PHYSICAL CHEMISTRY

A. I. GEL’BSTEIN and M. I. TEMKIN

ON DETERMINING THE ORDER OF REACTIONS WITH RESPECT TO ACIDITY

(Presented by Academician A. N. Frumkin on 23 VII 1957)

In catalysis by concentrated acids, the degree of conversion of the substrate B into the protonated form BH\(^+\) is determined by the acidity of the medium, i.e., by its ability to donate a proton, and also by the basicity of the substrate:

\[ \frac{C_{\mathrm{BH}^{+}}}{C_{\mathrm{B}}}=Kh_{0}. \tag{1} \]

Here \(h_{0}\) is the acidity of the medium, and \(K\) is the equilibrium constant of the reaction

\[ \mathrm{B}+\mathrm{H}^{+}\rightleftarrows \mathrm{BH}^{+}, \tag{2} \]

expressed in terms of activities, characterizing the basicity of B.

According to Hammett \((^{1})\), if the first stage of the reaction is the establishment of equilibrium (2) and the rate-limiting step is the subsequent unimolecular transformation of the protonated form BH\(^+\), then, provided that the degree of protonation is small (\(C_{\mathrm{BH}^{+}}\ll C_{\mathrm{B}}\)), the observed rate constant \(k\) is proportional to \(h_{0}\).

It is conceivable, however, that cases may occur in which the rate-limiting stage is preceded by the establishment of equilibrium

\[ \mathrm{B}+2\mathrm{H}^{+}\rightleftarrows \mathrm{BH}_{2}^{2+}. \tag{3} \]

In this case \(k\) should be proportional to \(h_{0}^{2}\) (if \(h_{+}\)—the acidity determined with respect to a singly positively charged base—is proportional to \(h_{0}\)—the acidity with respect to an uncharged base; in what follows, for simplicity, we take \(h_{+}=h_{0}\)).

Therefore the dependence of \(k\) on \(h_{0}\) is represented by the equation

\[ k=\mathrm{const}\cdot h_{0}^{\,n}. \tag{4} \]

The exponent \(n\) is usually found graphically by plotting \(\log k\) against the acidity function \(H_{0}=-\log h_{0}\).

In this way, for the decomposition of benzoylformic acid in sulfuric acid it was found that \(n=2\), and the conclusion was drawn that the reaction includes the addition of two protons \((^{2})\).

From the rates of decomposition of other carboxylic acids in sulfuric acid, values of \(n\) were obtained that were close to 1, close to 2, and also varied with the acid composition from 1 to 2 and exceeded 2; to explain this, special assumptions concerning the reaction mechanism were put forward \((^{2})\).

We have investigated the kinetics of the decomposition of formic acid into carbon monoxide and water \((^{3})\) in the media H\(_2\)SO\(_4\)—H\(_2\)O (from 80.7 to 98.2% H\(_2\)SO\(_4\)) and P\(_2\)O\(_5\)—H\(_2\)O (from 72.4 to 83.3% P\(_2\)O\(_5\), i.e., in the region of the so-called strong phosphoric acids). Measurements of \(H_{0}\) of these media were also carried out, including the temperature dependence of \(H_{0}\) \((^{4})\).

The acidity of the P\(_2\)O\(_5\)—H\(_2\)O system passes through a maximum at a P\(_2\)O\(_5\) content of about 80%, in contrast to the H\(_2\)SO\(_4\)—H\(_2\)O system, where the acidit-

monotonically changes with composition (4). This feature of the system \(\mathrm{P_2O_5—H_2O}\) makes it possible to reveal more clearly the relation between the reaction-rate constant and acidity.

In Fig. 1 the values \(-\log k\), extrapolated to \(20^\circ\) by the Arrhenius equation, are presented as a function of \(-H_0\).

For decomposition in sulfuric acid, the slope of the straight line corresponds to \(n = 1.6\), which does not make it possible to choose between mechanisms corresponding to \(n = 1\) and \(n = 2\).

The course of the curve for the system \(\mathrm{P_2O_5—H_2O}\) indicates the inapplicability of equation (4). The same acidity corresponds to substantially different values of the rate constant. It should be assumed\({}^{(3)}\) that the change in the rate constant with the composition of the medium is determined by the superposition of two effects—the influence of acidity and the influence of the nature of the solvent (the Menshutkin effect).

