Резюме. It is claimed that, in certain circumstances, initial-rate method (IRM) has advantages over the integral method (IM). But, the IRM will only be effective if the concentration change is less than \(5 \%\) and the rate constant \((k)\) obtained should be checked with IM. There are two main purposes of this research. First, to prove that the claim is invalid and the \(I R M\) can be used on broader concentration changes. In addition, the rate law model used in the claimed is invalid for reaction that takes place in two mechanisms. Second, to find a new form of \(I R M\) equation, which does not require \(I M\) checks. This study used the oxidation of formic acid with bromine as a reaction model. The values of \(n\) an d \(k\) of this reaction at two mechanisms were determined by the \(I M\), and then by the conventional IRM, the method used in the claim, and by the new IRM generated by this study. All of \(k\) values are compared by using IM as the standard. The proposed method is claimed valid if the error percentage of \(k\) is less than \(5 \%\). Despite the shortcomings, the results show that the new IRM is valid and more reliable.

Ключови думи: initial rate method; integral method; rate constant; reaction order

Introduction

The differential method (\(D M\) ) is believed to be the most reliable one for determining the rate law of a reaction (Laidler, 1987). By this method, the order, \(n\), with respect to a particular reactant can be directly determined; not by trial and error (as in the integral method, IM). However, the rate, \(v\), is not always proportional to every concentration. The initial rate of a reaction is defined as the rate of the reaction at the instant in which the reactants are first mixed. The rate at the beginning of a reaction is the most appropriate because at this condition, the rate is more linear with concentration. However, the exact initial rate \(v_{o}\), is not easily determined. A statistical method has been introduced. As long as the percent error of the reaction followed less than \(5 \%\), the setting up solution can be considered valid. It has been proved that under this condition, in certain circumstances, the initial rate method (IRM), has advantages over IM. For example, the IRM can be used to simultaneously determine \(k_{1}\) and \(k_{2}\) in a reaction of the type

(1) \[ v_{o}=k_{1}\left[C_{o}\right]+k_{2}\left[C_{o}\right]^{2} \]

whereas only \(k_{2}\) by \(I M\). The value of \(k_{1}\) is determined from the intercept and \(k_{2}\) from the slope of the linear graph of

(2) \[ v_{o} /\left[C_{o}\right]=k_{1}+k_{2}\left[C_{o}\right] \]

By Eq. (2) and using the diazotization of \(p\)-anisidine (initial concentration \(3 \times\) \(10^{-4} M\) ) in aqueous perchloric acid (initial concentration \(0.1 M\) ) as the study object, the rate equation is

(3) \[ v_{o}=k_{1}[\text { nitrite }]_{0}+k_{2}[\text { nitrite }]_{0}^{2} \]

and \(k_{1}=2.60 \times 10^{-5} \mathrm{~s}^{-1}\) and \(k_{2}=5.40 \times 10^{-3} M^{-1} \mathrm{~s}^{-1}\). (Casado et al., 1986).

There are two problems that arise related to the above statement. First, the recommendation to restrict the percentage of reaction should be followed by a maximum of \(5 \%\). It is expected that such limitation is caused by the use of graphical method for determination. However, graphical method is not recommended for determining the order, \(n\), and the rate constant, \(k\), of a reaction (Levine, 2003). The shelf life of a pharmaceutical product is \(t_{0.1 .}\) (Tan et al., 1994). It is not enough reason that the shelf life of a pharmaceutical product cannot be determined by IRM. A reaction which has the rate law such as in Eq. (1) will always take place in two different reaction mechanisms. The two mechanisms could take place simultaneously (competing) or in two successive steps (consecutive). Thus, the value of the two constants cannot be determined easily.

It is reported that the plot of \(v_{o} /[C]_{\mathrm{o}}\) v versus \([C]_{\mathrm{o}}\) gives both of \(k_{1}\) and \(k_{2}\) values simultaneously. The plot of \([C]^{-1}\) against \(t\) would overlook \(k_{1}[C]\). This is for sure not an overlooking. In \(I M\), the plot of \([C]^{-1}\) against \(t\) will never give the constants for first or other-order of a reaction. Any \([C]\) that appropriate for first-order is not suitable for other-orders. By concept, a first-order reaction means, doubling \([C]\) will make a reaction last \(2^{1}\) or 2 times faster and second-order reaction means, doubling [ \(C\) ] will make a reaction last \(2^{2}\) or 4 times faster. Thus, it is unlikely that \(v_{o}\) fo for first-order is used for second-order reaction (and conversely). In thermodynamic, to determine the absolute entropy of a liquid near the absolute temperature \(0^{\circ} K\), the curve of \(\mathrm{Cp} / \mathrm{T}\) versus T, only use data in the solid phase (Rastogi & Misra, 1980). It is more appropriate to use a model of integral equation that can give the two constant values simultaneously (Espenson, 1995; Brynildson et al., 1987). However, it is also possible that one found that for eq. (1), there is only one order, a non-integer, between 1 and 2, say 1.38. Though, the order is usually integers or half-integers (Levine, 2003).

