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Transformations between Extensive and Intensive Versions of Thermodynamic Relationships James G. Eberhart Department of Chemistry, University of Colorado at Colorado Springs, Colorado Springs, Colorado 80918 [email protected]

Most of the equations of thermodynamics can be written in two versions: one extensive and the other intensive. The intensive version is always somewhat simpler than the extensive version because it involves one less independent variable. The intensive version is also generally of broader interest because it presents information in a form that is independent of the size of the system under study. Because the first and second laws of thermodynamics and their immediate consequences are usually stated in extensive form (i.e., in terms of the total energy and entropy of the system), techniques for the transformation of thermodynamic equations from one format to the other are of considerable importance. This article will consider (i) the difference between extensive and intensive properties, (ii) the number of independent variables required to determine both types of properties, (iii) the extensive and intensive versions that are possible for a representative sample of thermodynamic relationships, and finally, (iv) a simple, calculusbased approach to transforming these equations from one version to the other. The types of equations considered are the differential expressions and the equations involving partial derivatives that form the heart of all thermodynamic expositions. Definitions and Notation Most thermodynamic properties of a system are either extensive or intensive (1-5). If a system is divided into several parts, then an extensive property of the system is one that is the sum of the properties of each of the parts. In a uniform and homogeneous phase these properties are proportional to the volume of the phase. Thus, extensive properties are often called global properties. Examples of these properties are energy, U, entropy, S, volume, V, and the amount of the ith substance in a mixture of C components, ni, where i = 1, 2, ..., C. The intensive properties of a system are not the sum of the properties of the parts of the system but can be defined at any point within the system. They are often referred to as local properties. In a homogeneous phase these properties have the same value throughout the phase. Examples of intensive properties are pressure, p, thermodynamic temperature, T, and the chemical potential of substance i, μi. Intensive properties for a phase can be created by dividing one extensive property by another. Thus, the density, F = M/V, is intensive, as is the mole or amount fraction of component i, xi = ni/n, where n = Σi ni is the total amount of all the C components. Other familiar molar properties are the molar volume, Vm = V/n, molar energy, Um = U/n, and molar entropy,

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Sm = S/n, of a phase. Partial molar properties are in the same category, except that they are differentiated with respect to ni rather than divided by n. Thus, the chemical potential of component i is μi = (∂U/∂ni)S,V,nj, where the subscript nj implies that all of the nj are held constant except for ni, that is, j 6¼ i. In the same fashion the partial molar volume of component i is Vi = (∂V/∂ni)T,p,nj. The notations used here for extensive and intensive thermodynamic properties are consistent with IUPAC conventions (6). Independent Variables Just as most thermodynamic properties are either extensive or intensive, the relationships between thermodynamic properties can usually be placed in an extensive or intensive format. The choice of format influences the number of independent variables required to describe the phase. Equations of state (such as the ideal gas law or van der Waals equation) relate these independent variables to a dependent variable. For a pure phase the (extensive) volume depends on three independent variables, T, p, and n, that is, V = V(T, p, n) = n f(T,p), while the (intensive) molar volume depends only on T and p, that is, Vm = V/n = Vm(T, p), because V is proportional to n (7). Here we see the loss of one independent variable as a result of transforming an extensive equation into the corresponding intensive equation. In these two versions of an equation of state, the dependent variable of the former is extensive because it is dependent upon at least one extensive independent variables, while the same in the latter is intensive because it is dependent upon only intensive variables. For a uniform phase at equilibrium, thermodynamics indicates that the volume of a phase that is a mixture of C substances or components depends on C þ 2 independent variables: V ¼ V ðT , p, n1 , n2 , :::, nC Þ

(1Þ

If eq 1 is divided by n, then the individual ni become mole or amount fractions, xi = ni/n. Although there are C different fractions in the resulting equation of state, they are not all independent because of the constraint X xi ¼ 1 (2Þ i

Thus, the molar volume is a function of only C þ 1 independent variables (8), which can be indicated via Vm ¼ Vm ðT , p, x1 , x2 , :::, xC - 1 Þ

