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An empirical method for the estimation of bond dissociation energies
Chemical Engineering Science, 1965, Vol. 20, pp. 529-532. Pergamon Press Ltd., Oxford. Printed in Great Britain.
An empirical method for the estimation of bond dissociation energies A. VAN TIGGELEN, J. PEETERS and R. BURKE University of Louvain, Belgium
Abstract--An empirical formula for the calculation of bond dissociation energies D(A--B) is suggested: D(A--B) ---- (1/2c0 [uAD(A--A) + uBD(B---B)] where uA and czB are constants characteristic of radicals A- and B. while u is eitherc,A orc~B,whichever has the smaller value. Values of c, for thirty-four organic and inorganic radicals have been determined and tabulated. From values of D(A--B), heats of reaction Q can be calculated. Activation energies E for substitution reactions can then be evaluated with the help of Polanyi's relation: E = (const.)x -- (const.)2. Q. INTRODUCTION
PAULING [1] has proposed a relation for the strength D' of a bond A---B:
F o r m u l a (2) is not restricted to A and B b e i n g a t o m s but it is valid for many types of radicals. It is convenient to introduce the symbols
D ' ( A - - B ) = ½ D ' ( A - - A ) + ½D'(B---B) + A' (1) where A', the extra-ionic resonance energy, depends on the electronegativities of A and B. E q u a t i o n (1) is very useful and of great theoretical importance. Unfortunately, a calculation of the dissociation energy D ( A - - B ) is not possible by m e a n s of (1) if A or B or both are not a t o m s but radicals. Indeed, in PAULING'S e q u a t i o n (1), the s y m b o l D' stands for the bond strength, e.g. D ' ( C - - H ) in methane, while often it is the bond dissociation energy, e.g. D ( C H 3 - - H), that is required. A l t h o u g h a theoretical expression for the bond dissociation energy is very desirable, wave-mechanical treatments have been helpful so far only in very simple cases. For other situations, a p u r e l y empirical approach suggests itself. An empirical relation that seems to yield excellent results is the following: D ( A - - B ) = (1/2~)[ctAD(A--A) + ~eD(B--B)] (2)
DA = ½ D ( A - - A )
(3)
O s = ½D(B--B)
(4)
Then equation (2) can be rewritten as D ( A - - B ) = (1/~XA)(0eADA + ceBDB)
if 0eA < a s (5)
D ( A - - B ) = (I/~e)(aAOA + ~BDs)
if 0es < 0eA (6)
Equations (5) and (6) can also be written in a form similar to PAULING'S e q u a t i o n (1): D ( A - - B ) = DA + DB + A
(7)
The meaning of A is obscure: A = (OBfiXA)(0~B -- 0CA)
if 0~A < $~B
(8)
A = (DA/OeB)(~A -- C~S) if ~B < ~A
(9)
Although equation (7) is of the same form as PAULING'S relation, its theoretical interpretation is not apparent. But, as will be seen shortly, it provides a satisfactory way of estimating bond dissociation energies.
where D ( A - - B ) , D ( A - - A ) and D ( B - - B ) are the dissociations energies of A B, A - - A and B--B respectively; ~A and ~s are constants characterizing radicals A and B, and ~ stands for the smaller of the latter two constants.
529
DETERMINATION OF EMPIRICAL CONSTANTS
The value of u for a hydrogen a t o m , u~i, is assumed arbitrarily to be equal to unity. This is
A. VAN TIGGELEN, J. PEETERS and R. BURKE
permissible since, as can be seen from equations (8) and (9), only the ratio of a-values enters into the calculation. The a-value ~R of any radical R is then calculated in the following way. Since, as will be verified, ~R > a . = 1
(10)
the bond dissociation energy of R - - H can be written following equation (5) as D ( R - - H ) = DH + aRDR
(11)
Now, the quantity D~ is well known. Further, experimental values for D ( R - - H ) and DR are frequently available [2-4]. Then ~R is the only unknown in equation (11) and its value can be obtained readily. If experimental values for D(R---H) or DR are not available, aR can still be obtained from known values of D ( R - - R ' ) and D(R--R"), provided that ~tR' and ~R,' have already been determined. Some values of ~R and 0tRDR calculated in this way are collected in Table 1. As can be seen, all values of ~ are indeed larger than or equal to unity as was assumed in relation (10) used in the derivation of equation (11).
