Research: Science and Education edited by
Advanced Chemistry Classroom and Laboratory
Joseph J. BelBruno Dartmouth College Hanover, NH 03755
A Generalized Deduction of the Ideal-Solution Model Teresa J. Leo* and Pedro Pérez-del-Notario Departamento Motopropulsión y Termofluidodinámica, E.T.S.I. Aeronáuticos, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros, 3, 28040-Madrid, Spain; *
[email protected] Miguel A. Raso Departamento Química Física I, Fac. CC. Químicas, Universidad Complutense de Madrid, 28040-Madrid, Spain
A single-phase mixture is generally considered to be an ideal solution if the chemical potential of every component in the mixture is a linear function of the logarithm of its mole fraction (1–5). This definition is essentially an extrapolation of the model of ideal-gas mixtures. The volume, enthalpy, entropy, and Gibbs energy of mixing are deduced from this assumption, and therefore a specific form for the chemical potential of the components is required to develop this model. Nevertheless, the Gibbs energy of mixing corresponding to the ideal-solution model is arrived at automatically through the relationships between the thermodynamic properties of mixing, Euler’s theorem on homogeneous functions, and knowledge of differential calculus. No specific model is required for the components. Deduction of the form of Gibbs energy takes into account the evidence that every extensive thermodynamic property expressed as a function of the pressure, temperature, and quantities of substance of the components is a homogeneous function of degree one on these quantities of substance. The deduction of the Gibbs energy of mixing for the ideal-solution model presented here is, then, a rational and general one. The authors believe this is a good way of teaching the thermodynamics of mixtures to advanced students, whose mathematical background enables them to grasp other disciplines in a general way. A macroscopic formulation of thermodynamics designed for advanced students is used. Since ideal solutions are mixtures, it is prudent to review some basic facts about mixtures before discussing the model. Firstly, all extensive thermodynamic properties of a mixture of N components, expressed as a function of the pressure, P, the temperature, T, and the amounts of substances, ni, of these N components are homogeneous functions (6) of the first degree in the corresponding ni. That is, if the ni are multiplied by a factor λ, the extensive property is multiplied by λ. Then, if F(P, T, n1, ..., nN) is an extensive thermodynamic property, Euler’s theorem (6) states that N
F ( P, T, n1 , ..., nN ) =
∑ ni
i =1
∂ F ∂ ni
The relationship between the derivatives of both functions F and f can be deduced and written as follows (see the Appendix) ∂ f ∂ F = f + ∂ ni P, T, n ∂ x i P, T, x j ≠ i
N
−
P ,T ,n j ≠ i
∑ ni Fi
i =1
(1)
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)
(2)
(3)
∂ f ∂ x j
P,T, xi ≠ j
where all the molar fractions are taken as independent variables. Obviously, the restriction ∑iNxi = 1 must be considered. This formulation is also used in ref 7. Finally, in general, for an extensive thermodynamic property F(P, T, n1, ..., nN), the corresponding property of mixing is defined as N
∆m F = F −
∑ ni fi *
N
=
i =1
∑ ni ( F i
− fi *
i =1
)
(4)
where fi * represents the corresponding molar property for the ith pure component, but at the pressure P and temperature T of the mixture and also in the same state of aggregation as this component is found in the mixture. Development of the Gibbs Energy of Mixing for the Ideal-Solution Model
