1881:
pH-buffered, aqueous solution. In more complex environments, where bound particles may be prevented from disassociation by their surroundings, or diffusion is slow or anomalous, the model of mass action does not always describe the behavior of the reaction kinetics accurately. Several attempts have been made to modify the mass action model, but consensus has yet to be reached. Popular modifications replace the rate constants with functions of time and concentration. As an alternative to these mathematical constructs, one school of thought is that the mass action model can be valid in intracellular environments under certain conditions, but with different rates than would be found in a dilute, simple environment .
2613:
2682:
The law of mass action forms the basis of the compartmental model of disease spread in mathematical epidemiology, in which a population of humans, animals or other individuals is divided into categories of susceptible, infected, and recovered (immune). The principle of mass action is at the heart of
1848:
In general many reactions occur with the formation of reactive intermediates, and/or through parallel reaction pathways. However, all reactions can be represented as a series of elementary reactions and, if the mechanism is known in detail, the rate equation for each individual step is given by the
1764:
in Europe until the 1890s). The derivation from the reaction rate expressions is no longer considered to be valid. Nevertheless, Guldberg and Waage were on the right track when they suggested that the driving force for both forward and backward reactions is equal when the mixture is at equilibrium.
1884:
The fact that
Guldberg and Waage developed their concepts in steps from 1864 to 1867 and 1879 has resulted in much confusion in the literature as to which equation the law of mass action refers. It has been a source of some textbook errors. Thus, today the "law of mass action" sometimes refers to
1880:
In biochemistry, there has been significant interest in the appropriate mathematical model for chemical reactions occurring in the intracellular medium. This is in contrast to the initial work done on chemical kinetics, which was in simplified systems where reactants were in a relatively dilute,
1472:
is the proportionality constant. Actually, Guldberg and Waage used a more complicated expression which allowed for interaction between A and A', etc. By making certain simplifying approximations to those more complicated expressions, the rate equation could be integrated and hence the equilibrium
1328:
The ratio of the affinity coefficients, k'/k, can be recognized as an equilibrium constant. Turning to the kinetic aspect, it was suggested that the velocity of reaction, v, is proportional to the sum of chemical affinities (forces). In its simplest form this results in the expression
267:
the "chemical affinity" or "reaction force" between A and B did not just depend on the chemical nature of the reactants, as had previously been supposed, but also depended on the amount of each reactant in a reaction mixture. Thus the law of mass action was first stated as follows:
2350:
2827:
Individuals in human or animal populations – unlike molecules in an ideal solution – do not mix homogeneously. There are some disease examples in which this non-homogeneity is great enough such that the outputs of the classical
3002:" … la décomposition du muriate de soude continue donc jusqu'à ce qu'il se soit formé assez de muriate de chaux, parce que l'acide muriatique devant se partager entre les deux bases en raison de leur action, il arrive un terme où leurs forces se balancent."
882:
3004:( … the decomposition of the sodium chloride thus continues until enough calcium chloride is formed, because the hydrochloric acid must be shared between the two bases in the ratio of their action ; it reaches an end at which their forces are balanced.)
614:
2983:
On pp. 404–407, Berthollet mentions that when he accompanied
Napoleon on his expedition to Egypt, he (Berthollet) visited Lake Natron and found sodium carbonate along its shores. He realized that this was a product of the reverse of the usual reaction
1751:
1042:
The rate expressions given in
Guldberg and Waage's 1864 paper could not be differentiated, so they were simplified as follows. The chemical force was assumed to be directly proportional to the product of the active masses of the reactants.
1031:
160:
independently came to similar conclusions, but was unaware of the earlier work, which prompted
Guldberg and Waage to give a fuller and further developed account of their work, in German, in 1879. Van 't Hoff then accepted their priority.
453:, the chemical force driving the forward reaction must be equal to the chemical force driving the reverse reaction. Writing the initial active masses of A,B, A' and B' as p, q, p' and q' and the dissociated active mass at equilibrium as
1098:
This is equivalent to setting the exponents a and b of the earlier theory to one. The proportionality constant was called an affinity constant, k. The equilibrium condition for an "ideal" reaction was thus given the simplified form
1621:. The equilibrium constant, K, was derived by setting the rates of forward and backward reactions to be equal. This also meant that the chemical affinities for the forward and backward reactions are equal. The resultant expression
153:, proposed the law of mass action in 1864. These papers, in Danish, went largely unnoticed, as did the later publication (in French) of 1867 which contained a modified law and the experimental data on which that law was based.
97:
between 1864 and 1879 in which equilibrium constants were derived by using kinetic data and the rate equation which they had proposed. Guldberg and Waage also recognized that chemical equilibrium is a dynamic process in which
749:
1323:
1844:
reactions that may not be concerted reactions. Guldberg and Waage were fortunate in that reactions such as ester formation and hydrolysis, on which they originally based their theory, do indeed follow this rate expression.
272:
When two reactants, A and B, react together at a given temperature in a "substitution reaction," the affinity, or chemical force between them, is proportional to the active masses, and , each raised to a particular power
1876:
expression so that the overall rate equation can be derived from the individual steps. When this is done the equilibrium constant is obtained correctly from the rate equations for forward and backward reaction rates.
1205:
1447:
426:. Active mass was defined in the 1879 paper as "the amount of substance in the sphere of action". For species in solution active mass is equal to concentration. For solids, active mass is taken as a constant.
3255:(Actually we understand by "active mass" only the quantity of substance inside the sphere of action; under otherwise identical conditions, however, the sphere of action can be represented by the unit volume.)
3253:"Eigentlich verstehen wir unter der activen Masse nur die Menge des Stoffes innerhalb der Actionsphäre; unter sonst gleichen Umständen kann aber die Actionsphäre durch die Volumeneinheit repräsentirt werden."