DIscusSION
Almost identical a-values are exhibited by radicals which have similar chemical properties, i.e. with the same atom carrying the free valence and similar atoms b o u n d to it. Examples are: R--CO
(CHO, CHACO, C6H5C0): = 1"00
R
CH2
(CHa, C2H5, n-CaHT, n-C4Hg): ct = 1"18-1"20
H2CX
(CH2CI, CH2Br):
~ = 1"22-1.23
X2(~H
(CHC12, CHBr2):
~ = 1.25
CX a
(CF3, CCla, CBr3):
ce = 1-24-1.29
I
Br
CI
F
1"20 2"5
1"48 2"8
1"75 3.0
4"57 4"0
From the a-values listed in Table 1, various dissociation energies have been calculated by means of equations (5) and (6). The results are compared in Table 2 with experimental data. Differences between calculated and experimental values do not exceed 2 or 3 kcal/gmole. This is within the limits of accuracy of most experimental results. The largest discrepancies occur for bonds between organic radicals and iodine. It is found experimentally that D ( R - - I ) is smaller than the sum DI + DR. Since for most organic radicals % ~- 1.2 and also cq = 1.2, it would be expected from formula (5) or (6) that D ( R - - I ) ~- D~ + DR. It is interesting to note that, similarly, according to PAULING'S relation (1), the strength of the C--I bond is equal to ½ [ D ' ( C - - C ) + D'(I--I)] since r c = r~ and therefore A = 0. An analogous case is that of bonds between radicals R and R' when D ( R - - R ) ,~ D ( R ' - - R ' ) . Then, as the difference between D ( R - - R ) and D ( R ' - - R ' ) becomes larger, the discrepancy between calculated and experimental values of D ( R - - R ' ) becomes also worse. An example is when R is .CN and R' is CH2 = CH--'~H2. Sometimes, departure between calculated and experimental values can be very large, e.g. F--C1, F - - B r , F - - I . Note however that good agreement is found for F - - R where R is an organic radical and also for Br--Cl. Another bad case is allylamine CH 2 = C H - - C H 2 - - N H 2 : D(C--Neale ' = 56 kcal/grnole whereas D(C--N)exvtt. = 64 kcal/ gmole. However, for other R - - N H 2 bonds, only small deviations are found and other bonds with the allyl radical behave normally.
APPLICATIONS
This does not mean that elements belonging to the same group of the periodic table have identical a-values. On the contrary, ~t increases with increasing electronegativity r :
With bond dissociation energies, it becomes possible to estimate heats of reaction. Thus for a substitution reaction of the type:
530
AB + C.-~ A. + BC
An empirical method for the estimation of bond dissociation energies
Table 1. Values of the constant ctR and of the product ~RDRfor various radicals R Radical R H CHO CHACO CeHsCO CN t-C4I-I9 CH------C. i-CaH7 SH Cell5 C2H5 CH2=CH CH3 n-Call7 n-C4I-I9 C6H5 -- CH2 I
0OR (dimensionless)
~RDR (kcal/gmolc)
Radical R
0OR (dimensionless)
0gRDR (kcal/gmolc)
1.00 1.00 1.00 1-00 1"05 1"05 1"06 1"10 1.15 1-18 1-18 1.19 1"20 1"20 1 '20 1 '20 1"20
51"6 26.5 31-5 23.0 60.0 31.5 69"4 36'5 39"1 60-0 46"0 52"4 49"8 43"3 44-4 28-2 21 "4
CHaCI CH2Br CBr3 CI-IBr2 CHCI~ CCla CF3 CH2 = CH---CH2 Br NO , NHa CI NO2 OH CHaO C2H50 F
1.22 1"23 1"24 1"25 1"25 1"29 1-29 1-32 1"48 1"60 1-70 1.75 2"00 2.45 2"95 3-33 4"57
45.8 43 "4 28.4 36"4 41 '6 37"4 51-4 24.4 33"7 7"7 49"5 50-0 16"0 62"5 54"5 52"5 82"4
Table 2. Comparison between calculated and experimental values of bond dissociation energy ( k c a l / g m o l ) R
-- R t
CHa--F CHa--OH CaHs--NHz C~H5--CI tert. C4Hg---CHa n-CaH~----SH n-C4Ha--CHa iso-CaHT--Br CaHsCH2---OH CsHsCH~-CH8 CHa = CH---CI CH2 = CHCH~---Br CeH5--OH CeHs--NH2 CHa----NO2 CHaO--NO C2HsO--NOa CHO--CHa CHO--C~H5 CHaCO--Br CHaCO--C1 CHaCO---CaH~ CaHsCO--Br CaHsCO---CI NO---C1 NO--Br CFa--C1
Dcalc.
