ARTICLE pubs.acs.org/IECR
The Gibbs-Duhem Equation, the Ideal Gas Mixture, and a Generalized Interpretation of Dalton’s Law R. Ravi Department of Chemical Engineering, Indian Institute of Technology, Madras Chennai, 600036, India ABSTRACT: The ideas used by Gibbs in defining the ideal gas mixture are brought to light in view of their fundamental significance and the fact that they have not received due attention in the literature. Specifically, the central role played by the well-known Gibbs-Duhem equation in this regard is explained. The concept of a fundamental equation and its importance in the definition of an ideal gas as well as an ideal gas mixture is elucidated. The manner in which these ideas enabled Gibbs to give a very general interpretation of Dalton’s law is examined in detail.
’ INTRODUCTION The Gibbs-Duhem equation is a well-known result in equilibrium thermodynamics and is found in almost every textbook on thermodynamics.1-6 It is most commonly written as a relation among the differentials of the temperature, pressure, and the chemical potentials although other forms of the equation appear especially within the context of phase equilibrium.5,6 Its importance in the analysis of the phase rule is also well recognized.1,2,5 But the fact that this equation played an important role in Gibbs’ definition7 of an ideal gas mixture (IGM) does not appear to have been noticed. It is the objective of this communication to bring to light Gibbs’s contribution in this regard. In particular, the importance given by Gibbs to the role of a fundamental equation in the definition of an ideal substance7 is emphasized. The central ideas used by Gibbs to define the ideal gas mixture are discussed in detail. Using this definition as the starting point, a host of other properties of an ideal gas mixture are derived. These properties led Gibbs to interpret Dalton’s law in a very general manner.7 Although the properties of the IGM are very well-known, I feel that Gibbs’s defining equation for the IGM deserves to be brought to the attention of the scientific community not only because of the physical and mathematical principles underlying it but also because it is possibly one of the earliest definitions of the IGM. ’ THE CONCEPT OF A FUNDAMENTAL EQUATION Fundamental Equation;Pure Substance. Gibbs8 suc-
cinctly expressed the laws of classical thermodynamics through the equation dE ¼ T dS - p dV
ð1Þ
where E and S are the intersnal energy and entropy of a homogeneous (fixed) mass of fluid, p, V, and T are its pressure, volume, and temperature. Gibbs was to call an “equation giving E in terms of S and V or more generally any finite equation between E, S, and V for a definite quantity of any fluid...as the fundamental thermodynamic equation of that fluid as from it by aid of equations...may be derived all the thermodynamic properties of the fluid (so far as reversible processes are concerned)”. At the r 2011 American Chemical Society
very outset of his second paper on thermodynamics,9 Gibbs regarded eq 1 as valid for “a given mass of fluid in a state of thermodynamic equilibrium”. In summary, an equation such as ^ðS, V Þ E ¼ E
ð2Þ
is to be regarded as a fundamental equation for a substance in a state of equilibrium. Equation 1 then enables us to determine all the thermodynamic (or more precisely, thermostatic) properties of that substance. It must be noted that E, S, and V correspond to a fixed mass or moles of a substance. If variation with respect to the extent of the fluid is to be considered, then eq 2 must be modified as ^ðS, V , NÞ E ¼ E
ð3Þ
dE ¼ T dS - p dV þ μ dN
ð4Þ
with eq 1 replaced by
where
μ
∂E ∂N
ð5Þ S, V
