2018
SAMUEL
[CONTRIBUTION FROM
H. MARON AND THE
VICTOR
K.LA MER
Vol. 61
DEPARTMENT OF CHEMISTRY, COLUMBIA UNIVERSITY ]
Calculation of Base Rate Constants from Acid Rate Constants BY SAMUEL H. MARON’ AND VICTORK. LA MER For the protolytic equilibrium kg
N+B=R+A
(1)
kA
where N is acid substrate which ionizes to yield a proton to a base, B is base (proton acceptor), R  is conjugate base of N, and A is conjugate acid of B, the equilibrium constant, K , is given by
However, the ionization constant of N , KN, corresponding to the reaction
+ HzO = Haof + R
N
The mechanism of this reaction has been discussed before. Again, the reisomerization of acinitroparaffin to the nitro form has been shown to be, under the proper conditions, a reaction between the acinitroparaffin ion and an acid5 (in the general sense) according to the scheme
+
R \ / ~ = ~ ~A ~ R’ R‘
If (9) and (10) be combined, we have the equilibrium condition :
(3)
is given by
with equilibrium constant
KN = [RI [HaO+I/[NI (4) while the ionization constant of A, KA, corresponding to the reaction A
+ H20 = H30+ + B
is given by K.4 =
P I
K =
(5)
(6)
l&O+l/IAl
Dividing now (4)by (6)
Equation (1%)is exactly analogous to equation (%),and hence it foIlows from (8) that
where KN is according to (3) and (4) the constant for the nitroparaffin dissociation
by eq. ( 2 ) . And hence
R
\CHN02
Knowing any three of the four constants K N , KA, k R , k A in (S), the fourth may be calculated. 11 is proposed to apply these equations to the prototropic isomerizatioti of nitroparaffins t(J calculate k B from available values of KN,K A ,and
n’
+ HzO
__
K
‘C.NO0
K’
+ 1130’
(14)
a11d
k ‘4. The primary and secondary nitroparaffins are acids which react with bases (defined in the general sense of Bronsted2) to yield an acinitroparaffin ion and a conjugate acid” according to the general scheme ‘>CHNO, R‘
+ B kB, R\C R’/
= NOO
+A
(9)
(1) Present address: Department of Chemical Engineering, Case School of Applied Science, Cleveland, Ohio. (2) Bronsted, Chem. Rev., 6, 231 (1928). (3) Pedersen, Del. Kgl. Videnskab. Selskab. Malh.fys. M e d d . , l a , 1 (1932) (in English); Junell. Z . p h y s i k . Chem., 141A, 7190 (1929); Dissertation, University of Uppsala, Sweden, 1935; Avkiv Kemi, 11B, No. 27 (1934): ibid., 11B, No. 34 (1934); Reitz, Z . p h y s i k . Chem.. 176A, 363387 (1936).
while KA is the ionization constant of the acid used and is defined by equations ( 5 ) and (0). The ionization constants K N of the nitro forms of nitromethane and nitroethane can be obtained from some measurements of Junell.6 Junell determined what he terms the “scheinbare Dissoziationskonstante” of these two substances by (4) Maron and La Mer, THISJOURNAL, 60, 2588 (1938). (5) Junell, (a) Arkiv K e m i , 11B, No.30 (1934); (b) Saensk K e m . Tid., 46, 125136 (1934); (c) Dissertation, Uppsala, 1935; (d) Maron and La Mer, THISJOURNAL, 61, 692 (1939). (6) Junell,a Dissertation, p p . 114124.
Aug., 1939
pp,00000g
determining a t high pH's the total concentration of aci form and calculating the ratio K ' N = [H30+] [aci form]/[nitroform]

o z p ~ 0F0W0 $ C + lo1$00l I I * opmggg
(16)
The highest H30f concentration used for the nitromethane measurements was 1.41 X IOpy, while the highest concentration of H30+for nitroethane was 5.38 X Now, since the ionization constant of acinitromethane is given by Junell' as 6 X lop4while that of nitroethane has been shown to be 4.1 X 105,s i t follows that the ratios of aci ion to aci acid a t equilibrium must be
[R]  [R
2019
BASERATECONSTANTS FROM ACIDRATECONSTANTS
6X
lo'
=
1.41

+
(7) Junell, Dissertation, ref. 3. (8) Maron and Shedlovsky, THISJOURNAL, $1, 753 (1939).