Dexptl.
110 93.5 80"9 81"3 77-4 71"5 75-2 63"8 75"6 65-0 86"0 44.0 104.0 92"8 54"8 38"8 34'2 76"3 72"5 65"2 81-5 77-5 56.7 73.0 36.0 28-0 78-6
107 90-5 78 80 74 72"1 77 63 73 63 86 47"5 107 94 57 36"4 36"4 75 71 67 82 77 57 73"6 37 28 80
531
Dcalc. -- DexptL
+3 +3 +2"9 + 1"3 +3'4 --0.6 -- 1'8 +0"8 +2-6 +2 0 -- 3"5 --3 -- 1-2 --2.2 +2-4 --2.2 + 1"3 + 1.5 -- 1"8 --0'5 +0'5 --0'3 --0"6 --1 0 -- 1"4
A. VAN TIC,OEI.EN, J. t~TERS and R. BURKE
From T a b l e 1 and formula (5) we have
the heat of r e a c t i o n Q counted as positive for an exothermie r e a c t i o n is given by
D ( C H a - - H ) = 51.6 + 49.8 = 101.4
a = D(B--C) - D(A--B)
D ( C 2 H s - - H ) = 51.6 + 46.0 =
With h e a t s of reaction, it b e c o m e s possible in turn to estimate energies of activation E by m e a n s of POLA~I'S relation:
97.6
The heat of the r e a c t i o n Q is Q = 101-4 - 94.9 = 3.8
E = (const)l -- (const)zQ
Therefore
w h i c h can be a p p l i e d to a variety of exothermic free r a d i c a l reactions [3]
E = 11.5 - 0 . 2 5 x 3.8 = 10.6 k c a l / g m o l e The experimental value of E is 10.4 [4].
with (const)l = 11.5 kcal/gmole (const)z = 0.25 F o r example, let us estimate the energy of activation of the r e a c t i o n ' C H 3 q- C 2 H 6 --~ CH4 + C 2 H 5"
AcknowledgemEnt--This work has been supported in part by the Office of Scientific Research of the United States Air Force (Grant AF-EOAR 42-63) through its European Office of Aerospace Research and by the Fends de la Recherche Scientifique Fondamentale Collective of the Belgian Government.
REFERENCES [1] [2] [3] [4]
PAtJLr~G L., The Nature of the Chemical Bond. Cornell University Press, New York 1960. COTrgELL T. L., The Strength of Chemical Bonds. Butterworths, L o n d o n 1958. SE~tqOV N. N., Some Problems of Chemical Kinetics and Reactivity (Translated by BOLrDAgT M.). Princeton University Press 1959. S~AClE E. W. R., Atomic and Free Radical Reactions. Reinhold, New York 1954.
R~sum~---L'auteur donne une formule empirique pour le calcul des 6nergies de dissociation des liaisons D(A -- B)
off ~ et ~n sont des c o m t a n t e s caract6ristiques des radicaux A° et B° tandis qu'~ est soit ~a, soit ~B,
quelle que soit leur plus petite valeur. Les valeurs de e pour 34 radicaux organiques et min&aux ont 6t6 d6tel'min6es et raises sous forme de tableaux. A partir des valeurs de D ( A - - B ) les chaleurs de r6action Q peuvent ~tre calcul6es. On peut ~valuer les 6nergies d'activation E pour les r~actions de substitution en utilisant la relation de POLAN~ E = (const.h -- (const.)2. Q.
