A Mathematical Verification of the Second Law of Thermodynamics from the Entropy of Mixing Michael J. Clugston Tonbridge School, Kent, England The usual derivation of the form of the equilihrium constant deoends on introducine the coucevt of a chemical potential. In this paper the equilibrium constant will he derived instead from the entropy .. of mixine, -. which is a much easier concept to visualize. A novel consequence of this approach is that i t is possible to uerify the second law of thermodynamics by proving mathematically that the Gibbs energy is a minimum at equilibrium. As will be proved helow, prouided that Stirlinji's approximotion is valid, the Gibbs energy will minimize (or, equivalently, the total entropy will maximize), simply because the mixed state is more probable. The proof rests only on the validitv of Stirline's avvroximation. This is well known to be .. exceptionally accurate so long as the numbers involved are laree. That is indeed the realm where thermodvnamics is trAitionally regarded as reliable. This proof fits in naturally with Lewis and Randall'saeneral form of the second law ( I ) : "Every system which is ieft to itself will, on the average, change towards a condition of maximum probability." As a coroliary, if the approximation is inaccurate, i.e., when the numbers are small, the second law too would be unhelpful: i t is known to he inaooro~riatefor oredictine" the behavior of individual atoms L d n;olecules. ' In a recent article. Loean (2) . . has shown that the entrovv of mixing crucially determines the position of homogen&us chemical eauilihrium. He did this bv choosina- a number of special cases. In this article, his conclusions will be reinforced bv ~ r o v i n atheaeneral case. I t will be shown that this approach can generate the isotherm (3),which controls the vosition of equilihrium quantitatively. A novel conclusion is that the form of the equilibrium constant is determined directly by the entropy of mixing. Before going into the detailed derivation, i t is worth identifying a couple of common misconceptions: first concerning the significant difference between the standard Gibbsenergy changeand ageneral Gibbs energy change, and second concerning the underlying reason for chemical change. The standard Gibbs energy change, AG,' is the change in Gibhs energy, per unit amount of substance, when pure reactants become pure products (4), a t a specified temperature. I t is very helpful, yet singularly uncommon, to draw this on a plot of the Gihbs energy as a function of the extent of reaction. The concept of extent of reaction, t, is rigorously defined (5). . .. see near ea.5.. so that € = 0 mol reoresents vure reactantsand ( = 1 mol represents pure products. Hence it is clear that on a d o t of Gibbs enerav extent of reaction -.against AG' is the slope of the dashed line in Figure 1joining the points a t 5 = 0 and 1mol. Any general Gibbs energy change, AG, is the change in Gibbs energy, per unit amount of suhstance, at a particular specified composition (4), at a specified temperature. This isalsoshown in Figpre 1, as theslope of the line at a particular extent of reaction. for example < = 0.2 mol. I t is immediately ohvious that AG is an unfortunate notation, for in reality this is a differential quantity, being evaluated at a varticular com~osition.while the notation leads us t o expect from common mathematical symbolism a finite change. I t would perhaps have been superior to have ~~
used a differential notation, such as aG/at, a t the risk of alienating some less mathematically inclined chemists. Indeed in some texts the neeative of this auantitv. the affinitv -aG/at, is introduced (6):~s a particular example, consid; the esterification equilibrium ~~
~
~
for which AGe = -3.5 kJ mol-' (at 298 K), Figure 2. It is common but misleading to say that "the reaction goes because the standard Gibbs energy change is negative." I t is true that an equimolar mixture of all four substances produces more ester and water a t the expense of acid and alcohol, because at ( = 0.5 mol the slope of aG/at is negative. Yet
Figure 1. The slope of the dashed line is AG'and equals -3.5 kJ MI-': me slope of the bold line is AG (at € = 0.2 moll. The minimum Gibbs energy of ~. -5.49 kJ occurs at t = 0.670 mol
~
Figure 2. ReaCliOns pmc& toward a minimum in Gibbs energy. This graph is appropriate for the esteriticatlonreaction, for which AG'= -3.5 kJ mal-'.