is the chemical potential and N the number of moles of the substance. Instead, we may regard a relation between the specific properties (defined on a per mole basis, say) ε, s, and v such as ε ¼ ^ε ðs, vÞ
ð6Þ
as a fundamental relation for a substance with eq 1 then replaced by dε ¼ T ds - p dv
ð7Þ
Special Issue: Ananth Issue Received: July 29, 2010 Accepted: January 7, 2011 Revised: January 3, 2011 Published: February 03, 2011 13076
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From eq 6, with the aid of eq 7, we can determine all the intensive and specific variables of a homogeneous substance in equilibrium. With the specification of just one more variable N, all the extensive properties may be determined as well. The advantage with this approach is that it can be easily extended to treat nonhomogeneous states of a substance. This would include nonequilibrium states where a local equilibrium approximation may be valid.10 That is, eqs 6 and 7 are assumed to hold at each point in the system in which case ε, s, v, T and p are locally defined quantities which may in general vary continuously with location in the system. Such a continuum hypothesis is needed even for systems in equilibrium under the influence of external forces such as gravity, and Gibbs himself adopted that viewpoint in such cases. Further, the intensive and specific variables may be regarded as the true indicators of the homogeneous state of a substance. For instance, 1 kg of water at 1 atm and 25 °C has essentially the same properties as 1 g of water at the same temperature and pressure, and it is these properties that are tabulated. In his treatment of the phase rule, which concerns the characterization of a multiphase, multicomponent system in equilibrium, Gibbs only considers intensive variables. In fact, he defines7 the phase of a substance thus: “We may call such bodies as differ in composition or state different phases of the matter considered, regarding all bodies which differ only in quantity and form as different examples of the same phase.” The phase here refers to a homogeneous part of a fluid. We thus adopt eqs 6 and 7 as the starting point for our discussion. Applying the idea of Legendre transforms4 other well-known fundamental relations involving the specific Helmholtz free energy (a), the specific enthalpy (h), and the specific Gibbs’ free energy (g) may be obtained. This leads to a ¼ ^aðT, vÞ;
h ¼ ^hðs, pÞ;
g ¼ ^g ðT, pÞ
ð8Þ
associated with component j. Any set of (C - 1) components can be chosen to form the composition (mole-fraction) vector. Correspondingly, the generalized chemical potential is not defined for the component whose mole fraction is not specified explicitly. For a homogeneous state of a mixture, the total internal energy E is given by ^ðS, V , N1 , :::, NC Þ E ¼ N^ε ðs, v, z1 , :::, zC - 1 Þ ¼ E where N ¼
C X
s ¼ S=
Nj ,
j¼1
z i ¼ Ni =
C X
Nj ,
h ε þ pv;
g ε þ pv - Ts
C X
da ¼ - s dT - p dv;
dh ¼ T ds þ v dp;
dg ¼ - s dT þ v dp
ð10Þ
For the homogeneous state of a pure substance in equilibrium, it can be proved that4 μ ¼ g ε þ pv - Ts
ð11Þ
In summary, any of the relations in eqs 6 and 8 can serve as a fundamental equation for a pure substance in a homogeneous state. Fundamental Equation;Mixtures. For a mixture with C components, a natural extension of eq 6 is11 ε ¼ ^ε ðs, v, z1 , :::, zC - 1 Þ C -1 X j¼1
μj dzj
i ¼ 1, :::, C - 1
Nj ,
∂E ∂ε ¼ ¼ T ∂S V , N ∂s v, z
∂E ∂V
μi ¼ ε þ pv - Ts þ
¼ S, N
∂ε ¼ -p ∂v s, z
-1 ∂^ε CX ∂^ε zj , ∂zi j ¼ 1 ∂zj
μC ¼ ε þ pv - Ts where
μi
∂E ∂Ni
μi ¼
ð16Þ
i ¼ 1, :::, C - 1 ð17Þ C -1 X j¼1
zj
∂^ε ∂zj
ð18Þ
, S, V , fNj6¼i g
∂^ε ¼ μi - μC , ∂zi
i ¼ 1, :::::C
ð19Þ
i ¼ 1, :::, C - 1
ð20Þ
Substituting eq 20 into eq 17 leads to the well-known Gibbs’ relation7 C X i¼1
where (z1, z2, ..., zC-1) are the (C - 1) independent mole fractions and μ*j can be regarded as a generalized chemical potential
