The Entropy Criterion for Chemical Equilibrium An Exercise in the Second Law Norman 0. Smith Fordham University, Bronx, NY 10458 The two most useful quantitative statements of the second law of thermodynamics for chemists are dSu,v B 0 and d G p , ~ < 0, where S is entropy, U internal energy, V volume, G Gihhs free energy, P the pressure, and T the temperature of aclosed system. The first~statementmeans, of course, that the approach to chemical equilibrium in an isolated system (fixed I-J and reeardless of the direction of approach, is accom- V). ~,, ~ panied by an entropy increase, so that at a Grit of equilibrium S is a maximum. The second statement means, analogously, the existence of a minimum in the Gibhs free energy of aclosed system a t a point of equilibrium, provided P and T a r e kept constant. Although the free energy minimum (P,T constant) and the increase in S (U, V constant) in a spontaneous process are often illustrated in texts, the existence of the entropy maximum is rarely, if ever, shown. This omission is unfortunate because a demonstration of the maximum makes concrete an abstract concept and complements a demonstration the direct connection beof the minimum in tween entropy and probability in statistical thermodynamics ma!;es such an illustration desirable. I t is the purpose of this article to illustrate the entropy maximum with a chemical examp1e.l The author believes this to be both an instructive and interesting exercise, and i t exposes the student to the comparative novelty of working with an isolated, chemically reactive system. I t also provides an opportunity to show that the entropy of a non-isolated system can pass through a maximum hut does so at a nonequilihrium point.
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The Model System The example chosen for this study is the system 2HI(g) =+ Hz(g) Iz(g). Two moles of HI(g) a t 750.0 K are placed in a vessel of volume 50.0 dm3 and completely isolated from the rest of the universe. The HI decomposes to some extent until equilibrium in reached. All three gaseous species are to be treated as ideal, and any dissociation of diatomic iodine is to he ignored. Any Hz and Iq will always he present in equimolar quantities. Since the reaction as written is endothermic the temperature falls as the system equilibrates. The extent of reaction, [, defined as the number of moles of HZ (or of 12) formed, initially zero, would increase to unity if the reaction were to go to completion. It is to be shown by calculation from existing data that the entropy of the isolated system passes through a maximum as E increases from zero (or decreases from unity) by considering all possible values of [. I t is important to reco(tnizr that, as changei,sodt,H (the enthalpy), .i C, , 1'. T, and ,I (the Helmholtz free e n e r a ) , because the svstem is isolirtrd. Onh. ( ' a n d \ ' arr runsfant. For consistencv ail the necessary thermodynamic data were taken from thk same reference work.2 These are the molar heat ca~acitiesat constant pressure, the standard molar entropies and Gihhs functions (referred to 298 K), the values of H7m - HSW, and the standard enthalpies of formation of the three s p e c s . The heat capacities a t constant pressure were converted to con-
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' The idea from which this study developed originated on reading Everdell, M. H., "Statistical Mechanicsand its Chemical Applications." Academic Press. London. 1975, pp 4-5. Still, Daniel. R., Westrum. Edgar F.. Jr., and Sinke, Gerard C., "The Chemical Thermodynamics of Organic Compounds," John Wiley & Sons, Inc., New York, 1969, pp. 209, 212. and 228. 50
Journal of Chemical Education
stant volume, Cv,and fitted by the author to a polynomial of the form a bT cT2 over the temperature r&ge of interest, as were the entropies. The resulting data are given in Tahle
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Calculation of the Entroov . . of the Mixtures The entropy of the mixture is now to he calculated for various values of h For this the composition of the mixtures are required. Although and their temperatures and approximate temperatures can he found by assuming that Cv for the mixture is independent of both T and E, i t is more satisfvine . . to find them accuratelv. This entails recognizing that the heat capacities of the mixtures are temperature deen dent. Since the enerm required by the reaction comes from ihe thermal energy, andio p~oducescooling,we equate AU7x (which is the same as AH750 for the system under consideration) to $$M Cv(mixt)dT, from which T may he found. For example, if [ = 0.300 the mixture consists of 0.300Hz. 0.300Iz, and 1.400HI so, from Tahle 1,
+ 1.80 X 10WTT + 0.300(27.84 + 3.46 X 10-3T - 1.82 X 10W6T2) + 1.400(20.33- 5.80 X 10-'T + 5.68 X 1 0 - 6 T 2 ) ] d T which leads to 2.65 X 10-6T3 - 5.35 X 10-5T2+ 43.127 - 29,620 = 0 for which the only pertinent root is T = 669.0K. (Solvingthe cubic equation can readily he accomplished by, for example, the Newton-Raphson method.) With this procedure the values of T were determined for various Fs. They are presented in Tahle 2, which includes the total pressure,P, given by 2RTIV (where V = 50.0 dm3). T would he a strictly linear function of E if Cv were independent of T. Table 1. Thermodynamic Data from Stull el al. (footnote2) and Derlved Quantities
296 600 700 800
...