532
An empirical method for the estimation of bond dissociation energies A. VAN TIGGELEN, J. PEETERS and R. BURKE University of Louvain, Belgium
Abstract--An empirical formula for the calculation of bond dissociation energies D(A--B) is suggested: D(A--B) ---- (1/2c0 [uAD(A--A) + uBD(B---B)] where uA and czB are constants characteristic of radicals A- and B. while u is eitherc,A orc~B,whichever has the smaller value. Values of c, for thirty-four organic and inorganic radicals have been determined and tabulated. From values of D(A--B), heats of reaction Q can be calculated. Activation energies E for substitution reactions can then be evaluated with the help of Polanyi's relation: E = (const.)x -- (const.)2. Q. INTRODUCTION
PAULING [1] has proposed a relation for the strength D' of a bond A---B:
F o r m u l a (2) is not restricted to A and B b e i n g a t o m s but it is valid for many types of radicals. It is convenient to introduce the symbols
D ' ( A - - B ) = ½ D ' ( A - - A ) + ½D'(B---B) + A' (1) where A', the extra-ionic resonance energy, depends on the electronegativities of A and B. E q u a t i o n (1) is very useful and of great theoretical importance. Unfortunately, a calculation of the dissociation energy D ( A - - B ) is not possible by m e a n s of (1) if A or B or both are not a t o m s but radicals. Indeed, in PAULING'S e q u a t i o n (1), the s y m b o l D' stands for the bond strength, e.g. D ' ( C - - H ) in methane, while often it is the bond dissociation energy, e.g. D ( C H 3 - - H), that is required. A l t h o u g h a theoretical expression for the bond dissociation energy is very desirable, wave-mechanical treatments have been helpful so far only in very simple cases. For other situations, a p u r e l y empirical approach suggests itself. An empirical relation that seems to yield excellent results is the following: D ( A - - B ) = (1/2~)[ctAD(A--A) + ~eD(B--B)] (2)
DA = ½ D ( A - - A )
(3)
O s = ½D(B--B)
(4)
Then equation (2) can be rewritten as D ( A - - B ) = (1/~XA)(0eADA + ceBDB)
if 0eA < a s (5)
D ( A - - B ) = (I/~e)(aAOA + ~BDs)
if 0es < 0eA (6)
Equations (5) and (6) can also be written in a form similar to PAULING'S e q u a t i o n (1): D ( A - - B ) = DA + DB + A
(7)
The meaning of A is obscure: A = (OBfiXA)(0~B -- 0CA)
if 0~A < $~B
(8)
A = (DA/OeB)(~A -- C~S) if ~B < ~A
(9)
Although equation (7) is of the same form as PAULING'S relation, its theoretical interpretation is not apparent. But, as will be seen shortly, it provides a satisfactory way of estimating bond dissociation energies.
where D ( A - - B ) , D ( A - - A ) and D ( B - - B ) are the dissociations energies of A B, A - - A and B--B respectively; ~A and ~s are constants characterizing radicals A and B, and ~ stands for the smaller of the latter two constants.
529
DETERMINATION OF EMPIRICAL CONSTANTS
The value of u for a hydrogen a t o m , u~i, is assumed arbitrarily to be equal to unity. This is
A. VAN TIGGELEN, J. PEETERS and R. BURKE
permissible since, as can be seen from equations (8) and (9), only the ratio of a-values enters into the calculation. The a-value ~R of any radical R is then calculated in the following way. Since, as will be verified, ~R > a . = 1
(10)
the bond dissociation energy of R - - H can be written following equation (5) as D ( R - - H ) = DH + aRDR
(11)
Now, the quantity D~ is well known. Further, experimental values for D ( R - - H ) and DR are frequently available [2-4]. Then ~R is the only unknown in equation (11) and its value can be obtained readily. If experimental values for D(R---H) or DR are not available, aR can still be obtained from known values of D ( R - - R ' ) and D(R--R"), provided that ~tR' and ~R,' have already been determined. Some values of ~R and 0tRDR calculated in this way are collected in Table 1. As can be seen, all values of ~ are indeed larger than or equal to unity as was assumed in relation (10) used in the derivation of equation (11).