ð15Þ
is the chemical potential of component i, N = (N1, ..., NC) and z = (z1, ..., zC-1) . Unlike the generalized chemical potential, μi is defined for each of the C components. Equations 17 and 18, which may be regarded as the mixture analogues of eq 11, are central to the theory of mixtures. Subtracting eq 18 from eq 17, and taking note of eq 13, we get
ð12Þ
ð13Þ
Nj ,
j¼1
In the above equations, Ni values are the mole numbers of the species. Application of the chain rule to eq 14 leads to
with the corresponding general equation given by dε ¼ T ds - p dv þ
C X
j¼1
ð9Þ
We note that corresponding to the fundamental eq 6 is the general eq 7 which enables one to determine the equilibrium properties of the substance once the fundamental equation for that substance is known. The general equations corresponding to the fundamental eqs 8 are given by
v ¼ V=
j¼1
where a ε - Ts;
ð14Þ
zi μi ¼ ε þ pv - Ts ¼ g
ð21Þ
which may also be regarded as the mixture analogue of eq 11. Equation 13 is a local relation whose validity can be extended to nonhomogeneous states as well. However, the same equation but with μj* replaced by (μj - μC) becomes valid only for a homogeneous state at equilibrium. In fact, eqs 15-18 and 13077
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eqs 20 and 21 are effectively conditions of homogeneity. Equations 20 and 21 will play an important role in subsequent sections. As in the case of a pure substance, additional fundamental relations can be obtained from eq 12 using Legendre transforms. For instance,
and that its specific internal energy is a function of temperature alone; that is,
g ¼ ^g ðT, p, z1 , :::, zC - 1 Þ
While the former is based on early experiments on gases, the latter is partly a fundamental consequence of the laws of classical thermodynamics.12 We may now ask: Are the two properties specified by eqs 29 and 30 sufficient to completely characterize an ideal gas? Equivalently, we may ask: Do eqs 29 and 30 lead to a fundamental equation for the ideal gas? To answer this question, we follow Gibbs in considering an ideal gas of constant specific heat cv, which is assumed to be given. Thus,
ð22Þ
is a fundamental relation with the corresponding general equation given by dg ¼ - s dT þ v dp þ
C -1 X j¼1
μj dzj
ð23Þ
with μj* given by eq 20 for a homogeneous state. Further, by successively eliminating the mole fractions in favor of the generalized chemical potentials, we get gðC - 1Þ g -
C -1 X j¼1
μj zj ,
dgðC - 1Þ ¼ - s dT þ v dp -
C -1 X j¼1
zj dμj
ð24Þ
Substituting eqs 20 and 21 into eqs 24, we obtain gðC - 1Þ ¼ μC
ð25Þ
and dμC ¼ -
-1 s v 1 CX dT þ dp zj dμj zC zC zC j ¼ 1
ð26Þ
The general eq 26 implies that ^C ðT, p, μ1 , :::, μC - 1 Þ μC ¼ μ
ð27Þ
Equation 27 has been obtained by the usual process of successive Legendre transformation starting with the fundamental equation, eq 12, and is thus also a fundamental equation. Rearrangement of eq 26 leads to C X zj dμj ¼ 0 ð28Þ s dT - v dp þ j¼1
which is the well-known Gibbs-Duhem equation. The significance of the equation becomes more transparent in the form of eq 26 than when expressed as the vanishing of the sum of the differentials as in eq 28. This equation played an important role in Gibbs’ definition of the ideal gas mixture as well as in his treatment of the phase rule.7 Since one of the main aims of this article is to bring to the fore Gibbs’s conception of an IGM, the derivation of this equation has been given a somewhat detailed treatment. Before we proceed to discuss the ideal gas mixture, we briefly summarize the properties of the ideal gas (IG) since the properties of the pure ideal gas play an important role in the definition of the ideal gas mixture.