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at { = 0.9 mol the slope of dG/a( is positive: at this composition water ~ -ester - and ~ ~oroduce ~ ~ acid ~ and alcohol! To the left of the minimum acid and alcohol react, whereas to the right of theminimum ester and water react. The direction of natural change is towards the minimum in Gibbs energy (3), which has been indicated by using arrow in Figure 2. At the minimum, no further net change occurs: the system has reached chemical eauilibrium. The position of chemical equilibrium therefore Eorrespnds to-the composition of minimum Gibbs energy, i.e., where aG/at = 0. Second, the underlying reason for chemical change is often totally misunderstood. The introduction to Warn's hook on chemical thermodynamics (7)is well written, yet it contains a grave error. He implies on p 2 that there are two laws eovernine change: one that enerw is minimized and the other that entropy is maximized. fi is crucial to realise that there is no minimization of energy principle, in either physics or chemistry, however siductivethis illusion may appear. Consider a hall rolling in a valley between two hills. I t is obvious that in the r e d world the ball gradually slows down and eventually stops in the valley. Yet in the ideal world (much beloved of the mathematicians) of a totally frictionless ball and surface, the ball would simply oscillate, rolling down and up forever. I t does not seek out a position of lower potential energy. The reason why the hall stops in the valley is that friction soreads out the enernv -- of the ball, dissipating it into the surr&ndings. The only law governing change is that entropy, which quantifies the spreading of energy (41, is maximized. I t is therefore important to realize that the minimization of Gibhs energy is only a conventional, albeit very convenient, picture. The deep explanation is, however, that tending to a minimum in Gihhs energy is disguising the true reason-the total entropy of the system and its surroundings is tending to a maximum. In order to derive the expression for the equilibrium constant, the Gibhs energy m i s t first be writtenbut as the sum of two terms: the first term is the extent of reaction multiplied by the standard Gibbs energy change. The second term is due to the mixing of reactants and products (2,3). (The entropy of mixing term is absent in the case of a heterogeneous equilibrium and so for such systems, as Logan points out (2),c&nplete reaction can occur, unlike for the present homogeneous equilibrium.) ~
~~
~
~
(1) G = FAGe - TASmi, I t is clear that, even in the special case when AG0is zero and for a simple homogeneous equilibrium such as A B =C + D, change can occir due to the entropy of mixing term. The physical picture behind this is that a mixture of A and B is more favorable entropically than either A or B on its own. The Gihbs energy would vary as in Figure 3. The minimum
+
occurs a t = 0.5 mol, as expected intuitively. Differentiating eq 1with respect to yields (at constant temperature)
The general expression for the ideal entropy of mixing ultimately comes from Boltzmann's equation S = k i n W ,using the number of ways of arranging distinguishable particles followed by Stirling's approximation. This is very clearly explainedon page 281 of Lewis and Randall's classic text ( I ) . (We also assume constant pressure and temperature.) No other assumptions or approximations are made in the present derivation. Their equation (21.5) can be rewritten as
where R is the gas constant, nj is the amount of substance (in moles) of each component j , and n is the total amount of suhstance, the sum of all the individual nj. To calculate the second term in eq 2, we require the chain rule' for partial differentiation of a function f:
Thesecond bracket is easy to work out from the definition of theextent of reaction (5), d t = dni/uj, where ui is the stoichiometric coefficient of component i (e.g., +2 for NH3, -1 for Nz, -3 for Hz in the Haber-Bosch synthesis Nz 3Hz * 2NH3):
+
Using the form off in eq 3, and differentiating with respect to a fixed n,, the first bracket in eq 4 can be evaluated
The final neat form of this equation uses the definition of the mole fraction xi = niln. Substituting ehs 5 and 6 into eq 4 gives the following result, which can then he rewritten using standard rules fo;manipulating logarithms:
Combining eqs 2,3, and 7 produces the final result that
As stated above, chemical equilibrium occurs when the Gihbs energy is a minimum, i.e., dG/at = 0. From eq 8, that occurs when (9) AG~=-RT~~K with
Equation 9 is the isotherm (3). The new feature to emerge, however, is that the form of the equilibrium constant in eq 10 is shown to depend directly on the entropy of mixing. (The normal form of K, for a gaseous reaction is obtained using Dalton's law for ideal gases, pi = xip, noting that the constant total pressure necessary for defining the standard Gihhs energy change is unit pressure, e.g., 1bar.)
+
+
Figwe 3. The Glbbs energy change for the equilibrium A B + C D, with AGe= 0. The minlmum Gibbs energy of -3.43 kJ occurs at € = 0.500 mol. 204
Journal of Chemical Education
' MOst common math texts describe this.