(3Þ

The same number of variables are, of course, found in the two versions of the fundamental equation of thermodynamics,

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r 2010 American Chemical Society and Division of Chemical Education, Inc. pubs.acs.org/jchemeduc Vol. 87 No. 3 March 2010 10.1021/ed800074g Published on Web 02/09/2010

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which follows from the first and second laws of thermodynamics, namely U ¼ U ðS, V , n1 , n2 , :::, nC Þ

(4Þ

which is extensive, and, dividing by n Um ¼ Um ðSm , Vm , x1 , x2 , :::, xC - 1 Þ

(5Þ

which is intensive. Again, the loss of one independent variable accompanies the transformation of a relationship from extensive to intensive format.

G ¼ U þ pV - TS

Many of the key relationships of thermodynamics are expressed in differential form. Probably the most important is the Gibbs (or characteristic) equation for the energy U X dU ¼ T dS - pdV þ μi dni (6Þ i

which is a consequence of eq 4 and the first and second laws. The equation can be transformed into its intensive counterpart via the substitutions U = Umn, S = Smn, V = Vmn, and ni = xin. The result is X dðUm nÞ ¼ T dðSm nÞ - pdðVm nÞ þ μi dðxi nÞ i

or, applying the product rule for differentials

X ndUm þ Um dn ¼ nðT dSm - pdVm þ μi dxi Þ i X þ ðTSm - pVm þ μi xi Þdn i

Equating the coefficients of n and dn on both sides of this equation yields the respective results X dUm ¼ T dSm - pdVm þ μi dxi (7Þ

i

G ¼

X

Um ¼ TSm - pVm þ

μi xi

(8Þ

where eq 7 is the intensive form of eq 6, and eq 8 is the integrated version1. of eq 7. It may appear at first glance that eq 7 does not display the expected loss of one independent variable. However, according to eq 2 the dxi are not all independent because Σidxi = 0. Thus, one of the terms in the summation, say dxj, can be expressed in terms of the others, that is, dxj = -Σi6¼jdxi and thus μjdxj = -Σi6¼jμjdxi. Substitution into eq 7 then yields X dUm ¼ T dSm - pdVm þ ðμi - μj Þdxi (9Þ i

where the summation constraint i 6¼ j can be dropped because μi - μj = 0 when i = j. Interestingly, there are very few physical chemistry or thermodynamics texts that present the intensive form of the Gibbs equations (9). In the case of a binary mixture, eqs 6 and 9 can be written as (10Þ dU ¼ T dS - pdV þ μ1 dn1 þ μ2 dn2 and dUm ¼ T dSm - pdVm þ ðμ2 - μ1 Þdx2

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μi ni

Vol. 87 No. 3 March 2010

Gm ¼ μ1 x1 þ μ2 x2 ¼ μ1 þ ðμ2 - μ1 Þx2

(15Þ

As mentioned above, the development from eqs 6-15 can be thought of as providing an intensive version of eq 6 (namely eq 7) as well as an integrated form of eq 6 (namely eq 14). Many texts on thermodynamics accomplish the integration portion of this process through the application of Euler's theorem on homogeneous functions (10, 11), and the extensive and intensive properties are identified as being of degree one and zero, respectively. The approach taken in this article does not require Euler's theorem and, thus, should be more accessible to students of physical chemistry and thermodynamics. In the above considerations of the Gibbs equation for U, extensive versions have been transformed into intensive versions. The opposite transformation can be carried out by similar methods. In eq 6, the intensive Gibbs equation for Um is presented in a form appropriate for a phase with C components. When a phase is a pure substance; however, its intensive form is the familiar (16Þ

This differential expression can be made extensive with the substitution Um = U/n, Sm = S/n, and Vm = V/n. Then the quotient rule for differentials yields dUm = (n dU - U dn)/n2, dSm = (n dS - S dn)/n2, and dVm = (n dV - V dn)/n2. Substitution into eq 16 and multiplication by n2 then yields ndU - U dn ¼ nðT dS - pdV Þ - ðTS - pV Þdn or rearranging dU ¼ T dS - pdV þ