’ FUNDAMENTAL EQUATIONS FOR AN IDEAL GAS Two of the most common properties of the ideal gas are its equation of state ðpvÞIG ¼ RT
εIG ¼ εIG ðTÞ
ð30Þ
εIG ¼ εIG ðTÞ ¼ cIG v T þ εo
ð31Þ
Rewriting eq 7 as ds = dε/T þ p/T dv, using eqs 29 and 31 and integrating, we get ! ε - εo IG IG s ¼ so þ cv ln þ R ln v ð32Þ cIG v so that while εo represents the specific internal energy of the gas at T = 0 units, so is its specific entropy at T = 1 unit and v = 1 unit.7 Equation 32 is of the form s = ^s(ε,ν) which is obtained by just inverting ε = ε̂(s,v) and hence is also a fundamental equation provided the two constants εo and so are specified. This implies specification of the zero of energy and the zero of entropy. The former is usually regarded as arbitrary, while the latter is regarded as specified by the third law of thermodynamics. However Callen4 points out “In the thermodynamic context, there is no a priori meaning to the absolute value of the entropy”. Gibbs too appears to have regarded both εo and so as arbitrary when he says “we may choose independently for each simple substance the state in which its energy and entropy are both zero.”7 He also reiterates this notion within the specific context of an equation such as eq 32 for the ideal gas. Denbigh3 defines an ideal gas through the following equation: μIG ¼ μo ðTÞ þ RT ln p
ð33Þ
Although Denbigh3 proves that eqs 29 and 30 can be obtained from eq 33, he does not emphasize the fact that eq 33 is a fundamental equation. Instead his main aim appears to be to demonstrate that the properties of the ideal gas, the ideal gas mixture, and the ideal solution can be defined through very similar equations. The connection between eqs 32 and 33 can be established by using eqs 29, 31, and 32 in eq 11. This leads to IG μIG ¼ εo - Tso þ ðcIG v þ RÞT - Tðcv ln T þ R ln RTÞ
þ RT ln p
ð34Þ
which yields IG μo ðTÞ ¼ εo - Tso þ ðcIG v þ RÞT - Tðcv ln T þ R ln RTÞ
ð35Þ Inversion of eq 32 leads to ε
ð29Þ 13078
IG
¼
εo þ cIG v
s - so - R ln v exp cIG v
! ð36Þ
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Since a = μ - pv, we obtain the fundamental equation for a as aIG ¼ μo ðTÞ þ RT ln RT - RT - RT ln v ¼ εo - Tso þ cIG v Tð1 - ln TÞ - RT ln v The fundamental equation for hIG can be shown to be ! hIG - εo 1 ln IG ¼ IG ½s - so þ R lnðp=RÞ cv þ R cv þ R
ð37Þ
ð38Þ
Equations 34, 36, 37, and 38 yield the four fundamental equations introduced in eqs 6 and 8 and specialized to an ideal gas. They are analogous to the relations arrived at by Gibbs, the only difference being that Gibbs expresses them in terms of extensive variables.
’ THE IDEAL GAS MIXTURE The Fundamental Defining Equation of Gibbs. Gibbs’ approach7 is to invert eq 34 to obtain an expression for the pressure of the (pure) ideal gas corresponding to species i: ! μi IG - εo, i ð1þcIG p ¼ Ai exp T v, i =RÞ , RT so, i Ai ¼ R exp - ð1 þ cIG =RÞ ð39Þ v, i R
The governing principle used by Gibbs to define the IGM is the following: “The pressure in a mixture of different gases is equal to the sum of the pressure of the different gases as existing each by itself at the same temperature and with the same value of the potential.” By “the same value of the potential”, Gibbs implies that each term pi in the sum for the pressure of the mixture must be evaluated at the value of the chemical potential that the particular IGM in component has in the mixture, that is, by replacing μIG i by μi eq 39. This results in ! C C X X μIGM - εo, i ð1þcIG i IGM p ¼ pi ¼ Ai exp T v, i =RÞ ð40Þ RT i¼1 i¼1 Equation 40 is the centerpiece of Gibbs’ treatment of the IGM. It expresses the pressure of an IGM in terms of the temperature and each of the C chemical potentials. That such an equation is a fundamental equation is seen by noticing that it is obtained by inversion of the fundamental equation, eq 27. The corresponding general equation is dp ¼