Besides, the term of \(v_{o} /[C]_{\mathrm{o}}\) in Eq. (1) is basically give the accurate value of \(k\) for first-order reaction \(\left(k_{1}\right)\). This is because the accurate value of \(v_{o} /[C]_{\mathrm{o}}\) is o obtained if \([C]_{\mathrm{o}} \rightarrow 0\). This case is similar with the method for the determination molecular weight of gases by limiting density (or Regnault) method (Crockford & Nowell, 1975). The consequences, the term of \(k_{2}[C]_{0}\) is ambiguous. It is correct mathematically but meaningless from chemical point of view. It is \(v_{o} /[C]_{0}{ }^{2}\) that gives the accurate value \(k_{2}\), that is if \([C]_{\mathrm{o}} \rightarrow 0\). It stands to reason; the \([C]_{\mathrm{o}}\) data used must be from the condition of second-order reaction.

Second, the \(I R M\) is believed to give the exact \(n\) value, but it is uncertain \(k\) value. The \(k\) value obtained must be checked with the exact value obtained from IM (Laidler, 1987). In fact, save for zeroth-order reaction, the calculated value of the apparent \(k\) obtained by the \(I R M\) method are always differ with that from IM. There is a missing link here. Both IM and \(I R M\) are derived from the same rate expression. Therefore, it is unlikely that a \(k\) value obtained from \(I R M\) (and then given the notation \(k_{\text {IRM }}\) ) at a certain condition, is uncertain. It is expected just caused by experimental technique and/or mathematical reasoning. However, \(I M\) itself is impractical in some aspects. The true \(n\) can only be determined by trial and error, using some integral equations to find the best fit equation.

In continuation of our previous reports in the field of chemical kinetics (Patiha & Firdaus, 2016; Firdaus & Patiha, 2017), we discuss here an improved technique for the initial rate method. Based on the above statement, there are two goals of this research. The first goal is to prove that, there is no need in initial rate method to restrict the percentage of reactions followed to a maximum of \(5 \%\). Also, to provide evidence that the CM (Casado Method), the method used as reported by Casado et al. (1986), cannot be used to simultaneously determine the two constants of the reaction that takes place in two mechanisms. The second is to introduce an alternative \(I R M\) model, a valid and reliable method to determine the correct order and the accurate value of \(k\) and no need for \(I M\) checks.

Methodology

The two goals will be confirmed by using the oxidation of formic acid by bromine as the model.

(4) \[ \mathrm{HCOOH}+\mathrm{Br}_{2} \rightarrow \mathrm{CO}_{2}+2 \mathrm{Br}+2 \mathrm{H}^{+} \]

At room temperature, this reaction is believed to take place by the following reaction mechanism {Hodgson et al., 1998):

(5) \(\begin{gathered} k_1\quad\quad\quad\quad\\ \mathrm{kCOO}_{1} \Leftrightarrow \mathrm{HCOO}^{-}+\mathrm{H}^{+} \\ k_{-1}\quad\quad\quad\quad \\ \\ k_{2}\quad\quad\quad\quad \\ \mathrm{HCOO}^{-}+\mathrm{Br}_{2} \rightarrow \mathrm{CO}_{2}+\mathrm{H}^{+}+2 \mathrm{Br}^{-} \end{gathered}\)

At first sight, the reaction is of the second order. But in highly acidic media, such as in the presence of \(\mathrm{HClO}_{4}\), the reaction is found to be of the first-order with respect to \(\left[\mathrm{Br}_{2}\right]\) :

(6) \[ -d[\mathrm{HCOOH}] / d t=\tfrac{k[\mathrm{HCOOH}]\left[\mathrm{Br}_{2}\right]}{\left[\mathrm{H}^{+}\right]} \]