106:. In order to derive the expression of the equilibrium constant appealing to kinetics, the expression of the rate equation must be used. The expression of the rate equations was rediscovered independently by
424:
1592:
2940:
The law of mass action is universal, applicable under any circumstance... The mass action law states that if the system is at equilibrium at a given temperature, then the following ratio is a constant.
2608:{\displaystyle n_{o}p_{o}=\left(N_{C}e^{-{\frac {E_{C}-E_{F}}{k_{\text{B}}T}}}\right)\left(N_{V}e^{-{\frac {E_{F}-E_{V}}{k_{\text{B}}T}}}\right)=N_{C}N_{V}e^{-{\frac {E_{g}}{k_{\text{B}}T}}}=n_{i}^{2}}
262:
2662:
describe dynamics of the predator-prey systems. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this rate is evaluated as
2060:
1944:
1940:
337:
1769:
of forward and backward reactions to be equal. The generalisation of the law of mass action, in terms of affinity, to equilibria of arbitrary stoichiometry was a bold and correct conjecture.
2687:, which provide a useful abstraction of disease dynamics. The law of mass action formulation of the SIR model corresponds to the following "quasichemical" system of elementary reactions:
1093:
646:
The third paper of 1864 was concerned with the kinetics of the same equilibrium system. Writing the dissociated active mass at some point in time as x, the rate of reaction was given as
3600:
2057:
2744:
2816:
2773:
887:
The overall rate of conversion is the difference between these rates, so at equilibrium (when the composition stops changing) the two rates of reaction must be equal. Hence
3450:
1980:
1835:
2145:
2103:
2342:
2248:
2211:
2178:
1627:
444:
2302:
2275:
1910:
1874:
1470:
1491:
636:
471:
1509:
In the 1879 paper the assumption that reaction rate was proportional to the product of concentrations was justified microscopically in terms of the frequency of
137:
Two chemists generally expressed the composition of a mixture in terms of numerical values relating the amount of the product to describe the equilibrium state.
2000:
3358:
3420:
760:
479:
1756:
is correct even from the modern perspective, apart from the use of concentrations instead of activities (the concept of chemical activity was developed by
652:
638:
represents the amount of reagents A and B that has been converted into A' and B'. Calculations based on this equation are reported in the second paper.
1524:). It was also proposed that the original theory of the equilibrium condition could be generalised to apply to any arbitrary chemical equilibrium.
3326:
893:
3395:
2919:
62:. Specifically, it implies that for a chemical reaction mixture that is in equilibrium, the ratio between the concentration of reactants and
1105:
1223:
3617:
2824:
A rich system of law of mass action models was developed in mathematical epidemiology by adding components and elementary reactions.
1765:
The term they used for this force was chemical affinity. Today the expression for the equilibrium constant is derived by setting the
348:
3653:
3302:
3472:
3486:
3313:
1530:
3577:
2899:
3461:
3000:↓ and therefore that the final state of a reaction was a state of equilibrium between two opposing processes. From p. 405:
2833:
2684:
1335:
69:
Two aspects are involved in the initial formulation of the law: 1) the equilibrium aspect, concerning the composition of a
2646:
to this quasichemical representation of diffusion. Mass action law for diffusion leads to various nonlinear versions of
3366:
278:
3026:
1510:
3370:
180:
3427:
1772:
The hypothesis that reaction rate is proportional to reactant concentrations is, strictly speaking, only true for
1761:
1493:
could be calculated. The extensive calculations in the 1867 paper gave support to the simplified concept, namely,
1049:
157:
107:
3643:
3337:
3638:
3462:
General
Chemistry Online: FAQ: Acids and bases: What is the pH at the equivalence point an HF/NaOH titration?
3383:
3090:
3068:
2009:
1932:
1926:
3046:
3502:
2953:
2711:
20:
3249:
3214:
3095:
Forhandlinger I Videnskabs-selskabet I Christiania (Transactions of the
Scientific Society in Christiania)
3073:
Forhandlinger I Videnskabs-selskabet I Christiania (Transactions of the
Scientific Society in Christiania)
3051:
Forhandlinger I Videnskabs-selskabet I Christiania (Transactions of the
Scientific Society in Christiania)
1217:
etc. are the active masses at equilibrium. In terms of the initial amounts reagents p,q etc. this becomes
3265:
Guggenheim, E.A. (1956). "Textbook errors IX: More About the Laws of
Reaction Rates and of Equilibrium".
2891:
Mathematical Models of
Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models
2832:
and their simple generalizations like SIS or SEIR, are invalid. For these situations, more sophisticated
2659:
2635:
3648:
3131:
C.M. Guldberg and P. Waage, "Experiments concerning Chemical Affinity"; German translation by Abegg in
122:
2793:
2750:
3341:
3180:
3557:
3146:
1497:
The rate of a reaction is proportional to the product of the active masses of the reagents involved.
3093:[Lecture on the laws of affinity, especially the influence of time on chemical processes].
2837:
138:
90:
3399:
2927:
1958:
2643:
1782:
16:
Empirical law; a chemical reaction rate is proportional to the product of reactant concentrations
2108:
2066:
1746:{\displaystyle K={\frac {{\left}^{\alpha '}{\left}^{\beta '}\dots }{^{\alpha }^{\beta }\dots }}}
2639:
1936:
2314:
2220:
2183:
2150:
429:
3147:"Application of the Law of Chemical Equilibrium (Law of Mass Action) to Biological Problems"
3109:
2889:
2864:
2849:
2280:
2253:
1888:
1852:
1757:
1455:
450:
150:
118:
114:
103:
74:
55:
43:
3291:
3091:"Foredrag om Lovene for Affiniteten, specielt Tidens Indflydelse paa de kemiske Processer"
1885:
the (correct) equilibrium constant formula, and at other times to the (usually incorrect)
1476:
621:
456:
8:
1773:
86:
63:
59:
3569:
3538:
3518:
2854:
1985:
1952:
1766:
1521:
877:{\displaystyle \left({\frac {dx}{dt}}\right)_{\text{reverse}}=k'(p'+x)^{a'}(q'+x)^{b'}}
126:
3483:
1951:. This constant depends on the thermal energy of the system (i.e. the product of the
609:{\displaystyle \alpha (p-\xi )^{a}(q-\xi )^{b}=\alpha '(p'+\xi )^{a'}(q'+\xi )^{b'}\!}
3573:
3022:
2972:
2895:
2869:
2859:
78:
70:
39:
3542:
3565:
3528:
3274:
3226:
3192:
3161:
3014:
2631:
1517:
1513:
3165:
744:{\displaystyle \left({\frac {dx}{dt}}\right)_{\text{forward}}=k(p-x)^{a}(q-x)^{b}}
3490:
2674:
is the number of predator. This is a typical example of the law of mass action.