C X
s 1 dT þ zj dμmix j v v j¼1
which implies that ∂p s ¼ , ∂T μmix v
∂p ∂μmix j
j ¼ 1, 2, :::, C
sat pmix ¼ psat A þ pB sat where psat A and pB are the saturation pressures of A and B at the given temperature. Gibbs’ postulate of the additivity of pressures for an IGM appears to be based on the above consequence of idea (i) which in turn could be viewed as a special case of Raoult’s law. Now, the potential of the liquids when they are separately in equilibrium with their respective gases is to be evaluated at the respective saturation pressures while that of the liquids when they are in equilibrium with the gas mixture is at pmix. Idea (ii) of Gibbs is to regard these potentials as “very nearly the same”. Gibbs’ approximation is justified by the fact that the variation of the chemical potential with pressure is given by the specific volume which, apart from being quite low for liquids, is also a weak function of pressure. Gibbs’ generalized interpretation of Dalton’s law. Applying the second set of equations in eqs 42 to eq 40, we obtain ! μIGM - εo, i ð1þcIG yi Ai pi i ¼ exp , T v, i =RÞ ¼ v RT RT RT
i ¼ 1, ::::, C
ð43Þ
where, following convention for a gas mixture, we denote the mole fraction of a species i in the IGM by yi. Summing eq 43 over all species and using eq 40, we get vIGM ¼
ð41Þ
¼
RT ð ¼ vIG Þ p
or equivalently pIGM ¼
! T , μmix i6¼j
In view of the fundamental significance accorded by Gibbs to eq 40, we digress to discuss briefly the physical principles that motivated Gibbs to introduce this equation. The two main ideas are the following7 (i) “if several liquid or solid substances which yield different gases or vapors are simultaneously in equilibrium with a mixture of gases......the pressure in the gas mixture is equal to the sum of the pressures of the gases yielded at the same temperature by the various liquid or solid substances taken separately”. (ii) “Now the potential in any of the liquids or solids for the substance which it yields in the form of the gas has very nearly the same value when the liquid or solid is in equilibrium with the gas mixture as when it is in equilibrium with its own gas alone.” To explicate the above ideas we consider two gases A and B which are separately in equilibrium with their liquids and also in equilibrium with the liquids as a mixture, all at a given temperature T. In the latter case, the pure liquids are to be assumed to be separated by a partition but simultaneously in equilibrium with the gas mixture. Further gas A does not absorb into liquid B, and gas B does not absorb into liquid A. Gibbs’ idea (i) leads to
C X i¼1
zj , v
Ni
RT V
ð44Þ
From eqs 43 and 44 we infer that pi = pyi. Dividing eq 43 by eq 39 and rearranging, we obtain ð42Þ
mix where μmix = μmix 1 , ..., μC . In eqs 41 and 42, the superscript “mix” has been appended to μj to distinguish it from the corresponding pure component chemical potential of species j.
μIGM ðT, p, yÞ ¼ μIG i i ðT, pÞ þ RT ln yi ,
i ¼ 1, :::, C ð45Þ
where y is the mole fraction vector (y1, ..., yC-1). To be consistent with the definition of y, one needs to express yC in the last of eqs 45 in terms of y. 13079