Actually, by the steady-states approximation, its rate law is

(7) \[ -d[\mathrm{HCOOH}] / d t=\tfrac{k_{1}[\mathrm{HCOOH}]\left[\mathrm{Br}_{2}\right]}{\left(k_{-1} / k_{2}\right)\left[\mathrm{H}^{+}\right]+\left[\mathrm{Br}_{2}\right]} \]

In an excessive of [ \(H C O O H\) ], the rate law of the reaction is

(8) \[ -d[\mathrm{HCOOH}] / d t=k^{\prime}[\mathrm{HCOOH}]^{0.5}\left[\mathrm{Br}_{2}\right] \]

The reaction is halved-order relating to [HCOOH] and first-order with regard to \(\left[\mathrm{Br}_{2}\right]\) (Patiha & Firdaus, 2] (Patiha & Firdaus, 2016).

Theoretically, when the initial \(\left[\mathrm{Br}_{2}\right]_{\mathrm{o}}\) is slightly greater than \(\left(k_{-1} / k_{2}\right)\left[H^{+}\right]\), at the beginning starting point, the reaction is zeroth-order relating to \(\left[\mathrm{Br}_{2}\right]\) and, end before the final stage, becomes first-order. The first step of Eq. (5) takes place very fast and thus therefore \(\left(k_{-1} / k_{2}\right)\left[H^{+}\right] \gg\left[B r_{2}\right]\), and the reaction is first-order relating to \(\left[B r_{2}\right]\). Because this step is reversible, then after a while, \(\left[B r_{2}\right]\) will be higher than \(\left(k_{-1} / k_{2}\right)\left[H^{+}\right]\), and the reaction becomes zeroth-order. However, it is also possible that at equilibrium, \([\mathrm{HCOOH}]_{\text {eq }}=\left[\mathrm{Br}_{2}\right]_{\text {eq }}\) and the reaction becomes one and halved-order; Eq. (8). However, the reaction is always first-order, whenever at the early stage, \(\left[\mathrm{Br}_{2}\right]_{\mathrm{o}}\) is lower than \(\left(k_{-1} / k_{2}\right)\left[H^{+}\right]\). Thus, the reaction will take place in two different mechanisms. For a rate law with a summation, there are two possible reaction types; competing or consecutive. Since this reaction is consecutive, the initial rate law of the reaction is

(9) \[ v_{o}=k_{1}\left[A_{\mathrm{o}}\right]+k_{2} \]

The \(k_{1}\) value can be calculated from the slope and \(k_{2}\) from the intercept of the linear plot of \(v_{o}\) v versus \(\left[A_{\mathrm{o}}\right]\). The kinetics o ]. The kinetics of the reaction is followed by monitoring the decrease of \(\left[\mathrm{Br}_{2}\right]\) at \(400\left(\varepsilon=160 M^{-1} \mathrm{~cm}^{-1}\right) \mathrm{nm}\) and room temperature. The required time each \(\mathrm{Br}_{2}\) reacts by \(10 \%\) (or constant of fraction-left \(=\beta=90 \%\) ) are then recorded.

The data were obtained from a single run for each experiment which was taken at the same \(\beta\), and the observed value at the end of one period is regarded as the initial value for a new interval. The difference between each pair was divided with the time difference, and denote as \(v_{o}\). The data (from the two conditions) and their setting are depicted in Tables 1 and 2.

The data in Tables 1 and 2. will be analyzed using the integral equation of:

(10) \({[A]-\left[A_{o}\right]=k_{A} t \text { for } \mathrm{n}=0} \)

(11) \({[A]^{0.5}=\left[A_{o}\right]^{0.5}-0.5 k_{A} t \text { for } \mathrm{n}=0,5}\)

(12) \(\ln ^{[A]}{\mskip -5mu/\mskip -3mu}_{\left[A_{o}\right]}=-k_{A} t \text { for } \mathrm{n}=1 \)

(13) \({[A]^{-0.5}=\left[A_{o}\right]^{-0.5}+0.5 k_{A} t \text { for } \mathrm{n}=1,5} \)

(14) \({[A]^{-1}=\left[A_{o}\right]^{-1}+k_{A} t \text { for } \mathrm{n}=2} \)

The true order \(n\) and the accurate value of \(k_{A}\) is that from the equation with regression coefficient closer to \(\pm 1.0000\). The values obtained are used as the standard.