Graphically it is intuitively clear that the Gibbs energy goes throueh a minimum; yet the argument so far demonitrates only that it goes through a tu&ingpoint. A loophole needs to be filled: the second differential must be calculated because, to prove that the Gibhs energy is indeed a minimum a t equilibrium, the second differential must be shown to be positive. As Deumi6 et al. have commented in this Journal (a), this vital step is rarely taken. In the detailed mathematical argument that follows, it will he shown that the second differential is indeed positive in all circumstances. This conclusion is mathematically guaranteed, given the form of the entropy of mixing, and is independent of any physical law, including the second law of thermodynamics. Although theshape of the graph is mathematically predictable, i t is the second law that predicts that reactions tend toward the minimum Gibbs energy. From eq 8, i t can he seen that
Notice that this result is independent of the magnitude and sign of the standard Gibbs energy change. The only contributory term comes from the entropy of mixing. As RT cannot he negative, proving that a2f/Jt2 is positive will therefore constitute a proof that the Gibhs energy goes through a minimum. From eq 7
Once again the chain rule for differentiation is needed. For clarity the sum this time will be taken over subscript j:
The first bracket in eq 13 can be found easily, remembering that n includes nj:
have a t least one reactant and one product! Inserting inequality 16 into eq 15 gives
The Cauchy-Schwarz inequality (9),which is valid for real numbers a, and bj, is as follows:
Using the identification a j = (Ivjllfi) and bj = equality 18 can be applied to give
fi the in-
noting trivially that ivjlz = uj2. As the second bracket in this expression is simply the total amount of substance n, dividing by n leads to
Putting expression 20 into the inequality 17 shows unambiguously that the term a2f/a$2is positive and hence that the Gibbs energy is a minimum a t equilibrium. The Gihhs energy has been shown rigorously t o be a minimum, starting from the ideal entropy of mixing, eq 3, the only significant assumption being the use of Stirling's approximation. Perhaps his contemporaries were far-sighted when they engraved his equation on Ludwig Boltzmann's tombstone. Giuen his equation, it is mathematically certain that the Gibbs enerm will minimize in a chemical reaction. I t is appropriat;L leave the last remark to Roltzmann ( I ) , who ascribed the fundamental idea to Gihbs: "the impossibility of an uncompensated decrease of entropy seems to be reduced to an improbability." Acknowledgment
The second bracket in eq 13 can be found by the method of eq 5. Combined with eq 14, this gives
I am most grateful to three of my former students, Rajesh Jena, Michael Phillips, and Julian Scarfe, for help writing the computer program that drew the figures. I am deeply grateful to Nick Lord for his crucial contribution to the solution using the Cauchy-Schwarz inequality and to Richard Compton for his useful advice. Literature Cited
In the last step, i t has been recognized that the two summaare identical. I t is the second term in eq tions & and 15 that will now be manipulated. First, though, an important, albeit apparently trivial, change is made.
zjvj
> .
,
> .
,
This will he true so long as a t least two of the v j values have opposite signs and this is certain, as all chemical reactions
1. Lewis, G. N.; Randall. M. Thermodynamics; revised by Pitzor and Brewer; McGraw-
Hill: New York, 1961:pp281,91,and92. 2. Lo8an.S. R. Educ. Chem. 1988. (Msrehl.44. 3. Atkins, P. W. Physical Chemistry, 3rd ed.; Oxford: Oxford, 1986; Sections 10.1 and 8.2. 4. Atkins. P. W.: Clumfon. M. J.: Frazor, M. J.; Jones, R. A. Y. Ch~misfrv:Lonrman: London. 1968: ~ e i t i o n11.2 and Appendix 11.2. 5. ASEChemical Nomenclsture,Symbolr andTerminology: 1985:Seetlon 2.4.Seealso ref 3,section lo.,. 6. Smith. E. B. Basic Chemical Th~rrnodynornics.3rd ed.: Ciarendon: Oxford, 1982: p 5. 7. Warn. J. R. W. Concise Chemical Thairnodvnomies: Van Nostrand: London. 1969: seetion 1.1. 8. Deumie. M.: Boulil. B.;Henri-Rousseau.0. J. Chem.Educ. 1987.64.201. 9. Abramowitz. M.; Stegun, I. A. Handbook of Mothernotical Functions; Dover: New York, 1965: thmrern3.2.9.
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