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Gibbs then proceeds to derive a new principle that helps him organize the properties of an IGM. His starting point is eq 40 and its consequence eq 44. Gibbs refers to this equation as displaying “the familiar principle that the pressure in a gas mixture is equal to the sum of the pressures which the component gases would possess if existing separately with the same volume at the same temperature.” An additional qualification, namely, that the component gas while “existing separately” must have the same number of moles as it does in the mixture, is implicit in Gibbs’ statement. Gibbs then extends this rule to apply to extensive quantities such as entropy. For instance, from the first of eqs 42 applied to the IGM, we obtain C X sIGM pi 1 IGM ðμ ¼ ðcIG þ RÞ ε Þ o, i T i v RT v, i i¼1
note that eqs 21 and 44 lead to
i¼1
Ni ðμIGM - RTÞ; i
H IGM ¼ TSIGM þ
C X i¼1
GIGM ¼
C X i¼1
Ni μIGM ¼ EIGM þ NRT; i
Ni μIGM i
ð53Þ
We now gather together the expressions for the pure component extensive properties:
ð46Þ
IG AIG i ðT, V , Ni Þ ¼ Ni ½εo, i - Tso, i þ cv , i Tð1 - ln TÞ
- RT lnðV =Ni Þ
From eqs 29, 34, 43, and 45, we obtain
ð54Þ
IG EIG i ðT, Ni Þ ¼ Ni ½εo, i þ cv , i T;
IG μIGM ¼ ðεo, i - Tso, i Þ þ ðcIG i v , i þ RÞT - cv, i T ln T
- RT lnðv=yi Þ
C X
AIGM ¼
ð47Þ
HiIG ðT, Ni Þ ¼ Ni ½εo, i þ ðcIG v , i þ RÞT
ð55Þ
.Substituting for (μIGM - εo,i) from eq 47 into eq 46, we obtain i sIGM ¼
C X i¼1
IG IG GIG i ðT, V , Ni Þ ¼ Ni ½εo, i - Tso, i þ ðcv , i þ RÞT - cv , i T ln TÞ
yi ½so, i þ cIG v , i ln T þ R lnðv=yi Þ
ð48Þ
From eqs 31 and 32, we obtain the specific entropy of the pure component ideal gas at a specified T and v as IG sIG ð49Þ i ¼ so, i þ cv , i ln T þ R ln v PC IG Clearly, sIGM ¼ 6 i = 1si . In fact, the difference would correspond to the entropy of mixing term at constant temperature and pressure in which case vIGM equals vIG. (This feature will also appear in similar relations involving the Helmholtz and Gibbs free energies. We do not discuss this aspect here as it is widely discussed in the literature.) However, Gibbs’ rule is to be applied to an extensive property in general (pressure being an exception) written in terms of the extensive variables: T, V, and the mole numbers of the species, {Ni}. Thus we may write eq 48 as
SIGM ¼
C X i¼1
Ni ½so, i þ cIG v , i ln T þ R lnðV =Ni Þ
- RT lnðV =Ni Þ
Equation 54 follows from eq 37, eq 55 from eqs 9, 29, and 31, while eq 56 is a consequence of eqs 11, 29, and 34. By a suitable combination of eqs. 47, 50, 53, and eqs 54-56, one can establish the following results: AIGM ðT, V , NÞ ¼
¼
C X i¼1
H IGM ðT, NÞ ¼
C X i¼1
EIGM ðT, NÞ ¼
C X i¼1
ð50Þ
GIGM ðT, V , NÞ ¼
Ni ½so, i þ cIG v, i
ln T þ R lnðV =Ni Þ
SIGM ðT, V , NÞ ¼
C X i¼1
SIG i ðT, V , Ni Þ
ð52Þ
where N = (N1, N2, ..., NC). Similar relations may be proved for the other extensive properties E, A, H, and G. Of these, the relation for A has a special significance because it is a fundamental equation though expressed in terms of extensive variables. To prove Gibbs’ assertions for the other extensive properties, we first
ð57Þ
EIG i ðT, Ni Þ
C X
GIG i ðT, V , Ni Þ
ð58Þ
ð59Þ
In addition eq 44 may be written as pIGM ðT, V , NÞ ¼
C X i¼1
ð51Þ
Equations 50 and 51 clearly imply that
AIG i ðT, V , Ni Þ
HiIG ðT, Ni Þ;
i¼1
If the ideal gas i, in its pure state, is at the same temperature, occupies the same volume, and has the same number of moles as it has in the mixture, then its total entropy is, from eq 49, SIG i
ð56Þ
pIG i ðT, V , Ni Þ
ð60Þ