The data is then analyzed with conventional \(I R M\) equation:

(15) \[ v_{o}=k_{I R M}\left[A_{o}\right]^{n} \]

where \(v_{o}, k_{I R M}\), kIRM, and \(n\) are the initial rate, the rate constant (for \(I R M\) ), and the reaction order respectively. From the slope, the reaction order, \(n\), can be determined and the rate constant \(k\) can be obtained from the intercept of a straight line of double-logarithmic plot of \(\log v_{o}\) versus \(\log \left[A_{o}\right]\).

\( \log v_{o}=\log k_{I R M}+n \log \left[A_{o}\right] \) (16)

The data is also analyzed by Eq. (9) and the equation that will be introduced. A method is claimed valid if, relative to the standard (IM), its error percentage (EP) of \(k_{A}\) is less than \(5 \%\).

The first goal

If an experiment is done in a single run and each observation is taken at constant fraction-left:

(17) \[ \beta=[A] /\left[A_{o}\right] \]

the time, \(t_{i}\), needed in each observation is:

(18) \[ t_{\beta-i}=(\beta)^{(i-1)(1-n)} t_{\beta} \]

where \(t_{\beta-i}, \beta\), β, and \(t_{\beta}\) are the interval-time, fraction-left, and the first interval-time, respectively (Patiha, 2011). If the initial observation is \(\left[A_{o}\right]\) then \(\left[A_{1}\right]=\beta\left[A_{o}\right],\left[A_{2}\right]\) \(=\beta^{2}\left[A_{o}\right]\), or \(\left[A_{i=1} \beta^{i}\left[A_{o}\right]\right.\).

\(v_{o-i}=\left(\left[A_{o}\right]-\left[A_{i}\right]\right) / t_{i}=k_{A-i}\left[A_{o}\right]^{n} \)

(19) \(v_{o-i}=\left(1-\beta^{i}\right)\left[A_{o}\right] / t_{i}=k_{A-1}\left[A_{o}\right]^{n} \)

(20) \(\left(1-\beta^{i}\right)\left[A_{o}\right]^{(1-n)} / t_{i}=k_{A-1}\)

Based on the basic of initial rate concept (the rate of the very beginning of a reaction), the example of the effect of changing the fraction reacted on the value of \(k_{A}\) will be given fro] - [A1]) and ([A of (\(\left[A_{o}\right]-\left[A_{1}\right]\) ) and (\(\left[A_{o}\right]-\left[A_{2}\right]\) ).

From Eq. (18): \(t_{\beta-1}=t_{\beta}=t_{1}\) and \(t_{\beta-2} \beta^{(1-n)} t_{\beta}\); tβ-2 = β(1-n) tβ; then by Eq. (20), for ([Aor (\(\left[A_{o}\right]-\left[A_{1}\right]\) ):

(21) \[ (1-\beta)\left[A_{o}\right]^{(1-n)} / t_{\beta}=k_{A-1} \]

and for ([Aor \(\left(\left[A_{o}\right]-\left[A_{2}\right]\right)\)

\[ \left(1-\beta^{2}\right)\left[A_{o}\right]^{(1-n)} / t_{2}=k_{A-2} \]

Because \(t_{2}=t_{\beta-1}+t_{\beta-2}=t_{\beta}+t_{\beta-2}\), then

(22) \[ \left(1-\beta^{2}\right)\left[A_{o}\right]^{(1-n)} /\left(t_{\beta}+t_{\beta-2}\right)=k_{A-2} \]

But by Eq. (18), \(t_{\beta-2}=\beta^{(1-n)} t_{\beta}\), t then

(23) \[ t_{\beta-2=} t_{\beta}\left(1_{+} \beta^{(1-n)}\right) \]

Inserting Eq. (23) into Eq. (22), after some steps gives

(24) \[ \left\{(1+\beta)(1-\beta)\left[A_{o}\right]^{(1-n)}\right\} / t_{\beta}\left\{1+(\beta)^{(1-n)}\right\}=k_{A-2} \]

As an example, for zeroth-order reaction: Eq. (24) becomes

(25) \[ \begin{aligned} & (1+\beta)(1-\beta)\left[A_{o}\right] / t_{\beta}(1+\beta)=k_{A-2} \\ & \quad(1-\beta)\left[A_{o}\right] / t_{\beta}=k_{A-2} \end{aligned} \]

and Eq. (21) becomes

(26) \[ (1-\beta)\left[A_{o}\right] / t_{\beta}=k_{A-1} \]

Comparing Eq. (25) with Eq. (26) gives \(k_{A-2}=k_{A-1}\). For zeroth-order reaction, there is no influence in the changing of the amount fraction reacted. That is not so for first-order reaction. But, both values are comparable and by Eq. (18), directly proportional to the fraction-reacted \((=1-\beta)\) and inversely to the time. This fact is then used to formulate a new model of IRM which is need no IM checks.

