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Statistical Factors in Complexation Reactions Chung-Sun Chung
National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China
Statistical factors have been used extensively in correlating equilibrium constants (1, 2) and rate constants ( 3 , 4 ) . In comparisons of equilibrium constants with each other or of rateionstants amone a set of rate constants. it is the intrinsic or chemical constank which should be compared and not the observed constants 1.5 .1. These chemtcal constants.. Krhamlor kchem),are related to the observed constant K (or k ) by the relation: Kchem= KIK, (or kch, = klK!) where K , and K! are statistical factors (5). The statistical factor should be evaluated as the sum of symmetry and isomer effects 16):however, scientists often use the statistical factors obtained hv the Rninstrd method ( I , 2),which are not general ( 6 ) . he complexation reaction MA.BN-,
Statistical Factor
Reaction MB,+AeMABg+B MAE3 A MA& MA& A s MA.8 M A s B + A e MA,+B
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+B
+B
Statisticd Factor
Reaction
k-1
Case 1 In this case, all of the (N - dB's in the reagent, MA,,BN-,, and all of the ( n 1)A's in the product, MAn+lB~-n-l,are structurally or energetically equivalent. Ligands are structurally or energetically equivalent if they can be interchanged bv a svmmetrv operation of the molecule or have the same piohability of &ing changed by another ligand. The suiii&xl factors for the forward and the reverse rate constants, Kl. and KT,,, are (N - ~ ) ( u A / u T )and ( n 1) ( U B ~ U T )respect&ely, , where a A , UB, and UT are the symmetry and isomer factors for A, B, and the activated complex of this reaction, respectively ( 6 ) .Thus, the statistical factor for the equilibrium constant, which is Ki,IK!l,, is equal to [(N - n)l(n I ) ] ( ~ A / u B ) .
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Case 2 In this case, the B's in the reagent are not structurally or energetically equivalent, while all of the A's in the product are structurally or energetically equivalent. The statistical factor for the forward rate constant, Kf,, is smaller than ( N - n ) ( u A / ( I T ) ; the statistical factor forthe reverse rate constant, KI,,, is equal to ( n + ~ ) ( ~ B / N SoTthe ) . statistical factor for theequilihrium constant, Kt,, issmaller than I(N - n ) / h +
Case 3 In this case, the A's in the product are not structurally or energetically equivalent, while all of the B's in the reagent are structurally or energetically equivalent. The statistical factor for the forward rate constant, Kt,,,, is equal to ( N - n ) ( a ~ l a=);the statistical factor for the reverse rate constant, KT1,, is smaller than ( n l ) ( u g l u ~Thus ). the statistical factor for the equilibrium constant, Kl,, is larger than [(N - n)l(n I)] '(LYA/~B).
4 (adus) 2 / 5 (a,/u.) 116 (adus)
'l7-e ssIwcal fa& d e p d on me rehliueamxlntr of me isomerscd born he reagsnt. MAnB,+n.andlh3 product. MAm,B,+,,.
Case 4 In this case, the A's in the product and the B's in the reagent are not structurally or energetically equivalent. The statistical factors for the forward and the reverse rate constants, Ki, and KL1,, are smaller than ( N - ~ ) ( u A / u T and ) ( n l)(u$aT), respectively. In this case, both the reagent, MA,BN-,, and the product MA,,+IBN-,,-~ have isomers, and the statistical factor for the equilibrium constant, Kl,, depends upon the relative amounts of these isomers ( 6 ) .Therefore, the term obtained by the Br6nsted method, [(N- n)l(n+ I ) ] ( ( Y A ~ U B ) , is meaningless in this case. Some examples are given in Tables 1-3. The statistical factors listed in these tables are the same as those derived from symmetry and isomer concepts ( 6 ) .
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Literature Cited
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(61 Chung, C-S.,Inorg.Chem.. 18,1318(19791.
Volume 62 Number 2
February 1985
107