The Gibbs Energy of Mixing: A General Formulation All the information needed to calculate mixing properties is contained in the Gibbs energy of mixing, ∆mG(P, T, n1, ..., nN), the thermodynamic potential expressed in its natural variables. Obviously, d(∆mG )is an exact differential. From the definition of the properties of mixing given in eq 4, the Gibbs energy ∆mG of mixing can be written as N
∆ mG =
∑ ni (µ i
i =1
– where F i is the corresponding partial molar property. Secondly, the corresponding molar thermodynamic property is then expressed as a function of the molar fractions xi (= nin) F ( P, T, n1 , …, nN ) = f ( P, T, x1 , ..., xN n
∑ xj j =1
N
=
j ≠ i
− µi *
)
(5)
where µi* is the chemical potential of the ith pure component at the mixture pressure P and temperature T and also in the same state of aggregation as in the mixture. Then, as thermodynamics teaches, the enthalpy, ∆mH, and the volume, ∆mV, of mixing can be expressed as ∆ m H = ∆ mG − T
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∂ ( ∆ mG ) ∂ T
(6a) P, ni
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∆ mV =
∂ ( ∆ mG ) ∂ P
(6b) T, ni
When the pressure tends to zero, there are no molecular interactions and hence no energetic interactions. The second term on the right hand side of eq 11 is then zero, so that
∂
∆ mG T ∂ T
∂
= −
∆m H
(7a)
T2
P, ni
T
= T,ni
∆ mV T
= −
2
0
∂
∆ mG T ∂ P
Ξ (P, T0 , n1, ..., nN ) dP
(7b)
T
P
T0
0
+
∂
+
T
∆ mG T ∂ T
P
(8)
dT
T0 0
P, ni
∂ Ξ ( P , T , n1 , ..., nN ) ∂ T T
=
∆ mG 0, T0, n1, …, nN T
T0
From eq 8, a general expression for ∆mG can then be obtained in the following way. Examining the first term on the right side of eq 8 and realizing that eq 7b is an extensive function, it can be written as ∆ mV = Ξ ( P, T, n1 , ..., nN ) T
(9)
The function Ξ(P, T, n1, ..., nN) is a homogeneous function of degree one in the ni. A similar expression can be attained for the second term of eq 8. Invoking the identity of the second-order (mixed) partial derivatives of ∆mGT (8, 9) gives ∆m H RT 2 ∂ P
∂ = − T, ni
dP dT P, ni
P
∂ ∂ T
Ξ ( P, T, n1 , ..., nN ) dP dT 0
(13b)
∆ mV RT ∂ T
(10)
P
Ξ (P, T, n1 , ..., nN ) dP −
= 0
P
T2
= − 0
+
146
∂
∆ mV T ∂ T
and the third term can be redefined as
Θ (0, T0 , n1 , ..., nN )
(11)
T2
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(13c)
∆ mG = T
P
Ξ (P, T, n1 , ..., nN ) dP + Θ(0, T0 , n1 , ..., nN ) (13d) 0
In summary, a general expression for the Gibbs energy of mixing takes the form P
dP
∆ m H (0, T, n1, n2,...., nN )
∆ mG 0, T0 , n1, ..., nN T
Substituting eqs 13b and 13c into 13a gives
P, ni
P, ni
Ξ( P , T0 , n1 , ..., nN ) dP 0
∆ mG = T Ξ ( P, T, n1 , ..., nN ) dP
because d(∆mGT ), like d(∆mG ), is an exact differential. Equation 10 allows us to express the enthalpy of mixing as
∆ mH
(13a)
∆ mG 0, T0, n1, ..., nN T
P
∂
dP dT P, ni
where the second term can be simplified as
dP
T0
+
∂ Ξ ( P, T , n1 , ..., nN ) ∂ T
T, ni
0 T
(12)
0
+ P
dP P, ni
P
∆mGT can be integrated as follows
∆ mG = T
∂ Ξ ( P, T, n1 , ..., nN ) ∂ T
Substituting eqs 9 and 12 into eq 8 and integrating from (0, T0) to (P, T ) it follows that ∆ mG = T
∆ mG T ∂ P
P
∆ mH
From the above expressions, the following relationships can be easily deduced
0
(13e)
+ T Θ(0, T0 , n1 , ...., nN ) where Θ(0, T0, n1, ..., nN) is also a homogeneous function of degree one in the ni, although at fixed pressure and temperature values. Both functions Ξ and Θ have to meet the specific mathematical conditions in eqs 1 to 3. The particularized solutions of eq 3 for Ξ and Θ will give the dependence of these functions on P, T, and the composition of the mixture:
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∂ Ξ ∂ ni ∂ Θ ∂ ni
N
∂ ξ ∂ x i
= ξ + P, T, nj ≠ i
∂ θ ∂ x i
= θ + P, T, n j ≠ i
∑ xj
−
j =1
P, T, xj ≠ i
N
∑ x j
−
j =1
P, T, x j ≠ i
∂ ξ ∂ x j
P, T, xi ≠ j
∂ θ ∂ xj
P, T, xi ≠ j
ϕj xj
d
(14)
xj
(15)
⇒
= a (P, T )
dx j ∂ Ξ ∂ ni
dϕi − a (P, T ) dx i
= P ,T, n j ≠ i
where, according to eqs 2 and 3, ξ = Ξn and θ = Θn.