The validity of Eq. (9) will also be tested using the combined data of first- and zeroth-order (Tables 1 and 2). The equation is invalid if the \(E P\) of \(k_{1}\) and \(k_{2}\) values with that obtained from IM is higher than 5%.

The second goal: the new method (NM)

Based on the previous discussion, there are three factors that affect the value of \(k\) and cause it different from \(I M\) : the fraction-reacted \((=1-\beta)\) the time, and the general form of IM. Because the time variable is already included in \(v_{o}\) the time factor is excluded. So, if the apparent rate constant for initial rate is given the symbol of \(k_{I R M}\) , to get the real value of \(k_{A}, k_{I R M}\) must be divided by (\(1-\beta\) ) and crossed with general form of \(I M, F(\beta)\). This is given in the following equation:

(27) \(k_{A}=\left( ^{F(\beta)} {\mskip -5mu/\mskip -3mu} _{ (1-\beta)} \right) k_{I R M}\)

or

(28) \(k_{I R M}=\left(^{(1-\beta)} {\mskip -5mu/\mskip -3mu} _{F(\beta)}\right) k_{A}\)

The first expression on the right side of Eq. (27) is the correction factor (\(C F\) ) to obtain \(k_{A}\) from \(k_{I R M}\). Conversely, the first expression on the right side of eq. (28) is the \(C F\) to obtain \(k_{I R M}\) from \(k_{A}\).

The integrated form of \(F(\beta)\) is obtained the following way. Generally, if, in a homogeneous reaction,

(29) \[ \mathrm{a} A+\ldots \ldots \rightarrow \mathrm{p} S+\ldots \ldots \]

the concentration of reactant \(A\) is made much less than the concentration of other species, the rate \(v\) at time \(t\) is experimentally found to be

(30) \[ \begin{gathered} \mathrm{v}=-1 / \mathrm{ad}[\mathrm{~A}] / \mathrm{dt}=\mathrm{k}[\mathrm{~A}]^{\mathrm{n}} \\ \mathrm{v}_{\mathrm{A}}=-\mathrm{d}[\mathrm{~A}] / \mathrm{dt}=\mathrm{k}_{\mathrm{A}}[\mathrm{~A}]^{\mathrm{n}} \end{gathered} \]

Dividing Eq. (31) with \(\left[A_{o}\right]\) gives

(32) \[ -\tfrac{d\left[[A] /[A]_{o}\right]}{d t}=k_{A}[A]_{o}^{(n-1)}\left[[A] /[A]_{o}\right]^{n} \]

Introducing Eq. (17) to Eq. (32) offers

(33) \(-d \beta / d t =k_{A}\left[A_{o}\right]^{(n-1)} \beta^{n} \)

\(-d\left(\tfrac{\beta}{\beta^{n}}\right) =k_{A}\left[A_{o}\right]^{(n-1)} d t \)

(34) \(-\int d\left(\tfrac{\beta}{\beta^{n}}\right) =k_{A}\left[A_{o}\right]^{(n-1)} \int d t \)

The general integral of the term in the left side of Eq. (34) is introduced as \(F\) (\(\beta\) ) or

(35) \[ F(\beta)=k_{A}\left[A_{o}\right]^{(n-1)} t_{\beta} \]

There are two solutions for \(F(\beta)\) :

(36) \[ F(\beta)=-\mathrm{h} \beta \text { for } \mathrm{n}=1 \]

and

(37) \[ F(\beta)=\tfrac{\left[A_{o}\right]^{(1-n)}}{(n-1)}\left[\beta^{(1-n)}-1\right] \text { for } \mathrm{n} \neq 1 \]

The difference between the IRM method and NM is in the determination of the rate constant. The constant for \(N M\) is determined by variance analysis. The value obtained is then multiplied with their corresponding \(C F\) (Eq. (36) or Eq. (37)):

(38) \[ k_{A=} k_{N M} \text { for } \mathrm{n}=0 \]

or,

(39) \[ k_{A}=k_{N M}\left[\tfrac{-\ln \beta}{(1-\beta)}\right] \quad \text { for } \mathrm{n}=1 \]

In this study \((\beta)=0.9\). Because the unit in both Table 1 and Table 2 is in absorbance then, in the unit of \(M\) (molarity), Eq. (38) becomes

(40) \[ k_{A=} k_{N M} / 160=k_{N M} 6.25 \times 10^{-3} M s^{-1} \text { for } \mathrm{n}=0 \]

and Eq. (39) becomes

(41) \[ k_{A}=k_{N M}\left[\tfrac{-\ln 0.9}{(0.1)}\right] s^{-1}=k_{N M} \times 1.0536 s^{-1} \quad \text { for } \mathrm{n}=1 \]

The \(N M\) is claimed valid if the \(E P\) of \(k_{A}\) values is lower than \(5 \%\) and reliable if its \(E P\) is lower than other methods.

