Gibbs' Paradox for Entropy of Mixing H. R. Kemp Royal Auotralian Naval College, Jervis Bay 2540, Australia The entropy of mixing of ideal gases is calculated from the relationship ( I )
where AS,i.i, is the change in entropy when several ideal gases with molecules A, B, . . . are mixed at constant temperature and the same initial pressure, nA, n ~. .,.are the amounts of gaseous molecules in the individual gases, VA, VB, . . . are the individual volumes, and Vf is the final volume of the mixture. This relation is used in the derivation of the equilibrium law and a number of other laws of mixtures. A puzzling feature of relation ( 1 ) is that it gives wrong answers when some of the gas snmples have molecules that are the same as those in the other s~unples.Consider, for example, the mixing of two ideal rases with identical molerules, volumes, temperatures and pressures. If we apply relation (1)we obtain
ideal gases in a mixture are truly indifferent to each other. Following van? Hoff, several authors ( 3 4 ) have derived relation (1) by considering the work required to separate a mixture of gases by way of hypothetical semi-permeahle membranes, but in so doing they introduced the assumption that when an ideal pure gas A is separated from an ideal gas mixture bv a membrane ~ermeahleto A onlv. then a t eouilihrium thk pressure of A equals the pa;Aal pressureof A in the mixture. Onlv Rwrv. ".Rice. and Ross recoenized this assumption explicitly ( 5 ) ,seeing it as an application of Dalton's law of partial pressures. Nevertheless. this is a considerable extension of ~ a l t o n ' slaw, which is no~mallyrestricted to gases that are freely mixed in one container. This eaualitv . of pressures across thk semipermeahle membrane can be deduced by means of chemical ~otentialsand in other wavs. but all of thk methods require p;ior clarification of the queition of entropy of mixing and so cannot be used a t this stage u
-
where nt is the total number of molecules. But since this mixing involves no change in thermodynamic state, the correct answer is AS,i,in,
=0
One problem is the deriving of relation (1) from classical thermodynamics in such a way that its limited application is made evident. Another problem is deciding when molecules are sufficiently different for relation (1) to he applicable. These problems will be discussed entirely within classical thermodynamics in accord with the view that this discipline should stand alone as a logical entity independent of statistical mechanics and quantum mechanics. It can be deduced (I) from the first and second laws of thermodynamics that for an ideal gas that changes from volume Vi to volume Vf at constant temperature
Now if we accept Gibhs' Theorem (2,3) that the entropy of each gas in an ideal gas mixture is independent of the presence of other gases, then relation (1) follows readily from relation (2) because each gas now occupies the total volume of the mixture. However, if the theorem is correct for mixing of molecules that are different, then it should he equally applicable when molecules are the same. That it does not cover hoth circumstances is shown by the lack of generality of relation (1). This inconsistency, known as Gibbs' paradox, throws doubt on the validity of the proposition that
F'gue 1. The mixing of several samplesof gas by d i f f m t p a h a t ccnstanttemperahre The overall pris step 0).The final volume of the mixture Vfis not necessarily equal tothe total volume V,of the separate samples'before mixing. Step (a) is the expansion of each gas to volume V,. Step (b) is the mixing of samples with identical mOleCUICS and reduction of their total volume to V,. Step (c) is the mixing of all gases without altering the total volume. Step (d) is the change of the mixture to the final volume V,. The total volume of the system at each stage is shown: x, is the total number of aiginal samples, r is the number of samples Ulat have the same kind of molecules. Step (i) is mixing of equi-volume samples at constant total volume. and step (f) is the reduction of the mixture to the original volume V,. Step (g) is the mixing of samples that are not equal in volume without change in total volume. The initial pressures of the samples are the same.
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without circularity. Bazarov (2) and Zemansky (4) deduced Gihhs' theorem hv van't Hoff's orocedure hut also found necessary the tacit assumption ofequality of pressures of a gas on both sides of a membrane permeable to i t only. All authors noted that the need for a semipermeable membrane restricted their derivations to the soecial case where the cas samples are distinguishable. Bazarov argued that Gihhs' paradox disappears when it is clearly recognized that the mixing of distinguishable particles is quite different from mixing of those that are indistinguishable. But this viewpoint does not throw light on the twin problems of the inconsistency of Gihhs' theorem and the degree of distinguishability that makes the theorem and relation (1) applicable.
The relation (6) is correct only if relation (1) is correct, so it is wrong when molecules in different samples are identical. This pinpoints the source of Gihhs' paradox to step (i). The steps (h), (c), and (e) are introduced to explore the effect of having two or more samples identical with resoect to kind of mol&ules and originally with the same pressure. In step (h) the samples with identical molecules are mixed and their total volume is reduced to Vt. In step (c) all molecules are mixed at constant volume rV+ where r L the numher of original samples that have different molecules. In step (d) the mixture is changed . to the final volume Vf.