+
0
0 G? W W N
z
,,??A'?!, I ,
z
s P
C
s X
N P
X
g' a0E0m G Z 8 8 ' 8 I00000E($

0
z
?L I
[N I
+ A +CHaN02 + B
P
0 0 3 0 0 3 0 0
. .   ,  o S l x
CH*=N00
C
C C C c l C  r C
N P a w  m 
These data are due to Junell.' Column (4) gives the equilibrium constant K calculated by Eq. (13), while column ( 5 ) gives the rate constant, Kg, calculated by (12) for the reaction : CH3NOz B + CHz=N02A where B is the conjugate base of the particular acid. It is possible to compare several of the calculated values of kB with observed results. Although no rate data are available for O", kinetic measurements are extant for the reaction with the bases H20, monochloroacetate, and acetate ions a t several temperatures, so that extrapolation to 0' and comparison with the calculated constants are possible. The data available for this purpose are summarized in Table 11. A
LT
xxxxxxxx;,
and
the ionization constants of the nitro forms. For nitromethane Junell reports from five measurewhile for nitroments KN = (2.6 * 0.2) X ethane he gives as a result of four measurements K N = (2.7 * 0.2) X These are the values, of KN employed in all calculations. Table I gives the observed quantities and calculations for nitromethane a t 0". Columns (l), ( 2 ) and (3) give the acids used, their ionization constants, and the rate constants, respectively, for the reaction (10)
$ 2
.N .P . .N c . m . . .a
4.3 X 106 for nitromethane
From these ratios i t is seen that compared to the concentrations of aci ion present the concentrations of aci acid are negligible, and hence (16) is essentially  [H@+1 [R 1 = K N K Nt (17)
500 oms
0
Z X X X X X X X ~
0
rp
SAMUEL H. MARON AND VICTORK. LA MER
2020
TABLE I1
RATECONSTANTS FOR CHaN02 Temp.
+B
THE
REACTION
kB f
CHFNOO
kB
+A Reference
Hz0 49.91 59.93 69.85 69.85
8.8 X lo' 2.5 X 6.8 X 7 . 4 x 106
11.2 20.0 25.0 35.0
5.1 X 1.50 104 2.35 X lo' 7.71 x 104
June118 Junel13 Junel16" Reitza
CHzClC00
11.0 20.0 22.0 25.0 35.0
x
CHICOO8.7 x 104 2.69 x 103 3.22 x 103 4.55 x 103 1.38 X
Reitz3 Pedersen3 ReitzY Reitz3
Vol. 61
in column (4), and calculated from Kcalcd. = KN/KA. The agreement again is good. And, finally, also by equation (13), KA = K N k A / k B ; values of KA thus calculated are given in column (8), and again show as good agreement with the observed values of KA in column ( 2 ) as can be expected. Table I11 gives the results of similar calculations for nitroethane. These calculations are again based on Junell's values for kA, and on his value of K N = 2.7 X for nitroethane. Columns (l), ( 2 ) and (3) have the same significance as before except that now the reaction involved is
+
C H ~ C H ~ N O ZB Reitz3 Pedersens Pedersen3 Reitz3 Reitzs
kB
+A
I _ CHsCH=NOOkA
while columns (4) and (5) give the calculated equilibrium constants of the reaction (Kcalcd.) and the rate constants for the forward reaction ( k B ) . Unfortunately no extended data on the plot of log k B vs. 1/T and extrapolation to 0' forward reaction are available to permit compariyields the constants given in column (6) of Table son of calculated and observed results in this case. I. The agreement between calculated and ob The only value of k~ available a t 0' is koH = served results, columns (5) and (6)' is good. 39.1 and due to Maron and La Mer,4 but no k A In case of the acid H$O+ the procedure was re observed is known to permit a calculation of kB. versed and kA calculated from the values of kB Hence again the procedure was reversed and the a t 0' measured by Maron and La Mer.4 known value of koH was used to calculate the value of k H I Oin column (3). However, a test of the calculated values of kB can be obtained in an1.00 4.00 other way. June11 showed that for both nitromethane and nitroethane a Q, plot Of log kA/$ ZJS. log (qKA/@)iS ;;i3.00 2.00> essentially linear, as would be exA 2 3 M pected if the Bronsted relation log 3 k A / p = log GA a log ( P I ~ K were A) o.oo obeyed. 