Description

Before entering the discussion, the following definitions are required. The initial rate method is the rate that depends on the initial concentration of the reagent. The conventional initial rate method (IRM) is the method that using the plot of log initial rate versus initial concentration for the determination of \(n\) and \(k\) or by using Eq. (16). The CM is the method used in the main reference. The new method (\(N M\) ) is the method that using correction factor, Eq. (38) or Eq. (39) in the calculation. This method uses variance analysis for the determination of \(n\) and \(k\). The true order is that with lower deviation standard.

The results of all calculations are presented in Tables 3 and 4. In all tables, the notation \(r\) is the regression coefficient and \(\boldsymbol{X}^{\sigma n}\) is the deviation standard for variance analysis which is introduced as NM.

There are two interesting points in Table 3. First, IM, IRM, and NM, show that, when the initial \(\left[\mathrm{Br}_{2}\right]_{\mathrm{o}}\) is greater than \(\left(k_{-1} / k_{2}\right)\left[H^{+}\right]\), the reaction is of the first-order with respect to \(B r_{2}\). By IM, the reaction is first-order (its regression coefficient is higher than other-order) with \(k_{1}=7.08 \times 10^{-3} M s^{-1}\). By. IRM, the order is \(1.03923 \approx 1=\) first-order. By \(N M\), it is also first-order (lower deviation standard) with \(k_{1}=7.11 \times 10^{-3} M s^{-1}\).

Second, by IRM, it is just assumed as first-order. This is because nearer to 1.00 with \(k_{1}=7.56 \times 10^{-3} M s^{-1}\). By \(C M\), there are two possible values of \(k\), but the first is meaningless because \(k\) must have positive value. Then, by Eq. (9), the reaction is of the first-order with respect to \(\mathrm{Br}_{2}\) with \(k_{1}=7.34 \times 10^{-3} \mathrm{M} \mathrm{s}^{-1}\).

There are also two interesting points in Table 4. First, IM, IRM, and NM, show that, when the initial \(\left[\mathrm{Br}_{2}\right]_{\mathrm{o}}\) is lower than \(\left(k_{-1} / k_{2}\right)\left[\mathrm{H}^{+}\right]\), the reaction is of the zerothorder with respect to \(\mathrm{Br}_{2}\). By IM, the reaction is zeroth-order (its regression coefficient is higher than \(o\) ther-order) with \(k_{2}=6.25 \times 10^{-7} M s^{-1}\). . By IRM, the order is \(-0.01361 \approx 0=\) zeroth-order with \(k_{2}=5.99 \times 10^{-7} M s^{-1}\). By NM, it is also zeroth-order (lower deviation standard) \(k_{1}=6.38 \times 10^{-7} M s^{-1}\).

Second, by IRM, it is also just assumed as zeroth-order. This is because nearer to 0.00. By \(C M\), there are also two possible values of \(k\), but the second is meaningless because \(k\) must have positive value. Then by Eq. (7), \(k_{2}=6.38 \times 10^{-7} M s^{-1}\).

Note: \(A\) is the combination of the first 3 data of Table 1 and the first 3 data of Table 2.

\(B\) is the combination of the first 3 data of Table 1 and the last 3 data of Table 2.

\(C\) is the combination of the last 3 data of Table 1 and the first 3 data of Table 2.

\(A\) is the combination of the last 3 data of Table 1 and the last 3 data of Table 2.

All combinations give two possible values of \(k\) but one of them are meaningless.