446:, a and b were regarded as empirical constants, to be determined by experiment.
3183:[The limit plane: a contribution to our knowledge of ester formation].
174:
In their first paper, Guldberg and Waage suggested that in a reaction such as
3196:
1776:(reactions with a single mechanistic step), but the empirical rate expression
3632:
3623:
3230:
2647:
2623:
1841:
1618:
1598:
99:
82:
47:
35:
3533:
3506:
1597:
The exponents α, β etc. are explicitly identified for the first time as the
3501:
2309:
754:
Likewise the reverse reaction of A' with B' proceeded at a rate given by
142:
113:
The law is a statement about equilibrium and gives an expression for the
94:
1026:{\displaystyle (p-x)^{a}(q-x)^{b}={\frac {k'}{k}}(p'+x)^{a'}(q'+x)^{b'}}
146:
3278:
2703:(removed individuals, or just recovered ones if we neglect lethality);
2829:
2627:
27:
3071:[Experiment for the determination of the laws of affinity].
1200:{\displaystyle k_{\text{eq}}_{\text{eq}}=k'_{\text{eq}}_{\text{eq}}}
2003:
3523:
1318:{\displaystyle (p-\xi )(q-\xi )={\frac {k'}{k}}(p'+\xi )(q'+\xi )}
51:
3118:] (in French). Christiania , Norway: Brøgger & Christie.
419:{\displaystyle {\ce {{alcohol}+ acid <=> {ester}+ water}}}
3019:
Affinity and Matter – Elements of Chemical Philosophy 1800–1865
2924:
Chemical reactions, chemical equilibria, and electrochemistry
2006:(the energy separation between conduction and valence bands,
3606:"Guldberg and Waage and the Law of Mass Action", E.W. Lund,
3507:"Quasichemical Models of Multicomponent Nonlinear Diffusion"
3181:"Die Grenzebene, ein Beitrag zur Kenntniss der Esterbildung"
2308:
crystal. Note that the final product is independent of the
1587:{\displaystyle {\text{affinity}}=k^{\alpha }^{\beta }\dots }
1501:
This is an alternative statement of the law of mass action.
641:
3603:. P. Waage and C.M. Guldberg; Henry I. Abrash, Translator.
2638:
and quasichemical interactions of particles and defects.
3185:
Berichte der Deutschen Chemischen Gesellschaft zu Berlin
345:
In this context a substitution reaction was one such as
102:
for the forward and backward reactions must be equal at
3239:, no. 104, Wilhelm Engleman, Leipzig, 1899, pp 126-171
1054:
385:
215:
169:
3135:, no. 104, Wilhelm Engleman, Leipzig, 1899, pp 10-125
2796:
2780:
If the immunity is unstable then the transition from
2753:
2714:
2353:
2317:
2283:
2256:
2223:
2186:
2153:
2111:
2069:
2012:
1988:
1961:
1891:
1855:
1785:
1630:
1533:
1479:
1458:
1442:{\displaystyle v=\psi (k(p-x)(q-x)-k'(p'+x)(q'+x))\!}
1338:
1226:
1108:
1052:
896:
763:
655:
624:
482:
459:
432:
351:
281:
183:
2887:
2788:
should be added that closes the cycle (SIRS model):
89:. Both aspects stem from the research performed by
3069:"Forsøg til Bestemmelse af Lovene for Affiniteten"
2810:
2767:
2738:
2607:
2336:
2296:
2269:
2242:
2205:
2172:
2139:
2097:
2051:
1994:
1974:
1904:
1868:
1829:
1745:
1586:
1485:
1464:
1441:
1317:
1199:
1087:
1025:
876:
743:
630:
608:
465:
438:
418:
331:
256:
2213:densities are equal, their density is called the
1438:
1084:
605:
393:
392:
375:
374:
332:{\displaystyle {\text{affinity}}=\alpha ^{a}^{b}}
223:
222:
205:
204:
3630:
2913:
2911:
2618:
3396:"Chemical equilibrium - The Law of Mass Action"
3178:
2981:] (in French). Paris, France: Firmin Didot.
2677:
1915:
257:{\displaystyle {\ce {A + B <=> A' + B'}}}
42:is directly proportional to the product of the
3495:
2920:"Chemical Equilibria - The Law of Mass Action"
1604:
3555:
3237:Ostwalds Klassiker der Exacten Wissenschaften
3212:
3133:Ostwalds Klassiker der Exacten Wissenschaften
3107:
3044:
2908:
1617:, of the 1879 paper can now be recognised as
125:. It can also be derived with the concept of
3562:Computational Modeling of Infectious Disease
1516:, as had been developed for gas kinetics by
121:. In modern chemistry this is derived using
3618:A simple explanation of the mass action law
3511:Mathematical Modelling of Natural Phenomena
2881:
1920:
1088:{\displaystyle {\mbox{affinity}}=\alpha \!}
3264:
2970:
2894:. Manchester University Press. p. 3.
2653:
3532:
3522:
3359:"Recap of Fundamental Acid-Base Concepts"
2731:
642:Dynamic approach to the equilibrium state
3088:
3021:. Gordon and Breach Science Publishers.
54:. It explains and predicts behaviors of
3505:, H.P. Sargsyan and H.A. Wahab (2011).