DIscusSION
Almost identical a-values are exhibited by radicals which have similar chemical properties, i.e. with the same atom carrying the free valence and similar atoms b o u n d to it. Examples are: R--CO
(CHO, CHACO, C6H5C0): = 1"00
R
CH2
(CHa, C2H5, n-CaHT, n-C4Hg): ct = 1"18-1"20
H2CX
(CH2CI, CH2Br):
~ = 1"22-1.23
X2(~H
(CHC12, CHBr2):
~ = 1.25
CX a
(CF3, CCla, CBr3):
ce = 1-24-1.29
I
Br
CI
F
1"20 2"5
1"48 2"8
1"75 3.0
4"57 4"0
From the a-values listed in Table 1, various dissociation energies have been calculated by means of equations (5) and (6). The results are compared in Table 2 with experimental data. Differences between calculated and experimental values do not exceed 2 or 3 kcal/gmole. This is within the limits of accuracy of most experimental results. The largest discrepancies occur for bonds between organic radicals and iodine. It is found experimentally that D ( R - - I ) is smaller than the sum DI + DR. Since for most organic radicals % ~- 1.2 and also cq = 1.2, it would be expected from formula (5) or (6) that D ( R - - I ) ~- D~ + DR. It is interesting to note that, similarly, according to PAULING'S relation (1), the strength of the C--I bond is equal to ½ [ D ' ( C - - C ) + D'(I--I)] since r c = r~ and therefore A = 0. An analogous case is that of bonds between radicals R and R' when D ( R - - R ) ,~ D ( R ' - - R ' ) . Then, as the difference between D ( R - - R ) and D ( R ' - - R ' ) becomes larger, the discrepancy between calculated and experimental values of D ( R - - R ' ) becomes also worse. An example is when R is .CN and R' is CH2 = CH--'~H2. Sometimes, departure between calculated and experimental values can be very large, e.g. F--C1, F - - B r , F - - I . Note however that good agreement is found for F - - R where R is an organic radical and also for Br--Cl. Another bad case is allylamine CH 2 = C H - - C H 2 - - N H 2 : D(C--Neale ' = 56 kcal/grnole whereas D(C--N)exvtt. = 64 kcal/ gmole. However, for other R - - N H 2 bonds, only small deviations are found and other bonds with the allyl radical behave normally.
APPLICATIONS
This does not mean that elements belonging to the same group of the periodic table have identical a-values. On the contrary, ~t increases with increasing electronegativity r :
With bond dissociation energies, it becomes possible to estimate heats of reaction. Thus for a substitution reaction of the type:
530
AB + C.-~ A. + BC
An empirical method for the estimation of bond dissociation energies
Table 1. Values of the constant ctR and of the product ~RDRfor various radicals R Radical R H CHO CHACO CeHsCO CN t-C4I-I9 CH------C. i-CaH7 SH Cell5 C2H5 CH2=CH CH3 n-Call7 n-C4I-I9 C6H5 -- CH2 I
0OR (dimensionless)
~RDR (kcal/gmolc)
Radical R
0OR (dimensionless)
0gRDR (kcal/gmolc)
1.00 1.00 1.00 1-00 1"05 1"05 1"06 1"10 1.15 1-18 1-18 1.19 1"20 1"20 1 '20 1 '20 1"20
51"6 26.5 31-5 23.0 60.0 31.5 69"4 36'5 39"1 60-0 46"0 52"4 49"8 43"3 44-4 28-2 21 "4
CHaCI CH2Br CBr3 CI-IBr2 CHCI~ CCla CF3 CH2 = CH---CH2 Br NO , NHa CI NO2 OH CHaO C2H50 F
1.22 1"23 1"24 1"25 1"25 1"29 1-29 1-32 1"48 1"60 1-70 1.75 2"00 2.45 2"95 3-33 4"57
45.8 43 "4 28.4 36"4 41 '6 37"4 51-4 24.4 33"7 7"7 49"5 50-0 16"0 62"5 54"5 52"5 82"4
Table 2. Comparison between calculated and experimental values of bond dissociation energy ( k c a l / g m o l ) R
-- R t
CHa--F CHa--OH CaHs--NHz C~H5--CI tert. C4Hg---CHa n-CaH~----SH n-C4Ha--CHa iso-CaHT--Br CaHsCH2---OH CsHsCH~-CH8 CHa = CH---CI CH2 = CHCH~---Br CeH5--OH CeHs--NH2 CHa----NO2 CHaO--NO C2HsO--NOa CHO--CHa CHO--C~H5 CHaCO--Br CHaCO--C1 CHaCO---CaH~ CaHsCO--Br CaHsCO---CI NO---C1 NO--Br CFa--C1
Dcalc.