Discussion

The first goal

By Eq. (21), for \(\left(\left[A_{o}\right]-\left[A_{l}\right]\right)\) in Table 2 (for zeroth-order) gives \(k_{A-1}=\{(1-0.9)\) \(\times 0.050\} / 50=1.00 \times 10^{-4} A s^{-1} / 160=6.25 \times 10^{-7} M s^{-1}\). By Eq. (20), for (\(\left[A_{o}\right]-\left[A_{2}\right]\) ) gives \(k_{A-2}=\left(1-0.9^{2}\right) \times 0.050 / 90=1.06 \times 10^{-4} A \mathrm{~s}^{-1} / 160=6.60 \times 10^{-7} M \mathrm{~s}^{-1}\). Their error percentage \(=\left[\left(6.60 \times 10^{-7}-6.25 \times 10^{-7}\right): 6.25 \times 10^{-7} \times 100 \%\right]=5.6 \%\). The error percentage is large. But this is not because the theory is wrong. This is an experimental error. By Eq. (18) \(t_{\beta-2}\) should be 45, then \(t_{\beta}=50+45=95 s\) and \(k_{A-2}=\left(1-0.9^{2}\right) \mathrm{x}\) \(0.050 / 95=1.00 \times 10^{-4} A \mathrm{~s}^{-1} / 160=6.25 \times 10^{-7} M \mathrm{~s}^{-1}\). The error percentage is \(0.00 \%\). Different fraction gives the same rate constant.

By Eq. (18), \(t_{\beta-1}=t_{\beta-2}\) and in Table \(1=15 s\). By Eq. (21), for first-order \(=\left(\left[A_{o}\right]\right.\) \(\left.-\left[A_{1}\right]\right)\) in Table 2 gives \(k_{A-1}=\{(1-0.9)\} / 15=6.67 \times 10^{-3} s^{-1}\). For \(\left(\left[A_{o}\right]-\left[A_{2}\right]\right)\) gives \(k_{A-2}=\left\{\left(1-0.9^{2}\right)\right\} / 30=6.33 \times 10^{-3} s^{-1}\). Their error percentage \(=\left[\left(6.66 \times 10^{-3}-6.33 \times 10^{-3}\right)\right.\) : \(\left.6.66 \times 10^{-3} \times 100 \%\right]=4.95 \%\). It is also large. But by Eq. (40), \(k_{A-1}=6.66 \times 10^{-3} \times(-\ln\) \(0.9 / 0.1)=7.02 \times 10^{-3}\) and \(k_{A-2}=6.33 \times 10^{-3} s^{-1} \times(-\ln 0.81 / 0.19)=7.02 \times 10^{-3}\). The \(E P\) is \(0.00 \%\). The same rate constant can be obtained at different fractions.

Due to the correction factor, by \(N M\) the value of error percentage are the same, including for first-order reaction. Thus, it is not a necessity to restrict the percentage of reaction followed to maximum \(5 \%\).

The model used in this study is slightly different from that of the main reference. However, by the same principle, both discuss the reaction that takes place in two different reaction mechanisms. Based on Table 3 and Table 4 the CM always gives twovalues of \(k\) but one of them is always negative.

In Table 3, the slope is positive which means the reaction is of the first-order. By Eq. (9), this value is from \(k_{1}\). The intercept is negative which thus the \(k_{2}\) value cannot be obtained; a constant must have positive value.

In Table 4 the intercept is positive which means the reaction is of the zerothorder. By Eq. (9), this value is the value of \(k_{2}\). This is in accordance with the theory: if \([C] \rightarrow 0\), plotting the intercept of \(v_{o}\) v versus \([C]\) gives the value of \(k\) of zeroth-order reaction and so must be positive then \(v_{o}=k_{2}\).

To be valid, \(C M\) should be able to give simultaneous value of \(k_{1}\) and \(k_{2}\) for every combination. In Table 5, there are two values, but once again, only one in each of them are positive; that is the slope. By Eq. (9) and in our model, this gives the value of \(k_{1}\). And, differ with that in Table 3. and Table 4 all with large \(E P\). One of the possible reason is that the data are not at the same level of concentration. But, whatsoever, the \(C M\) is not always suitable for data analysis of reaction with 2 successive mechanisms.

The second goal

There is a fundamental premise difference in the determination of a rate law by IM and by conventional IRM. By IM, the premise is that, for an experimental data, there must be a most probably fixed rate law. In contrast, in conventional IRM, the results of data analysis using Eq. (9) will give a certain order and its rate constant value. If the experiment is done correctly then the results of both methods (\(k_{A}\) and \(k_{\text {IRM }}\) ) will be attributable. For example, if from the calculation by Eq. (9), the order obtained, let say, 0.875, then it will be assumed to be first-order. This is because the order reaction usually integer or half-integer and 0.875 is nearer 1 than 0.5. The value of the calculated rate constant is the value for the \(0.875^{\text {th }}\)-order and not for first-order (as in IM). The greater the order difference (with that from \(I M\) ) the greater the difference. Thus, for this reason, most chemists recommend the need to check the IRM value.