Derivation of Entropy of Mixing of Ideal Gases Let us analyze the mixing of ideal gases by considering i t t o occur in several steps as is done in problems on Hess'law. This is valid because entropy is a state function. Figure 1 shows several paths. Although only five samples of gas are shown hv wav of illustration. the areument is aoolicable for any nun;ber."~ll samples have the same initial temperature and presslire and the temperature is kept constant through all changes. First consider steo (e) for mixinn a t constant total volume. This is equivalent tb tge sum of steps (a), (i), and (0. In step (a), each gas expands separately to the volume Vt where Vt is the total of the initial volumes V A ,V B ,. . . . In step (i) the gases are mixed a t constant total volume and step ( 0 is compression to the original total volume Vt. From relation (2)
where D, E, . . are identical molecules and Vt is still the original total volume of all of the samples. This relation, which follows from relation (2), can be written
This expansion step has the same entropy change a9 fm step (g) according to the relation (1). This means that if'rrlation ( I ) is correct AS. = AS,, so ASi + ASi = 0
(4)
However, from relation (2)
where at is the initial numher of samples of gas. Therefore, from relations (4) and (5) ASi = nJ( In at
ASb = (no + nE + . . .)Rin
.
vt VD+VE+
...
(7)
where ns is the total numher of molecules that are the same and os is the numher of samples with the same kind of molecules except that as = 1 when all samples have different molecules. After step (h) all of the samples are of equal volume and have different molecules, so from relation (1) AS, = ntR In r
(9)
where r is the numher of different kinds of molecules. From relation (Z), when at is the total numher of original samples,
Hence, from relations (81,(9),and (10) and Hess' law, This is the formula for the mixing of equi-volume samples of gas without change in temperature. Unlike relation (6) the relation is applicable whether or not samples of gas have different kinds of molecules. Because steps (a), (i), and ( 0 are equivalent t o step (g), it follows from relations (3), (51,and (11) that
(6)
This gives the entropy of mixing of ideal gases when the total volume does not change. T o obtain the entropy of mixing AS, when the total volume does change, we consider step (h). In this step, the volume of the mixed gases is changed to any given value Vf. From relations (2) and (12) and Hess' law
where ASi is the entropy of mixinn of several cases. The relation (f3) is general f i r constanttemperature; being applicable whether or not different samples have identical &olecules, and when the final volume of the mixture V f is different from the total original volume Vt of the individual samples. The relation (13) reduces to relation (1) when molecules A, B, C, D, E, . . . are all different, hut unlike relation (1) i t does not give rise to Gihhs' paradox. Fioure "- - 2. me mixino of ideal iiauids at constant temoerature and Dresswe bv vapaorason h o dlnerent p a b Step (0) ,s me rmpl.-te pocess Step ( m l s of each I qdld to a gas at large volume so Inat me gas oehaves ndea y Step (n) ISthe mixing of the gases wimout change in total volume: step (0) is the condensation of the gas mixtureto a liquid mixture at the Mme pressure as that of the original liquids.
48
Journal of Chemical Education
Derivation of Entropy of Mixing of ldeal Liquids The relation (13) for ideal gases can be used to derive the entropy of mixing at constant temperature of ideal liquids A, B, C, D, E, . . . ,where the molecules A, B, C, . . .may he the same or different. Different paths for the mixing are shown
~ at constant in Figure 2. Step (111 is the mixing of t h liquids nressure. Steu (m) is the va~orizationof each liquid separatrly a very large volume of vapor; step (n) is themixing of the gases at constant total volume Vl; step (0) is the condensation of the mixed gases to a liquid mixture a t the pressure pi of the separate liquids. We will assume that the pressures of the vapors are low enough for ideal gas behavior. Because ideal liquids mix at constant pressure and temperature without change in total volume and internal energy, there is no enthalpy change. Hence AH, = 0
(14)
where H is the enthalpy of the liquids, and, because ideal gases mix without change in internal energy a t constant temperature, there is no enthalpy change when the volume remains constant also. Hence AH.=O
(15)
The steps (m), (n), and (0) are equivalent to (q) and enthalpy is a state function. Hence, from relations (14) and (15) AH, = -AH.
(16)
From Gibbs' differential relations, at constant temperature and pressure for any system dH dS=(17) T Therefore, from relations (16) and (17) and the knowledge that entropy is a state function
Therefore, from relation (13) for mixing of ideal gases V'
V'
v:
v:,
As,=n~Rh-+n&in-+
nsRlnos
(19)
where V1 is the total volume of the vapors, and Vi, V i , . . .are the volumes of gaseous A, B, . . . before being mixed. This relation can he expressed in terms of mole fractions X A , X B , . . . since these are the same for both liquid and gaseous mixtures. Then (21) AS, = -n*R In r a - ~ B InR re - . . . - nsR in as This is the equation for the entropy of mixing of ideal liquids, to give an ideal liquid mixture, a t constant temperature and pr&ure. I t does not give rise to any paradox. For example, when A, B, . . . are all different, the last term disappears because ns = 0 and os = 1.On the other hand, if all of the molecules are the same, the relation gives the correct answer of zero change in entropy.