(19)
Therefore, the functional form of ϕi(P, T, xi ) will be
The Gibbs Energy of Mixing: A Particular Case There is a simple case from a mathematical point of view whereby particular solutions can be obtained for eqs 14 and 15. In this case the change of the extensive function with respect to each amount of substance ni depends solely on the molar fraction of the ith component and not on the rest. This case is studied for both Ξ and Θ functions. Study of the Function Ξ(P, T, n1, ..., nN) Two conditions are needed to make (∂Ξ∂ni )P,T,nj≠i depend only on xi. First, assume the solution is in the form of a sum of the contributions ϕi(P, T, xi ), one per component depending only on the composition of such component; that is,
d xi
ϕi xi d xi
∑ ϕi (P, T, xi )
i =1
The integration constant Ci is obtained as follows. At the limit of the ith component on its own, the mixture disappears and, as deduced from eq 4, any property of mixing must be equal to zero. Therefore, lim Ξ = 0
(21a)
lim ξ = 0
(21b)
lim ϕ i = Ci = 0
(21c)
xi → 1
then xi → 1
(16)
(20)
⇒ ϕ i = a (P, T ) x i ln x i + C i xi
N
ξ ( P, T , x1, ..., xN ) =
= a ( P, T )
so
Thus, substituting into eq 14
x i → 1
and Ci = 0. In summary ∂ Ξ ∂ ni
= P, T, n j ≠ i
=
dϕ i + dx i
dϕ i − dx i
N
∑ ϕ j
− xj
j =1
N
∑ xj
xj
j =1
dϕ j
ϕ i = a (P, T ) xi ln x i
dx j
ϕj d x j
N
ξ = a ( P, T ) ∑ xi ln xi
(17)
i =1
Ξ = n a ( P, T ) ∑ x i ln x i
Note that the first term on the right-hand side only depends on the ith component whereas the second term depends on all the components. It would therefore be fair to say that this last term represents the interaction between the mixture components. Henceforth it is referred to as the “interaction term” denoted by I
I = − ∑ x j xj j =1
d
ϕj xj
(18)
dx j
ϕ 0i = C 0 x i ln x i
•
(22c)
Study of the Function Θ(0, T0, n1, ..., nN) The study of this function is analogous to that performed with Ξ, although as can be appreciated in eq 13, Θ depends only on the composition since it is defined at a fixed temperature T0 and at a very particular pressure, 0. Therefore, new functions ϕ0i(0, T0, xi ) and θ = ∑iNϕ0i(0, T0, xi ) are defined and introduced in eq 15. The particular solution obtained in this case is
Since the objective is for (∂Ξ∂ni )P,T,nj≠i to depend only on xi, a second condition is needed. The above-defined interaction term I must be a constant quantity with respect to the composition of the mixture, and it must be equal for all mixture components. This is achieved by introducing a function a(P, T )
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(22b)
N
dx j
i =1
N
(22a)
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(23a)
N
θ = C 0 ∑ xi ln xi i =1
(23b)
N
Θ = nC 0 ∑ x i ln x i
(23c)
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Note that in this case, instead of the function a(P, T ) appearing in eq 19, a constant C0 is chosen since Θ was defined for given P and T values.
The Gibbs Energy of Mixing When No Interaction Occurs: A Particular Case By substituting the particular solutions obtained for eqs 22c and 23c into the general expression of ∆mG, eq 13e, we obtain a particular solution for the Gibbs energy of mixing ∆mG P
N
a (P, T ) dP
∆ mG = nT
One particular case of mixtures is the ideal-gas mixture. This is the mixture of gases described by a well-known equation of state, PV = nRT, that does not distinguish between the various gases but uses a universal constant, R, for all of them. So, interaction between different gases described by this model is not expected. Therefore, the mixture of ideal gases can be considered a particular case of the model derived here, eq 25, that is equally valid for solids, liquids, or gases. When the ideal gases form a mixture at constant P and T, it is shown, independently from the previous equations, that the Gibbs energy of mixing takes the form
∑ xi ln xi
N
i =1
0
∆ mG = n RT ∑ x i ln x i
(24)
N
+ nC 0T ∑ xi ln xi i =1
The Gibbs Energy of the Ideal Solution: A Particular Instance of a Particular Case There is however a special particular instance of the particular solution given in eq 24 that merits special attention. If a(P, T ) = 0 at all possible values of P and T, a new expression for the Gibbs energy of mixing is found N
∆ mG = nC0T ∑ xi ln xi
i =1
Hence, it is easy to identify the general constant quantity C0 appearing in the model derived here (eq 25) with the universal constant R of gases, since the more general case must contain the particular case. By identifying C0 with R in the expressions of eq 26, the ideal-solution model is derived as a particular instance of a particular case of a general case. As derived, it is equally valid for gaseous, liquid, or solid mixtures and is characterized by the absence of interaction between its components. The thermodynamic properties of the ideal-solution model (id) are written as
(25)
N
i =1
This particular instance of the particular solution describes the situation where the “interaction term” is a constant quantity equal for all mixture components, independently of the mixture composition. This function represents a special model of mixture whose components do not interact with one another and whose Gibbs energy of mixing is independent of the pressure and always has the same simple functional form whatever the temperature. Equation 25 could therefore describe either a mixture of solids, liquids, or gases. The thermodynamic properties of this special particular model can be readily deduced from eq 6 and from ∆ m S = ᎑[∂(∆mG)∂T )]P,n. Hence, N
∆ mG id = n RT ∑ x i ln xi
(28a)
∆ mV id = 0
(28b)
∆ m H id = 0
(28c)
i =1
N
∆ m S id = −nR ∑ xi ln xi i =1
(28d)
It is important to note, as we stress below, that the mathematical form xi lnxi appears here in the general derivation of the model and not bound to the ideal-gas mixture, as is habitual in the normal study of mixtures.