2052:{\displaystyle E_{g}\equiv E_{C}-E_{V}}
368:
198:
3631:
3624:The Thermodynamic Equilibrium Constant
3208:
3206:
3144:
3127:
3125:
3013:
2739:{\displaystyle {\ce {S + I -> 2I}}}
2147:bands. When the equilibrium electron
3235:Reprinted, with comments by Abegg in
3084:
3082:
3066:
3062:
3060:
3040:
3038:
3007:
1933:implications in semiconductor physics
2706:The list of elementary reactions is
2685:compartmental models in epidemiology
3384:Chemical Equilibria: Basic Concepts
3203:
3122:
2917:
170:The equilibrium state (composition)
13:
3594:
3570:10.1016/b978-0-32-395389-4.00011-6
3367:Washington University in St. Louis
3213:Guldberg, C.M.; Waage, P. (1879).
3111:Études sur les affinités chimiques
3108:Guldberg, C.M.; Waage, P. (1867).
3079:
3057:
3045:Waage, P.; Guldberg, C.M. (1864).
3035:
473:, this equality is represented by
14:
3665:
3324:
3049:[Studies of affinities].
2811:{\displaystyle {\ce {R -> S}}}
2768:{\displaystyle {\ce {I -> R}}}
3217:[On chemical affinity].
2951:
1931:The law of mass action also has
3549:
3477:
3466:
3455:
3444:
3413:
3388:
3377:
3351:
3318:
3307:
3296:
3285:
3258:
3242:
3215:"Ueber die chemische Affinität"
3172:
3138:
2888:Péter Érdi; János Tóth (1989).
3338:San Francisco State University
3248:(Guldberg & Waage, 1879),
3116:Studies of chemical affinities
3101:
2964:
2945:
2801:
2758:
2725:
2331:
2318:
2250:as this would be the value of
2237:
2224:
2200:
2187:
2167:
2154:
2134:
2131:
2125:
2112:
2092:
2089:
2083:
2070:
1824:
1818:
1815:
1809:
1728:
1721:
1712:
1705:
1572:
1563:
1554:
1545:
1435:
1432:
1415:
1412:
1395:
1381:
1369:
1366:
1354:
1348:
1312:
1295:
1292:
1275:
1254:
1242:
1239:
1227:
1188:
1176:
1167:
1155:
1135:
1128:
1119:
1112:
1081:
1075:
1072:
1066:
1009:
991:
977:
959:
932:
919:
910:
897:
860:
842:
828:
810:
732:
719:
710:
697:
591:
573:
559:
541:
521:
508:
499:
486:
395:
370:
320:
311:
302:
293:
225:
200:
1:
3558:"Simple compartmental models"
3556:von Csefalvay, Chris (2023),
3473:law of mass action definition
3267:Journal of Chemical Education
3219:Journal für praktische Chemie
3166:10.1152/physrev.1938.18.4.495
2875:
2619:Diffusion in condensed matter
1927:Mass action law (electronics)
151:reversible chemical reactions
3654:Jacobus Henricus van 't Hoff
3564:, Elsevier, pp. 19–91,
3145:McLean, Franklin C. (1938).
2699:(infected individuals), and
2678:In mathematical epidemiology
2636:ensemble of elementary jumps
1975:{\displaystyle k_{\text{B}}}
1916:Applications to other fields
1760:, in the 1870s, but was not
117:, a quantity characterizing
108:Jacobus Henricus van 't Hoff
34:is the proposition that the
21:Mass action (disambiguation)
7:
3601:Studies Concerning Affinity
2843:
2695:(susceptible individuals),
2061:effective density of states
1830:{\displaystyle r_{f}=k_{f}}
1605:Modern statement of the law
1599:stoichiometric coefficients
10:
3670:
3221:. 2nd series (in German).
3179:van 't Hoff, J.H. (1877).
3047:"Studier over Affiniteten"
2979:Essay on chemical statics
2974:Essai de statique chimique
2691:The list of components is
2140:{\displaystyle (N_{C}(T))}
2098:{\displaystyle (N_{V}(T))}
1924:
132:
123:equilibrium thermodynamics
18:
3197:10.1002/cber.187701001185
2971:Berthollet, C.L. (1803).
2838:reaction-diffusion models
2683:the transmission term of
1609:The affinity constants, k
3231:10.1002/prac.18790190111
2660:Lotka–Volterra equations
1921:In semiconductor physics
139:Cato Maximilian Guldberg
3334:Geochemistry — GEOL 480
3089:Guldberg, C.M. (1864).
2670:is the number of prey,
2654:In mathematical ecology
2644:absolute reaction rates
2337:{\displaystyle (E_{F})}
2243:{\displaystyle (n_{i})}
2206:{\displaystyle (p_{o})}
2173:{\displaystyle (n_{o})}
1504:
1037:
439:{\displaystyle \alpha }
164:
147:Claude Louis Berthollet
3484:Lab 4 – Slow Manifolds
3314:The Law of Mass Action
2812:
2769:
2740:
2642:applied his theory of
2609:
2338:
2298:
2271:
2244:
2207:
2174:
2141:
2099:
2053:
1996:
1976:
1906:
1870:
1840:is also applicable to
1831:
1747:
1588:
1487:
1466:
1443:
1319:
1201:
1089:
1027:
878:
745:
632:
610:
467:
440:
420:
333:
258:
81:aspect concerning the
3644:Equilibrium chemistry
3534:10.1051/mmnp/20116509
3154:Physiological Reviews
3097:(in Danish): 111–120.
