1 that relate p, T, V, U, S, H, A , and G . We should keep in mind that these equations apply to a reversible process involving pressure–volume work only. This limitation does not restrict their usefulness, however. Since all of the thermodynamic variables are state functions, calculation of ΔZ (Z is any of these variables) by a reversible path between two states gives the same value as would be obtained for all other paths between those states. When other forms of work are involved, additions can be made to the equations to account for the additional work. The result is that thermodynamic equations can be derived to describe the effect of such things as surface area, electrical field, gravitational field, and magnetic field, on the thermodynamic variables.

  3.1 The Gibbs Equations

  Combining equations (3.4) , (3.5) , and (3.6) , with the equal sign (reversible process) used in equations (3.5) and (3.6) , gives

  (3.7)

  This is the first Gibbs equation.

  Next, d(pV ) is added to both sides of equation (3.7) to give

  (3.8)

  On the left side of the equation, we recognize that dU + d(pV ) = dH . On the right side, we expand d(pV ) to (p dV + V dp ). After cancelling terms we get

  (3.9)

  This is the second Gibbs equation.

  We now return to equation (3.7) and subtract d(TS ) from both sides. After recognizing that dU − d(TS ) = dA and expanding d(TS ) to (T dS + S dT

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  Thermodynamics

  Principles Characterizing Physical and Chemical Processes

  • Jurgen M. Honig, J. M. Honig(Authors)
  • 1999(Publication Date)
  • Academic Press

    (Publisher)

  (2.1.7); the equality among appropriate #'s provides interrelations among the mole fractions of any given component in the various phases. The totality of constraints therefore is (c + 2)(p- I). The number of degrees of freedom remaining is then f- p(c + I) - (c + 2)(p- I) - 2 + c- p. (2.1.8) Equation (2.1.8) specifies the famous phase rule of Gibbs (1875-1878). Knowing the number of components and phases in a given system, and assuming that T and P for the system as a whole are uniformly adjustable, Eq. (2.1.8) indicates how many state variables may be independently adjusted without altering the number of phases of the system. The ramifications of the phase rule will be discussed in Section 2.3. Further insight regarding the concept of the chemical potential may be obtained as follows Consider a two-phase, one-component system at fixed temperature and pressure for which G- ni#i + ni~i, and suppose that at some instant ~i > ~i- The system can then not be at equilibrium; instead, some spontaneous process must occur which ultimately results in the equalization of ~i and ~i. At constant T and P this can occur only by a transfer of matter from one phase to the other. Let there be a transfer of- dni - + dn i > 0 moles from phase I to phase 2; then dG - (~i - ~i)dni, where we have set dn i m dn~. Since we assumed ~i > ~i, the preceding relation shows that dG < 0 for this case; i.e., the transfer of matter from the phase of higher chemical potential to the phase of lower chemical potential occurs spontaneously. Thus, a difference in chemical potential represents a 'driving force' for transfer of chemical THE GIBBSPHASERULE 195 species, rather analogous to the difference of electrical potential that is a 'driving force' for electrically charged species. As is the case for the electrical potential, equilibrium is achieved only by an equality of the chemical potential for the species in question throughout the entire system.

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  Physical Chemistry

  • David Ball(Author)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  But for multiple- component systems, we also need to specify the relative amounts of each compo-nent, usually in terms of moles. Figure 7.1 illustrates this for a simple system. If a chemical equilibrium is present, then not all of the components are truly independent. Their relative amounts are dictated by the stoichiometry of the bal-anced chemical reaction. Before applying the Gibbs phase rule, we need to identify the number of independent components. This is done by removing the dependent component from consideration. A dependent component is one that is made from any other component(s) in the system. In Figure 7.1, the water and ethanol are not in any chemical equilibrium involving both these compounds, so they are independent components. However, for the equilibrium H 2 O ( , ) m H 1 (aq) 1 OH 2 (aq) the amounts of hydrogen and hydroxide ions are related by the chemical reaction. Thus, instead of having three independent components, we have System Degrees of freedom at equilibrium: • Temperature • Pressure • Amount (mole fraction) of one component (mole fraction of other can be determined) [ 3 degrees of freedom From Gibbs phase rule: F 5 2 2 1 1 2 5 3 degrees of freedom C P H 2 O H 2 O H 2 O C 2 H 5 OH C 2 H 5 OH C 2 H 5 OH C 2 H 5 OH F I G U R E 7.1 A simple multiple-component system of water and ethanol. The Gibbs phase rule applies to this system, too. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Unless otherwise noted, all art on this page is © Cengage Learning 2014.

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  A Problem-Solving Approach to Aquatic Chemistry

  • James N. Jensen(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  64 Chapter 3 Thermodynamic Basis of Equilibrium • Intensive thermodynamic properties are not additive within a system. • Thermodynamics can tell you which chemical reactions are possible (under a given set of conditions) and whether species concentrations are not time-dependent. • For a reaction to be at equilibrium, both the reaction and its reverse reaction must be spontaneous. • The First Law of Thermodynamics states that energy is conserved and that the internal energy changes only through heat exchange or work. • The change in enthalpy is the heat absorbed at constant pressure. • The First Law of Thermodynamics does not tell you if processes are spontaneous. • Gibbs free energy is a measure of reversibility: dG = 0 for all reversible processes (at constant temperature and pressure). • Gibbs free energy is a measure of spontaneity: dG ≤ 0 for spontaneous reactions (at constant temperature and pressure). • The chemical potential (i.e., the difference in Gibbs free energy) drives work done by chemical species. • G rxn is related to species activities. • Equilibrium corresponds to a minimum in the Gibbs free energy or ∆G rxn = 0 at equilibrium. • The equilibrium constant is the product of the product activities divided by the product of the reactant activities, each raised to the power of its reaction stoichio- metric coefficient. • In dilute solution, the equilibrium constant is approximately equal to the product of the product concentrations divided by the product of the reactant concentra- tions, each raised to the power of its reaction stoichiometric coefficient.

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