where pIG i is given by eq 29 with v = V/Ni. Equations 52 and 57-60 are summarized in Gibbs’ statement: These quantities (namely, p, S, E, A, H, G) “relating to the gas mixture may therefore be regarded as consisting of parts which may be attributed to the several components in such a manner that between the parts of these quantities which are assigned to any component, the quantity of that component, the potential for that component, the temperature, the volume, the same relations shall subsist as if that component existed separately. It is in this sense that we should understand the law of Dalton, that every gas is as a vacuum to every other gas.” 13080
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It is clear from the above statement that Gibbs has generalized the so-called Dalton’s law in more than one way. First, the Dalton’s law is commonly stated only for the pressure.3 Gibbs however envisaged that principle to apply for all the extensive properties as well. This principle has also been recognized to varying extents by others1,2,4 since Gibbs. Further, Gibbs also includes the chemical potential along with the temperature, volume, and mole number of a species. That is, while evaluating the contribution of a particular species to a certain property of the IGM, the chemical potential in the expression for the pure component must correspond to the value of the potential that the component has in the mixture. This principle is reflected in eq 40 for the pressure, which in fact was the fundamental rule on which Gibbs’ definition of the IGM was based. All other relations concerning the IGM right up to eq 60 follow from that. That the same principle applies to the extensive quantities can be easily proved. For instance, from eq 46, we have, SIGM
C X 1 IGM ðμ ¼ Ni ðcIG þ RÞ ε Þ o, i v, i T i i¼1
C X
ð62Þ
i¼1
IGM ^SIG , Ni Þ i ðT, μi
ð63Þ
, ..., μIGM where μIGM = μIGM 1 C . Equations 9, 11, and 29 for the ideal gas imply IG AIG i ¼ Ni ðμi - RTÞ
ð64Þ
C X i¼1
IGM ^ IG A , Ni Þ i ðT, μi
ð65Þ
Equations 58 remain as such because both HIGM and EIGM depend only on the temperature and mole numbers. Also, since IG GIG i = Niμi , the following result trivially follows from the last of eqs 53: GIGM ðμIGM , NÞ ¼
C X i¼1
IGM ^ IG , Ni Þ G i ðμi
ð66Þ
C X i¼1
ðiiiÞ
sIGM i
¼
sIG i
Ni εIG i ;
- R ln yi , i ¼ 1, :::, C
ð68Þ
’ SUMMARY The concept of a fundamental equation and its relevance to the definition of an ideal substance were discussed. The role played by the Gibbs-Duhem equation in Gibbs’ definition of the IGM was pointed out. Then, fundamental equations were obtained for the ideal gas. This was followed by a detailed discussion of the fundamental equation proposed by Gibbs for the ideal gas mixture. The body of results obtained from this equation and the generalized interpretation of Dalton’s law provided by Gibbs were discussed. Some alternate definitions of the IGM introduced by later authors were summarized.
C X i¼1
Corresponding Author
E-mail:
[email protected]. Tel.: 91-44-22574167. Fax: 91-4422570509.
’ DEDICATION This article is submitted as a contribution to the special issue dedicated to my teacher Prof. M. S. Ananth. It is intended as a tribute to his deep interest in scientific history as well as his respect for the classical texts on thermodynamics. ’ NOMENCLATURE
In addition, eq 40 may be rewritten as pIGM ðT, μIGM Þ ¼
ðiiÞ EIGM ¼
’ AUTHOR INFORMATION
Equations 53 and 64 result in AIGM ðT, μIGM , NÞ ¼
pV IGM ¼ NRT;
where hs IGM is the partial molar entropy of component i in the IGM. i It is to be noted that properties (i) and (ii) in eqs 68 are stated in terms of extensive properties. Nevertheless, the set of three properties in eqs 68 are sufficient to derive all other properties of the IGM.