2813:
2770:
2741:
2610:
2339:
2299:
2297:{\displaystyle p_{o}}
2272:
2270:{\displaystyle n_{o}}
2245:
2208:
2175:
2142:
2100:
2054:
1997:
1977:
1907:
1905:{\displaystyle r_{f}}
1871:
1869:{\displaystyle r_{f}}
1832:
1748:
1589:
1488:
1467:
1465:{\displaystyle \psi }
1444:
1320:
1202:
1090:
1028:
879:
746:
633:
611:
468:
441:
421:
334:
259:
3639:History of chemistry
2865:Equilibrium constant
2860:Disequilibrium ratio
2850:Chemical equilibrium
2834:compartmental models
2794:
2751:
2712:
2351:
2315:
2281:
2254:
2221:
2184:
2151:
2109:
2067:
2010:
1986:
1959:
1889:
1853:
1783:
1774:elementary reactions
1758:Josiah Willard Gibbs
1628:
1531:
1486:{\displaystyle \xi }
1477:
1456:
1336:
1224:
1106:
1050:
894:
761:
653:
631:{\displaystyle \xi }
622:
480:
466:{\displaystyle \xi }
457:
430:
349:
279:
181:
119:chemical equilibrium
115:equilibrium constant
104:chemical equilibrium
87:elementary reactions
19:For other uses, see
3075:(in Danish): 92–94.
3053:(in Danish): 35–45.
2958:Defects in Crystals
2604:
1982:, and temperature,
381:
211:
60:dynamic equilibrium
3489:2007-11-17 at the
3292:Law of Mass Action
3067:Waage, P. (1864).
2855:Chemical potential
2808:
2765:
2736:
2605:
2590:
2334:
2294:
2267:
2240:
2203:
2170:
2137:
2095:
2049:
2002:), as well as the
1992:
1972:
1953:Boltzmann constant
1902:
1866:
1827:
1767:chemical potential
1743:
1601:for the reaction.
1584:
1522:Boltzmann equation
1483:
1462:
1439:
1315:
1197:
1085:
1058:
1023:
874:
741:
628:
606:
463:
436:
416:
400:
329:
254:
230:
127:chemical potential
32:law of mass action
3649:Chemical kinetics
3579:978-0-323-95389-4
3327:"Acids and Bases"
3279:10.1021/ed033p544
3015:Levere, Trevor H.
2954:"Mass Action Law"
2930:on 3 October 2018
2901:978-0-7190-2208-1
2870:Reaction quotient
2806:
2800:
2763:
2757:
2734:
2724:
2718:
2583:
2576:
2516:
2509:
2442:
2435:
1995:{\displaystyle T}
1969:
1939:, the product of
1741:
1569:
1551:
1537:
1273:
1194:
1173:
1141:
1125:
1057:
957:
796:
787:
688:
679:
414:
407:
402:
363:
356:
317:
299:
285:
248:
237:
232:
193:
187:
100:rates of reaction
40:chemical reaction
3661:
3589:
3588:
3587:
3586:
3553:
3547:
3546:
3536:
3526:
3499:
3493:
3481:
3475:
3470:
3464:
3459:
3453:
3448:
3442:
3441:
3439:
3438:
3432:
3426:. Archived from
3425:
3417:
3411:
3410:
3408:
3407:
3398:. Archived from
3392:
3386:
3381:
3375:
3374:
3369:. Archived from
3355:
3349:
3348:
3346:
3340:. Archived from
3331:
3322:
3316:
3311:
3305:
3300:
3294:
3289:
3283:
3282:
3262:
3256:
3246:
3240:
3234:
3210:
3201:
3200:
3176:
3170:
3169:
3151:
3142:
3136:
3129:
3120:
3119:
3105:
3099:
3098:
3086:
3077:
3076:
3064:
3055:
3054:
3042:
3033:
3032:
3011:
3005:
2982:
2968:
2962:
2961:
2949:
2943:
2942:
2937:
2935:
2926:. Archived from
2915:
2906:
2905:
2885:
2817:
2815:
2814:
2809:
2807:
2804:
2798:
2774:
2772:
2771:
2766:
2764:
2761:
2755:
2745:
2743:
2742:
2737:
2735:
2732:
2722:
2716:
2632:condensed matter
2614:
2612:
2611:
2606:
2603:
2598:
2586:
2585:
2584:
2582:
2578:
2577:
2574:
2567:
2566:
2557:
2547:
2546:
2537:
2536:
2524:
2520:
2519:
2518:
2517:
2515:
2511:
2510:
2507:
2500:
2499:
2498:
2486:
2485:
2475:
2465:
2464:
2450:
2446:
2445:
2444:
2443:
2441:
2437:
2436:
2433:
2426:
2425:
2424:
2412:
2411:
2401:
2391:
2390:
2373:
2372:
2363:
2362:
2343:
2341:
2340:
2335:
2330:
2329:
2303:
2301:
2300:
2295:
2293:
2292:
2276:
2274:
2273:
2268:
2266:
2265:
2249:
2247:
2246:
2241:
2236:
2235:
2217:carrier density
2212:
2210:
2209:
2204:
2199:
2198:
2179:
2177:
2176:
2171:
2166:
2165:
2146:
2144:
2143:
2138:
2124:
2123:
2104:
2102:
2101:
2096:
2082:
2081:
2058:
2056:
2055:
2050:
2048:
2047:
2035:
2034:
2022:
2021:
2001:
1999:
1998:
1993:
1981:
1979:
1978:
1973:
1971:
1970:
1967:
1935:. Regardless of
1911:
1909:
1908:
1903:
1901:
1900:
1875:
1873:
1872:
1867:
1865:
1864:
1836:
1834:
1833:
1828:
1808:
1807:
1795:
1794:
1752:
1750:
1749:
1744:
1742:
1740:
1736:
1735:
1720:
1719:
1703:
1699:
1698:
1697:
1688:
1687:
1683:
1669:
1668:
1667:
1658:
1657:
1653:
1638:
1593:
1591:
1590:
1585:
1580:
1579:
1570:
1567:
1562:
1561:
1552:
1549:
1538:
1535:
1492:
1490:
1489:
1484:
1471:
1469:
1468:
1463:
1448:
1446:
1445:
1440:
1425:
1405:
1394:
1324:
1322:
1321:
1316:
1305:
1285:
1274:
1269:
1261:
1206:
1204:
1203:
1198:
1196:
1195:
1192:
1186:
1175:
1174:
1171:
1165:
1154:
1143:
1142:
1139:
1127:
1126:
1123:
1094:
1092:
1091:
1086:
1059:
1055:
1032:
1030:
1029:
1024:
1022:
1021:
1020:
1001:
990:
989:
988:
969:
958:
953:
945:
940:
939:
918:
917:
883:
881:
880:
875:
873:
872:
871:
852:
841:
840:
839:
820:
809:
798:
797:
794:
792:
788:
786:
778:
770:
750:
748:
747:
742:
740:
739:
718:
717:
690:
689:
686:
684:
680:
678:
670:
662:
637:
635:
634:
629:
615:
613:
612:
607:
604:
603:
602:
583:
572:
571:
570:
551:
540:
529:
528:
507:
506:
472:
470:
469:
464:
445:
443:
442:
437:
425:
423:
422:
417:
415:
412:
408:
405:
403:
401:
399:
398:
391:
383:
382:
380:
373:
365:
361:
357:
354:
338:
336:
335:
330:
328:
327:
318:
315:
310:
309:
300:
297:
286:
283:
263:
261:
260:
255:
253:
252:
246:
241:
235:
233:
231:
229:
228:
221:
213:
212:
210:
203:
195:
191:
185:
91:Cato M. Guldberg
3669:
3668:
3664:
3663:
3662:
3660:
3659:
3658:
3629:
3628:
3597:
3595:Further reading
3592:
3584:
3582:
3580:
3554:
3550:
3500:
3496:
3491:Wayback Machine
3482:
3478:
3471:
3467:
3460:
3456:
3449:
3445:
3436:
3434:
3430:
3423:
3419:
3418:
3414:
3405:
3403:
3394:
3393:
3389:
3382:
3378:
3357:
3356:
3352:
3344:
3329:
3323:
3319:
3312:
3308:
3301:
3297:
3290:
3286:
3273:(11): 544–545.
3263:
3259:
3247:
3243:
3211:
3204:
3177:
3173:
3149:
3143:
3139:
3130:
3123:
3106:
3102:
3087:
3080:
3065:
3058:
3043:
3036:
3029:
3012:
3008:
2999:
2995:
2991:
2987:
2969:
2965:
2950:
2946:
2933:
2931:
2916:
2909:
2902:
2886:
2882:
2878:
2846:
2840:may be useful.
2836:or distributed
2797:
2795:
2792:
2791:
2754:
2752:
2749:
2748:
2715:
2713:
2710:
2709:
2680:
2656:
2621:
2599:
2594:
2573:
2569:
2568:
2562:
2558:
2556:
2552:
2548:
2542:
2538:
2532:
2528:
2506:
2502:
2501:
2494:
2490:
2481:
2477:
2476:
2474:
2470:
2466:
2460:
2456:
2455:
2451:
2432:
2428:
2427:
2420:
2416:
2407:
2403:
2402:
2400:
2396:
2392:
2386:
2382:
2381:
2377:
2368:
2364:
2358:
2354:
2352:
2349:
2348:
2325:
2321:
2316:
2313:
2312:
2288:
2284:
2282:
2279:
2278:
2261:
2257:
2255:
2252:
2251:
2231:
2227:
2222:
2219:
2218:
2194:
2190:
2185:
2182:
2181:
2161:
2157:
2152:
2149:
2148:
2119:
2115:
2110:
2107:
2106:
2105:and conduction
2077:
2073:
2068:
2065:
2064:
2063:in the valence
2043:
2039:
2030:
2026:
2017:
2013:
2011:
2008:
2007:
1987:
1984:
1983:
1966:
1962:
1960:
1957:
1956:
1929:
1923:
1918:
1896:
1892:
1890:
1887:
1886:
1860:
1856:
1854:
1851:
1850:
1803:
1799:
1790:
1786:
1784:
1781:
1780:
1731:
1727:
1715:
1711:
1704:
1690:
1689:
1676:
1672:
1671:
1670:
1660:
1659:
1646:
1642:
1641:
1640:
1639:
1637:
1629:
1626:
1625:
1616:
1612:
1607:
1575:
1571:
1566:
1557:
1553:
1548:
1534:
1532:
1529:
1528:
1507:
1478:
1475:
1474:
1457:
1454:
1453:
1418:
1398:
1387:
1337:
1334:
1333:
1298:
1278:
1262:
1260:
1225:
1222:
1221:
1216:
1212:
1191:
1187:
1179:
1170:
1166:
1158:
1147:
1138:
1134:
1122:
1118:
1107:
1104:
1103:
1053:
1051:
1048:
1047:
1040:
1013:
1012:
1008:
994:
981:
980:
976:
962:
946:
944:
935:
931:
913:
909:
895:
892:
891:
864:
863:
859:
845:
832:
831:
827:
813:
802:
793:
779:
771:
769:
765:
764:
762:
759:
758:
735:
731:
713:
709:
685:
671:
663:
661:
657:
656:
654:
651:
650:
644:
623:
620:
619:
595:
594:
590:
576:
563:
562:
558:
544:
533:
524:
520:
502:
498:
481:
478:
477:
458:
455:
454:
431:
428:
427:
404:
394:
387:
386:
384:
376:
369:
367:
366:
364:
353:
352:
350:
347:
346:
323:
319:
314:
305:
301:
296:
282:
280:
277:
276:
245:
234:
224:
217:
216:
214:
206:
199:
197:
196:
194:
184:
182:
179:
178:
172:
167:
149:'s ideas about
135:
24:
17:
12:
11:
5:
3667:
3657:
3656:
3651:
3646:
3641:
3627:
3626:
3621:
3620:. H. Motulsky.
3615:
3604:
3596:
3593:
3591:
3590:
3578:
3548:
3517:(5): 184–262.
3494:
3476:
3465:
3454:
3443:
3412:
3387:
3376:
3373:on 2012-02-06.