U þ pV - TS dn n

Now according to eqs 12 and 14, U þ pV - TS = G = μn for a pure phase, so that the previous differential expression becomes dU ¼ T dS - pdV þ μdn

(17Þ

which is the extensive version of eq 16. All of the above extensive and intensive versions of the Gibbs equation for U have analogies in the equation of state of a phase. To provide just one example, the equation of state of a binary phase leads to the extensive differential expression

(11Þ

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(14Þ

For the two-component system illustrated above, eq 13 becomes

dUm ¼ T dSm - pdVm

i


X i

i

and

(12Þ

which is Gm = Um þ pVm - TSm when expressed in intensive form. Because eq 8 can be written as Um - pVm þ TSm = Σi μixi, it follows that X Gm ¼ ui x i (13Þ or in extensive form

Differential Expressions

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Returning to the second consequence of this extensiveto-intensive transformation, eq 8 is a familiar relationship that is usually expressed in terms of the Gibbs energy

pubs.acs.org/jchemeduc

dV ¼ V RdT - V Kdp þ V1 dn1 þ V2 dn2

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(18Þ

r 2010 American Chemical Society and Division of Chemical Education, Inc.


where R is the isobaric expansivity, κ is the isothermal compressibility, and V1 and V2 are the partial molar volumes of components 1 and 2. Just as this result is analogous to eq 10, the intensive counterparts of eqs 11 and 15 are dVm ¼ Vm RdT - Vm Kdp þ ðV2 - V1 Þdx2

(19Þ

and its integrated form Vm ¼ x1 V1 þ x2 V2 ¼ V1 þ ðV2 - V1 Þx2

(20Þ

Equations Involving Partial Derivatives The last category of thermodynamic relations considered here involves partial derivatives. These relations are illustrated with transformations on equations for a binary phase. The first illustration involves transforming the dependent variable in a partial derivative and can be illustrated through finding the relationship between the two derivatives (∂Vm/∂T)p, x2 and (∂V/∂T)p,n1,n2. Using V = Vmn and the fact that n1, n2, and thus n are constant, the constancy of n leads to dV = n dVm and DV DVm ¼n (21Þ DT p, n1 , n2 DT p, x2 Partial derivatives are, of course, closely related to exact differentials, and the same result can be obtained by comparing the differential coefficients in eqs 18 and 19, which leads to the relationship DV DVm DT p, n1 , n2 DT p, x2 ¼ R ¼ V Vm Combining this result with the fact that V = Vmn also yields eq 21 by an alternative path. The same type of reasoning handles a transformation of the independent variable that is illustrated by finding the relationship between (∂p/∂Vm)T,x2 and (∂p/∂V)T,n1,n2. Again the result dV = ndVm for constant n leads to Dp Dp ¼n (22Þ DVm T , x2 DV T , n1 , n2

is the relationship between V1 = (∂V/∂n1)T,p,n2 and the related derivative with corresponding intensive variables, (∂Vm/∂x2)T,p. First, the dependent variable is transformed via V = Vmn yielding (with the subscripts T, p, and n2 temporarily dropped) DV DðVm nÞ Dn DVm ¼ ¼ Vm þn V1 ¼ Dn1 Dn1 Dn1 Dn1 where the first of the two derivatives in the last member of this expression is Dn Dðn1 þ n2 Þ Dn1 Dn2 ¼ ¼ þ ¼ 1þ0 Dn1 Dn1 Dn1 Dn1 Substituting the second result into the first yields DV DVm ¼ Vm þ n V1 ¼ Dn1 Dn1

(24Þ

The second step of the procedure is to apply the chain rule to the partial derivative on the right-hand side of eq 24, which yields DVm DVm Dx2 ¼ Dn1 Dx2 Dn1 The last of the derivatives in this result is then n2 D Dx2 n2 n2 n1 þ n2 ¼ ¼ 2 ¼ - 2 Dn1 Dn1 n ðn1 þ n2 Þ When these results are substituted into eq 24 the expression becomes n2 DVm V1 ¼ Vm - n 2 n Dx2 or, simplifying