where the ^S is used to distinguish the variable from the functional relation SIG(T,V,Ni). From eqs 61 and 62, it follows that SIGM ðT, μIGM , NÞ ¼
ðiÞ
ð61Þ
For a pure ideal gas, from eqs 3, 11, and 29, we get 1 IG IG ðμ SIG ¼ N ðc þ RÞ ε Þ i o, i i v, i T i IG ¼ ^SIG i ðT, μi , Ni Þ
’ DEFINITION OF AN IGM;ALTERNATE PROPOSALS Denbigh3 uses eqs 45 as the defining equations of the IGM. From these equations, Denbigh systematically derives various other properties of the IGM. Tester and Model5 adopt a similar approach. Feinberg13 obtains Denbigh’s equations from a set of simple postulates about the variables upon which the partial molar properties of an ideal gas mixture should depend. Upon multiplying each equation in eqs 45 by the corresponding mole number Ni and summing, one may obtain a fundamental equation for the extensive variable G in terms of T, p, and N. Callen4 uses eqs 52 and 58 to define an IGM and refers to eq 52 as Gibbs’ theorem. Prigogine and Defay1 use eq 57 to introduce the IGM and derive other properties of the mixture from it. Guggenheim2 too adopts a similar approach. Sandler6 defines the IGM through the following properties:
IGM ^pIG Þ i ðT, μi
Abbreviations
ð67Þ
Equations 52 and 57-60, 63, 65-67 thus summarize Gibbs’ generalized interpretation of Dalton’s law. What is significant is that all of these results are derived from eq 40, the single defining equation used by Gibbs to introduce the IGM.
IG = ideal gas IGM = ideal gas mixture mix = mixture sat = saturation Notation
cIG v = specific heat at constant volume of the ideal gas a = specific Helmholtz free energy 13081
dx.doi.org/10.1021/ie101614s |Ind. Eng. Chem. Res. 2011, 50, 13076–13082
Industrial & Engineering Chemistry Research
ARTICLE
A = total Helmholtz free energy E = total internal energy g = specific Gibbs free energy G = total Gibbs free energy h = specific enthalpy H = total enthalpy Ni = mole number of species i p = pressure s = specific entropy S = total entropy T = temperature v = specific volume V = total volume yi = mole fraction of species i in the ideal gas mixture zi = mole fraction of species i in a general mixture ε = specific internal energy μ = chemical potential
’ REFERENCES (1) Prigogine, I. Defay, R. Chemical Thermodynamics (Translation, Everett, D. H.); Longmans and Green: London, 1954. (2) Guggenheim, E. A. Thermodynamics: An Advanced Treatise for Chemists and Physicists; North Holland: Amsterdam, 1959. (3) Denbigh, K. The Principles of Chemical Equilibrium: With Applications in Chemistry and Chemical Engineering; Cambridge University Press: Cambridge UK, 1961. (4) Callen, H. B. Thermodynamics and an Introduction to Thermostatistics; John Wiley and Sons: New York, 1985. (5) Tester, J. W.; Modell, M. Thermodynamics and Its Applications; Prentice-Hall: NJ, 1996. (6) Sandler, S. I. Chemical, Biochemical, and Engineering Thermodynamics; John Wiley and Sons: Hoboken, NJ, 2006. (7) Gibbs, J. W. On the equilibrium of heterogeneous substances. Trans. Connecticut Acad. 1876, III, 108–248; 1878, III, 343–524. (Reprinted in The Scientific Papers of J. Willard Gibbs; Dover Publications: New York, 1961; Vol. 1.) (8) Gibbs, J. W. Graphical methods in the thermodynamics of fluids. Trans. Connecticut Acad. 1873, II, 309–342.(Reprinted in The Scientific Papers of J. Willard Gibbs; Dover Publications: New York, 1961; Vol. 1.) (9) Gibbs, J. W. A method of geometrical representation of the thermodynamic properties of substances by means of surfaces. Trans. Connecticut Acad. 1873, II, 382–404.(Reprinted in The Scientific Papers of J. Willard Gibbs; Dover Publications: New York, 1961; Vol. 1.) (10) De Groot, S. R., Mazur, P. Nonequilibrium Thermodynamics; Dover Publications: New York, 1984. (11) Truesdell, C., Toupin, R. A. The Classical Field Theories. Handbuch der Physik; Flugge, S., Ed.; Springer-Verlag: Berlin, 1960; Vol. III/1, pp 226-902. (12) Truesdell, C. The Tragicomical History of Thermodynamics 1822-1854; Springer-Verlag: New York, 1980. (13) Feinberg, M. Constitutive equations for ideal gas mixtures and ideal solutions as consequences of simple postulates. Chem. Eng. Sci. 1977, 32, 75–78.
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