3350:
3347:on 2006-09-21.
3317:
3306:
3295:
3284:
3257:
3241:
3202:
3171:
3160:(4): 495–523.
3137:
3121:
3100:
3078:
3056:
3034:
3027:
3006:
2997:
2996:→ 2NaCl + CaCO
2993:
2989:
2985:
2963:
2952:Föll, Helmut.
2944:
2918:Chieh, Chung.
2907:
2900:
2879:
2877:
2874:
2873:
2872:
2867:
2862:
2857:
2852:
2845:
2842:
2822:
2821:
2820:
2819:
2803:
2778:
2777:
2776:
2760:
2746:
2730:
2727:
2721:
2704:
2679:
2676:
2655:
2652:
2620:
2617:
2616:
2615:
2602:
2597:
2593:
2589:
2581:
2572:
2565:
2561:
2555:
2551:
2545:
2541:
2535:
2531:
2527:
2523:
2514:
2505:
2497:
2493:
2489:
2484:
2480:
2473:
2469:
2463:
2459:
2454:
2449:
2440:
2431:
2423:
2419:
2415:
2410:
2406:
2399:
2395:
2389:
2385:
2380:
2376:
2371:
2367:
2361:
2357:
2333:
2328:
2324:
2320:
2291:
2287:
2264:
2260:
2239:
2234:
2230:
2226:
2202:
2197:
2193:
2189:
2169:
2164:
2160:
2156:
2136:
2133:
2130:
2127:
2122:
2118:
2114:
2094:
2091:
2088:
2085:
2080:
2076:
2072:
2046:
2042:
2038:
2033:
2029:
2025:
2020:
2016:
1991:
1965:
1949:at equilibrium
1947:is a constant
1945:hole densities
1925:Main article:
1922:
1919:
1917:
1914:
1912:rate formula.
1899:
1895:
1863:
1859:
1838:
1837:
1826:
1823:
1820:
1817:
1814:
1811:
1806:
1802:
1798:
1793:
1789:
1754:
1753:
1739:
1734:
1730:
1726:
1723:
1718:
1714:
1710:
1707:
1702:
1696:
1693:
1686:
1682:
1679:
1675:
1666:
1663:
1656:
1652:
1649:
1645:
1636:
1633:
1619:rate constants
1614:
1610:
1606:
1603:
1595:
1594:
1583:
1578:
1574:
1565:
1560:
1556:
1547:
1544:
1541:
1506:
1503:
1499:
1498:
1482:
1461:
1450:
1449:
1437:
1434:
1431:
1428:
1424:
1421:
1417:
1414:
1411:
1408:
1404:
1401:
1397:
1393:
1390:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1326:
1325:
1314:
1311:
1308:
1304:
1301:
1297:
1294:
1291:
1288:
1284:
1281:
1277:
1272:
1268:
1265:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1214:
1210:
1208:
1207:
1190:
1185:
1182:
1178:
1169:
1164:
1161:
1157:
1153:
1150:
1146:
1137:
1133:
1130:
1121:
1117:
1114:
1111:
1096:
1095:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1039:
1036:
1035:
1034:
1019:
1016:
1011:
1007:
1004:
1000:
997:
993:
987:
984:
979:
975:
972:
968:
965:
961:
956:
952:
949:
943:
938:
934:
930:
927:
924:
921:
916:
912:
908:
905:
902:
899:
885:
884:
870:
867:
862:
858:
855:
851:
848:
844:
838:
835:
830:
826:
823:
819:
816:
812:
808:
805:
801:
791:
785:
782:
777:
774:
768:
752:
751:
738:
734:
730:
727:
724:
721:
716:
712:
708:
705:
702:
699:
696:
693:
683:
677:
674:
669:
666:
660:
643:
640:
627:
617:
616:
601:
598:
593:
589:
586:
582:
579:
575:
569:
566:
561:
557:
554:
550:
547:
543:
539:
536:
532:
527:
523:
519:
516:
513:
510:
505:
501:
497:
494:
491:
488:
485:
462:
435:
411:
397:
390:
379:
372:
360:
343:
342:
341:
340:
326:
322:
313:
308:
304:
295:
292:
289:
265:
264:
251:
244:
240:
227:
220:
209:
202:
190:
171:
168:
166:
163:
145:, building on
134:
131:
83:rate equations
48:concentrations
15:
9:
6:
4:
3:
2:
3666:
3655:
3652:
3650:
3647:
3645:
3642:
3640:
3637:
3636:
3634:
3625:
3622:
3619:
3616:
3613:
3609:
3605:
3602:
3599:
3598:
3581:
3575:
3571:
3567:
3563:
3559:
3552:
3544:
3540:
3535:
3530:
3525:
3520:
3516:
3512:
3508:
3504:
3498:
3492:
3488:
3485:
3480:
3474:
3469:
3463:
3458:
3452:
3447:
3433:on 2007-06-15
3429:
3422:
3421:"Indiana.edu"
3416:
3402:on 2010-02-01
3401:
3397:
3391:
3385:
3380:
3372:
3368:
3364:
3363:Chemistry 152
3360:
3354:
3343:
3339:
3335:
3328:
3325:Leech, Mary.
3321:
3315:
3310:
3304:
3299:
3293:
3288:
3280:
3276:
3272:
3268:
3261:
3254:
3251:
3245:
3238:
3232:
3228:
3224:
3220:
3216:
3209:
3207:
3198:
3194:
3190:
3187:(in German).
3186:
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3163:
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3030:
3028:2-88124-583-8
3024:
3020:
3016:
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2629:
2625:
2624:Yakov Frenkel
2600:
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2062:
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2040:
2036:
2031:
2027:
2023:
2018:
2014:
2005:
1989:
1963:
1954:
1950:
1946:
1942:
1938:
1934:
1928:
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1005:
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824:
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772:
766:
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736:
728:
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714:
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694:
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681:
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664:
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447:
433:
409:
388:
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358:
324:
306:
290:
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275:
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249:
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188:
177:
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124:
120:
116:
111:
109:
105:
101:
96:
92:
88:
84:
80:
76:
72:
67:
66:is constant.
