deletions | additions       diff --git a/The_consecutive_two_step_reversible__.tex b/The_consecutive_two_step_reversible__.tex  index 29dd9cc..6a5d5d0 100644  --- a/The_consecutive_two_step_reversible__.tex  +++ b/The_consecutive_two_step_reversible__.tex 
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\begin{equation}  \frac{d[C]}{dt} = k_2[A]-k_{-2}[C]  \end{equation}   Note that, for the purposes of this analysis, the inter-conversion of the major isomers is ignored. This is a valid assumption if the rate of inter-conversion of $A $B  \longleftrightarrow C$ is small. We estimate from cursory analysis of the data that the rate constants for this process are at least two orders of magnitude smaller than the pathway that proceeds through the minor isomer. However, a steady-state approximation for the system of rate equations is not valid for the minor isomer because its concentration changes appreciably and is not significantly different from those concentrations of either major isomer. Thus, the differential equations in (2)-(4) are solved explicitly. The exact solutions to these equations are:  \begin{equation}  A(t) B(t)  = A_0[\frac{k_{-1}k_{-2}}{mn}+\frac{k_1(m-k_2-k_{-2})}{m(m-n)}\mathrm{e}^{-mt}+\frac{k_1(k_2+k_{-2}-n)}{n(m-n)}\mathrm{e}^{-nt}] B_0[\frac{k_{-1}k_{-2}}{mn}+\frac{k_1(m-k_2-k_{-2})}{m(m-n)}\mathrm{e}^{-mt}+\frac{k_1(k_2+k_{-2}-n)}{n(m-n)}\mathrm{e}^{-nt}]  \\ \end{equation}   \begin{equation}  B(t) A(t)  = A_0[\frac{k_{1}k_{-2}}{mn}-\frac{k_1(k_{-2}-m)}{m(m-n)}\mathrm{e}^{-mt}-\frac{k_1(k_{-2}-n)}{n(m-n)}\mathrm{e}^{-nt}] B_0[\frac{k_{1}k_{-2}}{mn}-\frac{k_1(k_{-2}-m)}{m(m-n)}\mathrm{e}^{-mt}-\frac{k_1(k_{-2}-n)}{n(m-n)}\mathrm{e}^{-nt}]  \\ \end{equation}   \begin{equation}  C(t) = A_0[\frac{k_{1}k_{2}}{mn}-\frac{k_1k_{2}}{m(m-n)}\mathrm{e}^{-mt}+\frac{k_1k_{2}}{n(m-n)}\mathrm{e}^{-nt}] B_0[\frac{k_{1}k_{2}}{mn}-\frac{k_1k_{2}}{m(m-n)}\mathrm{e}^{-mt}+\frac{k_1k_{2}}{n(m-n)}\mathrm{e}^{-nt}]  \end{equation}  Where,  \begin{equation} 
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\begin{equation}  q = (p^2-4(k_1k_2+k_{-1}k_{-2}+k_1k_{-2}))^\frac{1}{2}  \end{equation}  The values of \textit{m} and \textit{n} generally differ by less than an order of magnitude within the range of rate constant values expected for Reaction (1). Thus, fits to time-dependent data in the form of $y(t) = y_0+\Lambda\mathrm{e}^{-\lambda{}t}+K\mathrm{e}^{-\kappa{}t}$ tend to converge to the form $y(t) = y_0+\alpha{}\mathrm{e}^{-\beta{}t}$. We take advantage of this to approximate $\textit{k}_1$ by fitting data for major  isomer \textit{A} \textit{B}  with a single exponential such that equation (5) may be rearranged as: \begin{equation}  A(t) B(t)  = A_0[\frac{k_{-1}k_{-2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}+[\frac{k_1k_2+k_1k_{-2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}]\mathrm{e}^{-x_1t}] B_0[\frac{k_{-1}k_{-2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}+[\frac{k_1k_2+k_1k_{-2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}]\mathrm{e}^{-x_1t}]  \end{equation}  Similarly, the integrated rate laws for isomers B A  and C may be rewritten as: \begin{equation}  B(t) A(t)  = A_0[\frac{k_{1}k_{-2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}+[\frac{k_1k_{-2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}]\mathrm{e}^{-x_2t}]\\ B_0[\frac{k_{1}k_{-2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}+[\frac{k_1k_{-2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}]\mathrm{e}^{-x_2t}]\\  \end{equation}   \begin{equation}  C(t) = A_0[\frac{k_{1}k_{2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}+[\frac{k_1k_{2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}]\mathrm{e}^{-x_3t}] B_0[\frac{k_{1}k_{2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}+[\frac{k_1k_{2}}{k_1k_2+k_{-1}k_{-2}+k_1k_{-2}}]\mathrm{e}^{-x_3t}]  \end{equation}  Note that in equations (12) and (14), $x_1$ and $x_3$ are roughly equivalent to $k_1$ and $k_{-2}$, respectively. $x_2$ does not have an obvious analogue. At equilibrium, for a normalized data set, the following relations also hold:   \begin{equation}