DVm V1 ¼ Vm - x2 Dx2 T , p

(25Þ

where the subscripts indicating constant T and p are now returned to the equation. The transformation leading to eq 25 is certainly a lengthy procedure. However, the same equation can be obtained much more directly through the use of eqs 19 and 20. From eq 19 it follows that DVm ¼ V2 - V1 (26Þ Dx2 T , p

Transformations involving the independent variable in a partial derivative can also be carried out through the chain rule. Using the previous example, (∂p/∂V)T,n1,n2 = (∂p/∂Vm)T,x2(∂Vm/∂V)T,x2. Then, because ∂Vm/∂V = 1/n, eq 22 is again obtained. Sometimes it is necessary to transform both the dependent and independent variables in a partial derivative. An example where the process is very simple is the relationship between (∂U/∂S)V,n1,n2 and (∂Um/∂Sm)Vm,x2. Because U = Umn and S = Smn, it follows immediately from their differentials that DU DUm ¼ (23Þ DS V , n1 , n2 DSm Vm , x2

Conclusions

Of course, this result is also obvious from eqs 10 and 11 because both of these partial derivatives are equal to the temperature T. Sometimes the process of transforming both the dependent and independent variables is much more complicated. An example (12)

This article has defined extensive and intensive thermodynamic properties and provided techniques for the transformation of equations from extensive to intensive format or vice versa. The types of equations considered are differential


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Eliminating V2 - V1 from eqs 20 and 26 then yields eq 25 directly. Thus, we again see that use of appropriate differential expressions is often the shortest route to the transformation of a partial derivative. Equations 25 and 26 are important results for the analysis of the molar volume or density of binary mixtures and the determination of partial molar volumes from the method of intercepts (12).2.


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expressions and equations involving partial derivatives. The intensive version of thermodynamic relationships is seen to have one less independent variable than the extensive. Acknowledgment I thank Robert Scott of the UCLA Chemistry Department for introducing me to the transformation techniques described here. Notes 1. An alternative way of integrating eq 6 is to consider that a portion of the phase is removed so the relative volume change is dλ = dV/V. Then, because U, S, and ni are all proportional to V, dλ = dU/U = dS/S = dni/ni. Substituting dU = Udλ, dS = Sdλ, dV = Vdλ, and dni = nidλ into eq 6 and dividing by dλ yields the extensive version of eq 8. 2. See ref 3, pp 272-273, for an analogous result for the chemical potentials, μ1 and μ2, in a binary mixture.

Literature Cited 1. Guggenheim, E. A. Thermodynamics, 5th ed.; North-Holland Publishing Co.: New York, 1988; pp 18-19.

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2. Pippard, A. B. The Elements of Classical Thermodynamics; Cambridge University Press: New York, 1979; p 44. 3. de Heer, J. Phenomenological Thermodynamics; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1986; pp 26-27. 4. McGlashan, M. L. Chemical Thermodynamics; Academic Press: New York, 1979; pp 2-3. 5. Brainard, A. J. Chem. Educ. 1969, 46, 104–107. 6. Quantities, Units, and Symbols in Physical Chemistry, 2nd ed.; Mills, I., Cvitas, T., Homann, K., Kallay, N., Kuchitsu, K., Eds.; Blackwell Scientific Publications: Oxford, 1993; pp 48-54. 7. Levine, I. N. Physical Chemistry, 6th ed.; McGraw-Hill, Inc.: New York, 2009; pp 22-23. 8. Levine, I. N. Physical Chemistry, 6th ed.; McGraw-Hill, Inc.: New York, 2009; p 266. 9. de Heer, J. Phenomenological Thermodynamics; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1986; p 272. 10. McGlashan, M. L. Chemical Thermodynamics; Academic Press: New York, 1979; pp 327-328. 11. de Heer, J. Phenomenological Thermodynamics; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1986; pp 60-62. 12. Atkins, P. W. Physical Chemistry, 4th ed.; W. H. Freeman and Co.: New York, 1990; pp 182-183.


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