130.59 136.29 138.72 141.03
163.81 181.76 195.94
206.48 212.25 214.72 217.15
'me wluesfw idglare b a d on H& fa ids). merefae, torefer hem tohat f a ldg). A f ~ k & . g)/ Tor 62.4421TJ KK' mol-' musi be added to them. 1.1 X 10-lT+ 1.80 X 1 0 @ f Cy(H2.g)lJK~'m= l ~21.03~ &(Iz. g)/J K-' mi-' = 27.84 3.46 X iO-=T- 1.82 X 10@P HI. g)/J K-' mol-' = 20.33 - 5.80 X 1 r 4 T + 5.88 X 10"P SOW*, g)lJ K-' mi-' = 108.67 8.15 X 10-ZT- 3.51 X 10-5P SO(12. gXlJ K-'ml-' = 232.43 1.173 X 10-'T- 4.50 X 10-5T2 3.24 X 10-'P SO(H1, g(/J K-' mol-' = 184.76 9.00 X 10'T-
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(c, valuer valid in me ran@ 400 to 800 K, 9 valuer 500 to 800 K l A~H kdH2, 91' 0, A t ~ k d e agl , = 62.442 kJlmd, AfHHdHI. gl = 28.36 Wlmol i 0.05. Hz191 lzlgl. A H & = 0 82.442 - 2126.361 = 9.7 W i 0.1. Therefore, for 2Hllgl H, ; - H & (lnlerpolatedl: Hdg) = 13.226. Idg) = 79.352. Hllg) = 13.56 kJlmol i 0.05. H w e h w l w faldgl is b a d oo H b f a Ms), merefm, fa2Hllg)- Hdgl Idgl, AH;, = 13.226 79.352 2113.58) - ~(AIH;~IHI. gl = 12.7 W i 0.1.
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Table 2.
Properties of Mixtures Given by 2Hl(g), lnltlally at 750.0 K, in Isolated 50.0-dm9 Vessel a
T IK)
latm)
S IJIK)
U- U&.
t
IkJ)
H - H&8 IkJ)
0 0.100 0.190 0.200 0.210 0.220 0.250 0.300 0.500 0.750 1.000
750.0 722.7 696.3 695.6 693.0 690.3 682.3 669.0 617.3 555.6 497.7
2.462 2.372 2.292 2.263 2.275 2.266 2.240 2.196 2.026 1.624 1.634
453.1 456.4 457.1 457.1 457.1 457.1 457.0 456.7 453.3 444.7 428.5
19.6 19.6 19.6 19.6 19.6 19.6 19.6 19.6 19.6 19.6 19.6
27.2 26.7 26.3 26.3 26.2 26.2 26.0 25.8 25.0 23.9 23.0
P
G- G;w
A-A& IkJ)
IkJ) -189.5 -160.0 -169.6 -166.6 -167.4 -166.3 -162.7 -156.6 -131.7 -100.0 -67.2
-197.1 -167.1 -175.2 -175.2 -174.0 -172.8 -169.0 -162.8 -137.0 -104.3 -70.5
Uh,, H h . O h . and A & all refer to 2Hl(g)
The values of S may now he computed for each E, remembering to include the entropy of mixing and to convert to the proper pressure. With f = 0.300, for example, for which T = 669.0 K and therefore P = 2.196 atm,
Finally, values of A - Aig8are readily obtained from G - Gig8 - 2RT 2R(298.15). I t is evident from Table 2 that, as € increases, U remains constant, H decreases nearly linearly, and both G and A increase nearly linearly. Only S passes through a maximum.