Fig. 1

Without prejudging the question of the number of protons adding to the reacting molecule, and taking as the limiting stage a monomolecular transformation of the protonated form, it is easy, analogously to \({}^{(3)}\), to obtain the expression

\[ k = B h_0^n (x) e^{-A(x)/RT}. \tag{5} \]

Here \(h_0(x)\) is the acidity as a function of the acid concentration \(x\); \(n\) is the reaction order with respect to acidity; \(A(x)\) is the activation energy depending on \(x\), with \(A(x)=A_1(x)+\Delta H^0\), where \(A_1(x)\) is the activation energy of the limiting stage, and \(\Delta H^0\) is the standard change in heat content upon protonation.

The proportionality coefficient \(B=B_1 e^{\Delta S^0/R}\), where \(B_1\) is the pre-exponential factor of the rate constant of the limiting stage, and \(\Delta S^0\) is the standard change in entropy upon protonation. We assume that \(B_1\), and consequently also \(B\), do not depend on \(x\).

In order to determine \(n\), let us represent \(h_0\) by the equation

\[ h_0 = \beta(x)e^{-\alpha(x)/RT}, \tag{6} \]

where \(\beta(x)\) and \(\alpha(x)\) are certain functions of \(x\).

Equations (5) and (6) give:

\[ k = B\beta^n(x)e^{-[A(x)+n\alpha(x)]/RT}. \tag{7} \]

Consequently,

\[ k = F(x)e^{-E(x)/RT}, \tag{8} \]

where \(F(x)\), the observed pre-exponential factor, and \(E(x)\), the observed activation energy, are expressed as follows:

\[ F(x)=B\beta^n(x), \tag{9} \]

\[ E(x)=A(x)+n\alpha(x). \tag{10} \]

To determine \(n\), it is convenient to use the logarithmic form of equation (9):

\[ \log F(x)=\log B+n\log \beta. \tag{11} \]

The quantities \(\log \beta\) are calculated from the acidity function and its temperature dependence by means of the equation

\[ \log \beta=-H_{0}-T\left(\frac{\partial H_{0}}{\partial T}\right)_{x}, \tag{12} \]

which follows from (6).

Tables 1 and 2 give the values of \(\log \beta\) at \(20^\circ\), calculated from the experimental data \((^{4})\).

Table 1

System \(\mathrm{H_2SO_4—H_2O}\)

Table 2

System \(\mathrm{P_2O_5—H_2O}\)

The quantities \(\log \beta\) are somewhat less accurate than \(H_{0}\). The determination of \(F\) from kinetic data is also less accurate than the determination of \(k\). Therefore a graphical plot of \(\log F, \log \beta\) gives a greater scatter of points than a plot of \(\log k, H_{0}\).

Fig. 2

In Fig. 2 the points correspond to the experimental data on the decomposition of formic acid \((^{3})\), and the straight lines to the value \(n=1\). Despite the considerable scatter of points in the case of the \(\mathrm{P_2O_5—H_2O}\) system, it may be concluded that the reaction proceeds through a reversible protonation stage, in which only one proton is attached to the reacting molecule. This is confirmed by the constancy of the values of \(B\), calculated from equation (5) with \(n=1\) at different \(x\) \((^{3})\).

The significant deviation of the value of $n$ from 1, observed in a number of cases when $n$ was determined from equation (4), is apparently a consequence of neglecting the influence of the acid concentration on the activation energy of the reaction.

Determination of $n$ from equation (11) is, in principle, more correct than determination from equation (4), but it imposes considerably higher requirements on the experimental data.

Scientific Research
Physico-Chemical Institute
named after L. Ya. Karpov

Received
16 VII 1957

CITED LITERATURE

1. L. P. Hammett, A. J. Deyrup, J. Am. Chem. Soc., 54, 2721 (1932); L. P. Hammett, Physical Organic Chemistry, London, 1940.
2. W. Elliott, D. Hammick, J. Chem. Soc., 1951, 3402.
3. A. I. Gelbshtein, G. G. Shcheglova, M. I. Temkin, ZhFKh, 30, 2267 (1956).
4. A. I. Gelbshtein, G. G. Shcheglova, M. I. Temkin, Zhurn. neorg. khim., 1, 282 (1956); 1, 506 (1956); DAN, 107, 108 (1956).