9 and q are the statistical 5.00 factors to correct the respective measured acid and base constants KA and KB to a uniform basis (see ref. 1). 7.00 The values assigned p and 4 are given 8 6 4 2 in Tables I and 111. On theoretical Log ($?/pKA). grounds i t may be anticipated that a Fig. l.Plot of log k B / q and log k A / P ws. log ($?/PICA)for nitr.omethane a t 0 '. plot of log k B / g vs. log q/pKA should It is possible to make the comparison between also yield a straight line but with slope = /3 observed and calculated constants in two other and y intercept = log GB. Further a /3 should ways, namely, by calculation of K and KA from equal unity. The plot of both log kA/F and log observed data. From equation (13), K = kB/ k B / q vs. log q/pKA for nitromethane a t 0' is kA. Using the observed values of kA and k B shown in Fig. 1. The expected linearity for log given in columns (3) and (6) of Table I, K o b s d . k B / q vs. log q/pKA is verified. Whenever a point have been calculated and are given in column (7). is off the curve, the discrepancy is traceable to These values are to be compared with those given the corresponding value of k A , which is also off h pl
3

+
+
BASERATECONSTANTS FROM ACIDRATECONSTANTS
Aug., 1930
202 1
the curve. Further, from this plot we obtain nitromethane G B / G A = 1.98 X 10'/0.82 X = 8200, G B = 1.98 X lo', a = 0.39 and lo4 = 2.4 X 1Ol1, as against the observed value 0 = 0.62, and hence a P = 0.39 0.62 = 1.01 as against the expected sum vf unity. 1.00 2 00 A similar plot for nitroethane is shown in Fig. 2. Again the linear r; 3.00 \ A relation is obtained between log 0.00 d 5'00 M k B / p and log q / p K A . In this case G A = 32, G B = 1.05 X lo', $ 7,00 a = 0.36, ,8 = 0.65 and a 8 , = 0.36 0.65 = 1.01. 9.00 2 00 From the deductions of this paper may be derived another very 8 6 4 2 interesting and quite general relaLog (deKA). tionship* For reactions Of the Fig. 2.Plot of log kB/p and log k A / p vs. log ( q / p K ~for ) nitroethane at 0 " . type discussed here, and which of K N = 2.6 X lo". Similarly, for nitroethane lead to an equilibrium, we have in general G B / G A = 1.05 X 10'/32 = 3.3 X as against kA/P = G A ( ~ / ~ Kand A ) kB/q ~ = GB(P/~KB)~ the experimental value KN = 2.7 X Conwhere K B = l / K A , i. e., the ionization constant sidering the uncertainties in drawing the straight for the base conjugate to A. With these equations lines in Figs. 1 and 2 the agreement is satisfactory. we obtain finally for log k B / k ~
GA
+
+
h
5
+
 (a 3 P ) 1 log S/p GB log (a
1% ke/kA =
GA
Rut
s
+
kB/'kA
=
K and ( a
kg kA
Summary
+ P ) 1% KA
+ @)= 1.
log  = log K = log
Hence
GB  log K A GA
However, according to (8) log K log KN. Therefore
(18)
(19)
4 log KA =
log K N = log G B / G A
and K N = GB/GA
(20)
According to (20) the ratio G B / G A should be the ionization constant of the substrate involved in the basic reaction. This relation can be tested easily by utilizing the values of G B and G A found a t 0' for nitromethane and nitroethane. For
1. Equations are deduced for calculating rate constants from certain kinetic and thermodynamic data. 2. These equations are applied to calculate the base rate constants for the isomerization of nitromethane and nitroethane to the aci forms. 3. Wherever comparison between calculated and observed values is possible, the agreement is good. 4. The calculated constants obey the Bronsted relation (log kB/p = GB(p/pKB)@. 5 . An equation is derived relating acid strength of the substrate to the constants G A and G B in the Bronsted equations. The equation is shown to be valid for the cases discussed. NEWYORK,N. Y .
RECEIVED MAY1, 1939