CHEMICAL REACTIONS O F T H E THIRD ORDER* BY FRANK E. E. GERMANS
The general form of stoichiometric equations for reactions of the third order may be represented by A B C j R i Rz RB (1) A special case of a third order reaction may be represented by 3A e R i Rz RS (2) which states that three molecules of A react in such a way as to give the products on the right side of the equilibrium equation. According to the principle of the mass action law, we may write in the first case d(a - x) d(b - x) d(c - X) - ____ = k(a - x) (b - x) (c - x) (3) dt dt dt or dx,idt = k(a - x) (b - x) (c - x) (4) in which a, b and c represent the initial concentration in mols per liter of A, B and C respectively present in the solution a t zero time. z therefore represents the mols per liter of each compound, A, B and C, which hnve disappeared at time t. In the same way, we may write for the second case d(a - x) - dx (ka - x ) ~ dt dt This is also the form assumed in the first case, if the initial concentrations of A, B and C are equal. If, however, a f b = c, that is, if the initial concentrations of B and C are identical, but different from A, we have dx,/dt = k(a - x) (b - x ) ~ (6) Up to this point the discussion is equally valid whether we speak in terms of mols or equivalents per liter. Suppose now we have a third order reaction of the form A 2 B e R i Rz (7) If we are using equivalents per liter as the measure of concentrations, then we have dx/dt = k(a - x) (b - x)' (8) which is identical to (6). If, however, we are expressing our concentration in terms of mols per liter, it is obvious from the equation that two mols of B disappear for every mol of A that disappears. We must, therefore, now write dx/dt = k(a - x / 2 ) (b - x ) ~ (9)
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* Contribution from the Department of Chemistry of the University of Colorado by Frank E. E. Germann.
CHEMICAL REACTIONS O F THE THIRD ORDER
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A particular case of an actual reaction which was first carefully studied by Noyesl is represented by the equilibrium SnClz 2FeC13 zFeC1’ SnC14 (10) Since Xoyes expressed concentrations in terms of equivalents] he made use of equation (8), which on integration yielded 1 (a - b)x + a(b - x) k = - . -I (11) b(a - x) t (a - b)2 b(b - x) The tendency a t the present time is to make use of mols per liter in expressing all concentrations, since there is less danger of confusion. As a result, many errors have crept into treatises on chemistry] and as a necessary consequence, into the research work of the unsuspecting. Thus we find2 in Getman’s “Outlines of Theoretical Chemistry” the statement] “The following table gives the results obtained with 0.025 molar solutions of ferric chloride and stannous chloride.” As a matter of fact the table is a copy of Table 6 of Noyes taken from page 551 of the article referred to above, except for the fact that Noyes expressly labels his concentrations as “ 0 . 0 2 5 normal” in place of “0.025 molar” as given. I n connection with the same reaction,$ Taylol-l states specifically that concentrations are expressed in mols per liter and goes on to state (referring to the equation of the form of equation ( I ) ) “If the initial concentrations of A and B are equal, or if the stoichiometric equation has the form 2A C = one or more resultants] we have
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i ~
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dx/dt = k (a - x ) (c ~ - x) whence k = - . I-
1
t (c - a)’
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(c-a)x a(a - x) ~
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This is correct for the case of a = b, but not for the case of 2A C = RI Obviously equation (12) in the case of concentrations expressed in mols per liter, should be replaced by dx/dt = k (a - x)’(c - x/2) which is of the form of equation (9), and (13) should be replaced by the integrated form of equation (14). Although equations of the forms of (9) and (14) may have been used and integrated in the past, they have successfully escaped the attention of the writer. It may, therefore, be of interest t o develop this point more fully. For integration, equation (8) may be writkn dx kJdt=J (a - x) (b - x)’ Similarly, equation (9) may be written
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Soyes: Z. physik Chem., 16, 546 (1895). Getman: 4th Edition, p. 433 (1927). Taylor: “A Treatise on Physical Chemistry,” 2 , 872 Chapter written by Francis Owen Rice.
FRAKK E. E. GERMANK
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dx dx (16) (za - x) (b - x)' (a - x/2) (b - x ) ~ I t is, therefore. obvious that if equation ( I I ) gives the value of k as derived from equation (8),we have only to transform ( I 1) by introducing the factor z in the numerator and replace a by 2a to obtain a value of k from equation (9). Thus we have (za - b) x za(b - x) k = 2 .I In t ( 2 8 - b)' ( b(b - X) b(za - x) Similarly the equation a t the bottom of page 8 7 2 in Taylor should read kJdt
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In all other cases of ist, 2nd and 3rd order reactions, the equations developed for k are equally valid whether niols or equivalents are used. The numerical value of k will naturally depend on the units chosen. K h e n Forking with orders higher than the third, the equations will, in general, be different for concentrations expressed in equivalents or in niols. S o attempt has been made in the above to distinguish between different values of k by means of different symbols. Under the date of February 14,1928, Dr. F. 0. Rice called the attention of the author t o an error of the above type in Bodenstein's article. (Z. physik. Chem., 29, 6 7 7 (1899) ) in which the reaction 2 H,O 2 H, f 0 2 was under consideration. This eiror n'as pointed out and critically examined by Wegscheider: Z. physik. C'heni., 3 5 , 580 (1900); but no formula for k was given which can be readily compared with equations (1 7 ) or ( I 8).