Dexptl.
110 93.5 80"9 81"3 77-4 71"5 75-2 63"8 75"6 65-0 86"0 44.0 104.0 92"8 54"8 38"8 34'2 76"3 72"5 65"2 81-5 77-5 56.7 73.0 36.0 28-0 78-6
107 90-5 78 80 74 72"1 77 63 73 63 86 47"5 107 94 57 36"4 36"4 75 71 67 82 77 57 73"6 37 28 80
531
Dcalc. -- DexptL
+3 +3 +2"9 + 1"3 +3'4 --0.6 -- 1'8 +0"8 +2-6 +2 0 -- 3"5 --3 -- 1-2 --2.2 +2-4 --2.2 + 1"3 + 1.5 -- 1"8 --0'5 +0'5 --0'3 --0"6 --1 0 -- 1"4
A. VAN TIC,OEI.EN, J. t~TERS and R. BURKE
From T a b l e 1 and formula (5) we have
the heat of r e a c t i o n Q counted as positive for an exothermie r e a c t i o n is given by
D ( C H a - - H ) = 51.6 + 49.8 = 101.4
a = D(B--C) - D(A--B)
D ( C 2 H s - - H ) = 51.6 + 46.0 =
With h e a t s of reaction, it b e c o m e s possible in turn to estimate energies of activation E by m e a n s of POLA~I'S relation:
97.6
The heat of the r e a c t i o n Q is Q = 101-4 - 94.9 = 3.8
E = (const)l -- (const)zQ
Therefore
w h i c h can be a p p l i e d to a variety of exothermic free r a d i c a l reactions [3]
E = 11.5 - 0 . 2 5 x 3.8 = 10.6 k c a l / g m o l e The experimental value of E is 10.4 [4].
with (const)l = 11.5 kcal/gmole (const)z = 0.25 F o r example, let us estimate the energy of activation of the r e a c t i o n ' C H 3 q- C 2 H 6 --~ CH4 + C 2 H 5"
AcknowledgemEnt--This work has been supported in part by the Office of Scientific Research of the United States Air Force (Grant AF-EOAR 42-63) through its European Office of Aerospace Research and by the Fends de la Recherche Scientifique Fondamentale Collective of the Belgian Government.
REFERENCES [1] [2] [3] [4]
PAtJLr~G L., The Nature of the Chemical Bond. Cornell University Press, New York 1960. COTrgELL T. L., The Strength of Chemical Bonds. Butterworths, L o n d o n 1958. SE~tqOV N. N., Some Problems of Chemical Kinetics and Reactivity (Translated by BOLrDAgT M.). Princeton University Press 1959. S~AClE E. W. R., Atomic and Free Radical Reactions. Reinhold, New York 1954.
R~sum~---L'auteur donne une formule empirique pour le calcul des 6nergies de dissociation des liaisons D(A -- B)
off ~ et ~n sont des c o m t a n t e s caract6ristiques des radicaux A° et B° tandis qu'~ est soit ~a, soit ~B,
quelle que soit leur plus petite valeur. Les valeurs de e pour 34 radicaux organiques et min&aux ont 6t6 d6tel'min6es et raises sous forme de tableaux. A partir des valeurs de D ( A - - B ) les chaleurs de r6action Q peuvent ~tre calcul6es. On peut ~valuer les 6nergies d'activation E pour les r~actions de substitution en utilisant la relation de POLAN~ E = (const.h -- (const.)2. Q.
532