∆ mG = n C0 T ∑ xi ln xi
(26a)
Conclusion
∆ mV = 0
(26b)
∆mH = 0
(26c)
A new general procedure for deriving the Gibbs energy of mixing is developed through general thermodynamic considerations, and the ideal-solution model is obtained as a special particular case of the general one. This treatment does not begin with a concrete expression for the chemical potential of the components of the mixture, and the mathematical form xi lnxi appearing in the Gibbs energy of mixing when no interaction between components occurs is not bound to the ideal-gas mixture, as is normally the case in the study of mixtures. It is to be viewed as a rational deduction suitable for advanced students who appear to have an adequate mathematical background.
i =1
N
∆ m S = −nC0 ∑ xi ln xi i =1
(26d)
The reasoning leading to this expression requires that the constant C0 be equal for all possible components of the mixture, but no value is imposed. The question is whether it could it be assigned a specific value?
148
(27)
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Literature Cited 1. Levine, I. N. Physical Chemistry, 5th ed.; McGraw-Hill: New York, 2002. 2. Smith, J. M.; Van Ness, H. C.; Abbot, M. N. Introduction to Chemical Engineering Thermodynamics, 5th ed.; McGraw-Hill: New York, 1996. 3. Pitzer, K. S. Thermodynamics, 3rd ed.; McGraw-Hill: Singapore, 1995. 4. Bevan Ott, J.; Boerio-Goates, J. Chemical Thermodynamics: Advanced Applications; Academic Press: San Diego, 2000. 5. Wark, K., Jr. Advanced Thermodynamics for Engineers; McGraw-Hill: New York, 1995. 6. Courant, Richard. Differential and Integral Calculus; John Wiley & Sons: New York, 1988; Vol. 2, pp 108–110. 7. Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworths and Co Ltd.: Boston, 1982; Vol. 2, p 91. 8. Goodman, A. W. Modern Calculus with Analytic Geometry; The Macmillan Company: New York, 1968; p 328. 9. Courant, Richard. Differential and Integral Calculus; John Wiley & Sons: New York, 1988; Vol. 2, p 55.
(
∂ f P, T ,
n,
nN
…,
)
n
∂ ni N
=
P ,T ,nk ≠ i
(
n1
∂ f P, T ,
∂
j =1
nN
n , …,
n
nj
∂
)
(n n ) j
(A3)
n ∂ ni
nk ≠ j
P,T , n
nk ≠ i
Taking into account that ∂n∂ni|nj≠i = 1 ∂
(n n ) j
∂ ni
∂
(n n )
n
nk ≠ i
i
∂ ni
nj
= −
1 n − 2i n n
= nk ≠ i
∂ n ∂ ni
2
∂ n ∂ ni
1 nj n n
= − nk ≠ i
1 1 ni − n n n
= nk ≠ i
(A4)
(A5)
this gives
Appendix
Derivatives of Equation 3 Considering F is a homogeneous mathematical function in the (n1, n2, ..., nN) variables and that n = n1 + n2 + ... + nN, all n1, n2, ..., nN being independent variables, it can be written as
(
F ( P, T, n1 , ..., nN ) = n f P, T, n1 n ,…, nN n
)
= n f (P, T, x1, ..., xN )
∂ F ∂ ni
(
= f P, T, P, T, nk ≠ i
+ n
(
n1
n,
∂ f P, T ,
…,
n1
(
= f P, T, P, T, n j ≠ i
(
n,
n
)
…,
∂ n ∂ ni nN
n
(A1) N
− ∑
(A2)
∂ F ∂ ni
P ,T ,nk ≠ i
P ,T ,nk ≠ i
•
nj
n,
n
)
…, nN n
)
( n) ni
(
∂ f P, T ,
n
∂ nf ∂ ni
=
where
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n1
nN
nk ≠ i
(A6)
P ,T , n n1
∂
n
, …, nN n
(n n )
)
j
P, T,
nk ≠ j n
or, equivalently, nk ≠ i
)
∂ ni
n , …,
∂
j =1
nN
n1
∂ f P, T,
+
This derivative can be expressed as ∂ F ∂ ni
n1
Vol. 83 No. 1 January 2006
= f +
•
P ,T , nk ≠ i
∂ f ∂ xi
N
− P,T, x k ≠ i
∑ x j j =1
∂ f ∂ x j
(A7) P,T, xk ≠ j
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