Meaning of Identical and Different Particles Although there are degrees of difference in particles, the calculation of entropy of mixing requirrs a yes or no answer to the question. T h r matter has been discussed in rhe literature on Gibbx'parridox (3,5,7J. It isapparent that molrcules of various samples of fluid should be rrgarded as different if the rxperimenter has the ability and incention to distinguish brtween them and toseparace them ahrr mixing. On theother hand. if the molecules are different but the ex~erimenrerdws not have this ability and this intention, the question is more difficult to answer. In this latter event, however, no incorrect predictions should ever arise if the m~leculesare taken to be identical for the purpose of calculating entropy of mixing. The difficulty of making a yes-no decision highlights the problem of assigning an absolute value to the entropy of a substance.
The Basic Axioms of Ideal Thermodynamics If it is desired to see ideal classical thermodynamics as a logical structure, then it is important to identify the assumptions or axioms from which the various thermodynamic laws and relationships are deduced. The entropy of mixing cannot hededuced .--~ - ~ from the first and second laws alone. even if these laws are taken to include that entropy and internal energy are fixed by the thermodynamic state. The well-known relation (1) can he deduced if the additional assumntion is made that ideal eases in the same mace are indifferent to each other, each haiing its own therm"dynamic state. Rut thisassumption should he rejected hecause it leads tu Gibljs' paradox. In the previous discussion, relation ( I 1 ) isdeduced from the two lawsand relation (11. Examination of the argument will convince the readw that it ii equally valid toassume relntion (11) and deduce relation (1). Recause rrlation (1) is entangled with rrlation (21and because it is limited to the special m e of molecules A, H,.. .being d~fferent and furthermore, hecause relation (1) is associated with the paradox, relation (11) is the better choice as a basic axiom to replace the assumption of indifference. It is true that relation (11) is more difficult to remember than the assumption of indifference hut it has the virtue of giving the correct answer. ~
~~
~
~
~
~~~
~
~
~
~
Conclusions The general relations for entropy of ideal mixing of fluids a t constant temperature and identical initial pressures are relations (13) and (Zl), the former for gases and the latter for liquids. They are applicable whether or not the various samples have different molecules. The relations for ideal mixing cannot he deduced from the first and second laws of thermodynamics alone but can he deduced from the two laws, the general gas equation PHV = n&T, and relation (11) for mixing of ideal gases at the same or temperature. initial volume, without change in t& vol-G These four laws can he regarded as independent principles of ideal classical thermodynamics. I t should he noted that the gas equation in the above form refers to a constituent B; this form includes Dalton's law of partial pressures. Relation (11) could be replaced by certain other assumptions as a fundamental axiom (5) hut not by Gihbs' theorem of indifference for constituent gases of a mixture. This theorem gives rise to the incorrect notion that the constituents of a mixture can he assigned separate entropies. This is unsound whether or not the separate entropies happen to equal the entropy of the mixture. It opens the door to misinterpretation of partial molar entropies and other partial molar quantities. In considerine ideal mixing. it is a mistake to regard the individual sampres as several &terns and also the mixture as several svstems since after mixing the constituents have lost the indi;iduality of their thermobynamic states. When the sounder procedure is adopted of regarding mixing as a change in a single system then we note that there is no change in volume for ideal isothermal isobaric mixing. Hence, relation (1)is seen to give the entropy change associated with the molecules being different. Despite its form, it should not be viewed as giving the effect of the expansion of the individual substances.
Literature Clted S., and Brewer, Leo, (Lewis and Randall1 "Thermodynamics." M10r.w-Hill, New York, 1961.
(1) Pitzer, Kenneth
(21 Bszarov, I. P., "Thermodynamin."Pergamon Prrsa, 1964,pp.60.241-242.245-248. (31 terHaar,D.,andWerpeland, H. N.S.,"ElomentsofThermodvnami~s,"Addlson-Wmk. Reading, MA, 1966.p. 87. (41 &mansky, Mark W.,"Hest and Thcrmodvnsrnies."Sthd..MeOraw-Hill,NeuYork, 1968, pp. 559563. (51 Bemi, R.Stephen. Rie,Stuart A..snd Ross.John. "PhysicalChemistry."JohnWiley &Sons, New York, 1980. pp. 752-755. (61 Fong. Peter, "Foundations of Thermodynamics,"Oxford University Pleas. 1963, pp.
1-9 .. ... (71 I l 8 k . A . M., Amer J. Phya, A, 15[4], pl. 3 (19801.
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