There is an alternative way of determining the order. That is by variance analysis; and introduced as the new method: the NM. The spirit is resembled with IM. Both give the most possible rate law.

It is interesting to compare the values obtained by IRM, CM, and NM. In Table 3, the \(I R M\) gives \(E P=6.78 \%, C M\) gives \(E P=3.67 \%\), and \(N M=0.42 \%\) for \(k_{r}\). In Table 4, the \(I R M\) gives \(E P=4.16 \%, C M\) gives \(E P=2.08 \%\), and \(N M=0.00 \%\) for \(k_{2}\). The \(E P\) for \(N M\) are always lower than \(5 \%\). Then, it can be concluded that of the 3 models, the \(N M\) is the best. These values are obtained after using their correspondence correction factor; However, by Eq. (38) or Eq. (39), the value \(k_{\text {IRM }}\) can be determined if \(k_{A}\) is known. That is to say, that any \(k_{\text {IRM }}\) value is fixed. There is no need of \(I M\) checks for \(k\).

However, all of the above conclusions apply only to certain conditions; at the out-set or at the nearly-end of the reaction. Thus, for the reaction which takes place in two reaction mechanisms the IRM, CM, or CN cannot be used, in one step, to determine the rate constant. The initial rate method cannot be used to determine both constants simultaneously. Compared with \(I M\), this method is also impractical because rate is the difference of to concentrations. But, by IRM, a rate law can be determined directly from the change of product concentration.

Implementation

There are two cases that can be discussed further as the possible implication of this research.

First case

It is about enzymatic reaction which has the rate law:

(42) \[ v=\tfrac{v_{\max }[S]}{K_{M}+[S]} \]

where \(v, v_{\text {max }},[S]\), and \(K_{M}\) are the rate, the maximum rate, the substrate concentration, and the Michaelis-Menten constant, respectively (Michaelis & Menten, 1913). Mostly, \(K_{M}\) is determined by \(\operatorname{IRM}\) (Briggs & Haldane, 1925).

Theoretically, enzymatic reaction takes place in 2 successive reaction mechanisms. If the substrate concentration, \([S] \gg K_{M}\), the reaction is zeroth-order relating to \([\mathrm{S}]\). However, as the reaction proceed, \([S]\) will decrease and because \(K_{M}\) constant then, \([S] \ll K_{M}\), and the reaction becomes first-order with respect to [S]. That is to say, the reaction takes place in 2 successive mechanisms.

There are two equations that most commonly used in the determination of \(K_{M}\) : the Lineweaver-Burk equation and the Eadie–Hofstee equation (Lineweaver & Burk, 1934; Eadie, 1942; Hofstee, 1959). Both equations derived from eq. (42) and use \(I R M\) for \(K_{M}\) determination but, mostly give different values. Some even claimed that the second is better and superior than the first (Dowd & Riggs, 1965).

The first problem: there is no explanation of the source of the data used for the calculation, from the condition when the reaction is zeroth-order, first-order or mix. It is stand to reason that using data from one condition will not represent the reaction as whole. The second is that eq. (42) is resemble with Langmuir isotherm and the Lineweaver-Burk equation is very much resembled with Langmuir equation: which is unreliable for \(K_{d}\) d determination (Patiha et al., 2016).

Second case

As with \(I M\), the introduced method is impractical. Unified integral equations have been introduced (Tan et al., 1994). An improvement was also given (Patiha, 1998). Because, by the \(N M\) is the value of \(k\) is determined by variance analysis then it is impractical. Unified \(N M\) is open to be invented. It is possible to invent a new IRM modification formula for DM, with which, the true order and the accurate value of apparent rate constant obtained will always be the same with that from integral method.

Conclusions

The IRM is not suitable for simultaneous determination of constants of reaction which take places in 2 mechanisms. There is no necessary to restrict the percentage of reaction followed to a maximum of \(5 \%\). The \(C M\) cannot be (always) used to simultaneously determine the two constants of the reaction that takes place in two mechanisms. The \(N M\) is valid and more reliable and there is no need of \(I M\) checks for \(k\).

Recommendations

It is recommended to invent a new, valid, and reliable method for determining \(K_{M}\) of M Michaelis-Menten equation and to derive a unified form of \(D M\) equation, which does not require \(I M\) checks.

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