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U e of Spectroscopic Data
from Table 1.Therefore,
S = 0.300(154.17) t 0.300(290.76) + 1.400(230.47) - R[0.300 ln(0.30012.000)+ 0.300 ln(0.30012.000) + 1.400 ln(1.400/2.000)]+ 2R ln(112.196)= 456.67 J/K In this way the entropies listed in Table 2 were obtained. They are plotted against f in Figure 1. The entropy maximum is obvious and occurs a t E = 0.210 0.005, for which T = 693 K f 1.The equilibrium constant, Kp (which is given by [€/(2 2f)]2 when [ is the equilibrium value), is found to be 1.77 X 10-2. This is the point of final equilibrium.
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Variation of the Other Thermodynamic Functions
I t is enlightening also to determine how the remaining thermodynamic functions, U, H, G, and A vary with [. Their absolute values cannot, of course, be determined, so they must he computed relative to some reference state. For this purpose 2 mol of HI(g) at 298.15 K and 1atm has been chosen, to which Uig8, H;98, Gigs, and AigSrefer throughout Tahles 2 and 4. Since, for 2HI(g),
The entire exercise described above can, if desired, be translated into the statistical thermodynamic approach by replacing most of the data of Table 1by the thermodynamic properties calculated from molecular constants. The latter are given in Table 3. By applying the rigid rotor, harmonic oscillator approximations almost the same results as given above are obtained. The System at Constant Temperature and Pressure
I t has been shown above that starting with 2HI(g) in a completely isolated 50.0-dm3 vessel at 750.0 K the final state
750
T/K 617
693
I
I
460 -
556
498
I
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H m - Hiss = Z(13.58) = 27.2 kJ (see Table I ) , and (because H = U + P V = U + nRT), H7m - Hw = U ~ J-OU& + ZR(750 - 298) i t follows that However, since the system is isolated, this is also the internal energy for all values of f, and is so listed in Table 2. In a similar a t any temperature is given by manner H - HBa
- Gm = H - Hm - TS + 298.15(0.41296)kJ
For example, for [ = 0.300 ( T = 669.0 K),
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C F lgure 1. Entropy and Gibbs free energy as a function of extent of reaction and temperature for isolated system.
In this way the values of H - H;98 in Table 2 were obtained. T o find G - G& one requires Sissfor HI(g). This is identical to the free energy function for this substance as given in Table 1, namely 206.48 J K-I mol-I, or 412.96 for 2HI. I t follows that at any temperature T G
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Table 3.
Species H2 12
HI
Molecular Properties of Reacting Species
Molecular Mass (glmoi)
Characteristic Rotational Temperature IK)
Characteristic Vibrational Temperature IK)
2.016 253.61 127.91
87.5 0.0536 9.43
6986 306.6 3209
Volume 62
Number 1
January 1965
51
Properties ol Mixtures Given by 2Hl(g) in 50.0-dm3 Vessel at 693.0 K and 2.275 Atma
Table 4.
'Ub. H i s , G b . and A h a l l referto ZHligl.
is one for which E = 0.21, T = 693 K, and P = 2.275 atm. As a comvlementaw exercise we mas, instead of considering an isolated system, consider one wh&h is held at constant and P. By starting with 2HI(g) in a 50.0-dm3 vessel a t 693.0 K and 2.275 atm, one may calchate the thermodynamic properties of all mixtures resulting as E increases from zero to unity, keeping T = 693.0 K and P = 2.275 atm. Under these conditions any energy required to maintain constant temperature as E increases comes from the surroundings and not from the system itself. At the same time G should be a minimum a t f = 0.21 rather than S being a maximum at this poinebecause of the new constraints on the system. I t may be noted that since there haovens to be no volume chanee in the reaction under considekion V is also constant a t c k % m t T and P. One would exvect. therefore. that the second law statement d l v ,