65:
61:
57:
53:
49:
45:
41:
37:
33:
29:
22:
3611:
3607:
3583:, retrieved
3561:
3551:
3514:
3510:
3497:
3479:
3468:
3457:
3451:Berkeley.edu
3446:
3435:. Retrieved
3428:the original
3415:
3404:. Retrieved
3400:the original
3390:
3379:
3371:the original
3362:
3353:
3342:the original
3333:
3320:
3309:
3298:
3287:
3270:
3266:
3260:
3252:
3244:
3236:
3222:
3218:
3188:
3184:
3174:
3157:
3153:
3140:
3132:
3115:
3110:
3103:
3094:
3072:
3050:
3018:
3009:
3001:
2978:
2973:
2966:
2957:
2947:
2939:
2932:. Retrieved
2928:the original
2923:
2890:
2883:
2826:
2823:
2785:
2781:
2700:
2696:
2692:
2681:
2671:
2667:
2663:
2657:
2640:Henry Eyring
2626:represented
2622:
2305:
2214:
1948:
1930:
1883:
1879:
1847:
1842:second order
1839:
1771:
1762:widely known
1755:
1608:
1596:
1508:
1500:
1451:
1327:
1209:
1097:
1041:
886:
753:
645:
618:
448:
344:
266:
173:
155:
136:
112:
68:
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25:
3610:., (1965),
3608:J. Chem. Ed
3503:A.N. Gorban
3191:: 669–678.
2630:process in
2310:Fermi level
1511:independent
451:equilibrium
158:van 't Hoff
143:Peter Waage
95:Peter Waage
77:and 2) the
75:equilibrium
73:mixture at
3633:Categories
3614:, 548-550.
3585:2023-03-02
3437:2007-07-12
3406:2007-07-12
3225:: 69–114.
2876:References
2648:Fick's law
1514:collisions
44:activities
3524:1012.2908
2830:SIR model
2802:⟶
2759:⟶
2726:⟶
2628:diffusion
2554:−
2488:−
2472:−
2414:−
2398:−
2215:intrinsic
2180:and hole
2037:−
2024:≡
1738:…
1733:β
1717:α
1701:…
1692:β
1662:α
1582:…
1577:β
1559:α
1520:in 1872 (
1518:Boltzmann
1481:ξ
1473:quantity
1460:ψ
1385:−
1376:−
1361:−
1346:ψ
1310:ξ
1290:ξ
1252:ξ
1249:−
1237:ξ
1234:−
1064:α
926:−
904:−
726:−
704:−
626:ξ
588:ξ
556:ξ
535:α
518:ξ
515:−
496:ξ
493:−
484:α
461:ξ
434:α
396:⇀
389:−
378:−
371:↽
291:α
226:⇀
219:−
208:−
201:↽
56:solutions
52:reactants
28:chemistry
3543:18961678
3487:Archived
3017:(1971).
2844:See also
2666:, where
2004:band gap
1941:electron
1695:′
1681:′
1665:′
1651:′
1536:affinity
1423:′
1403:′
1392:′
1303:′
1283:′
1267:′
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1163:′
1152:′
1056:affinity
1018:′
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818:′
807:′
600:′
581:′
568:′
549:′
538:′
284:affinity
250:′
239:′
156:In 1877
71:reaction
64:products
2934:21 July
2306:perfect
795:reverse
687:forward
355:alcohol
133:History
79:kinetic
50:of the
3576:
3541:
3303:SC.edu
3250:p. 71:
3025:
2992:+ CaCl
2898:
2634:as an
2059:) and
1937:doping
1452:where
30:, the
3539:S2CID
3519:arXiv
3431:(PDF)
3424:(PDF)
3345:(PDF)
3330:(PDF)
3150:(PDF)
3114:[
2977:[
2304:in a
1613:and k
413:water
406:ester
38:of a
3574:ISBN
3023:ISBN
2936:2019
2896:ISBN
2658:The
2277:and
1943:and
1505:1879
1038:1867
362:acid
165:1864
141:and
93:and
85:for
36:rate
3566:doi
3529:doi
3275:doi
3227:doi
3193:doi
3162:doi
2784:to
1033:...
449:At
58:in
46:or
26:In
3635::
3612:42
3572:,
3560:,
3537:.
3527:.
3513:.
3509:.
3365:.
3361:.
3336:.
3332:.
3271:33
3269:.
3223:19
3205:^
3189:10
3158:18
3156:.
3152:.
3124:^
3081:^
3059:^
3037:^
2988:CO
2984:Na
2956:.
2938:.
2922:.
2910:^
2664:xy
2650:.
2344::
1955:,
1215:eq
1213:,
1211:eq
1193:eq
1172:eq
1140:eq
1124:eq
129:.
110:.
3568::
3545:.
3531::
3521::
3515:6
3440:.
3409:.
3281:.
3277::
3233:.
3229::
3199:.
3195::
3168:.
3164::
3031:.
2998:3
2994:2
2990:3
2986:2
2960:.
2904:.
2818:.
2805:S
2799:R
2786:S
2782:R
2775:.
2762:R
2756:I
2733:I
2729:2
2723:I
2720:+
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2701:R
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2672:y
2668:x
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2596:i
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2588:=
2580:T
2575:B
2571:k
2564:g
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2522:)
2513:T
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2448:)
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2430:k
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2084:(
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2071:(
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2019:g
2015:E
1990:T
1968:B
1964:k
1898:f
1894:r
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1858:r
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1822:B
1819:[
1816:]
1813:A
1810:[
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1725:B
1722:[
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1709:A
1706:[
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1674:[
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1644:[
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1156:[
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312:[
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298:A
294:[
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247:B
243:+
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192:B
189:+
186:A
23:.
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