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I was dissatisfied with the method my teacher taught me to solve rate equations, where you're given experimental values for the concentrations of reactants, and the rate of reaction, which given by $$r = k\,[\ce{A}]^n\,[\ce{B}]^m\,\ldots,\tag{1}$$

as I thought the method wasn't very computational. Instead I rearranged the system of equations and was able to create a general formula to solve using matrices for any reaction

$$\ce{R_1 + R_2 + \ldots + R_n -> P[ ]}\tag{R1}$$ $$r = k\,[\ce{R_1}]^{N_1}\,[\ce{R_2}]^{N_2}\,\ldots\,[\ce{R_n}]^{N_n}\tag{2}$$

$$ \begin{pmatrix} \ln k \\ N_1 \\ N_2 \\ \vdots \\ N_n \end{pmatrix} = \begin{pmatrix} 1 & \ln_1R_1 & \ln_1R_2 & \cdots & \ln_nR_n \\ 1 & \ln_2R_1 & \ln_2R_2 & \cdots & \ln_nR_n \\ 1 & \ln_3R_1 & \ln_3R_2 & \cdots & \ln_nR_n \\ \vdots & & & \ddots & \\ 1 & \ln_nR_1 & \ln_nR_2 & \cdots & \ln_nR_n \\ \end{pmatrix}^{-1} \begin{pmatrix} r_1 \\ r_2 \\ r_3 \\ \vdots \\ r_n \end{pmatrix}\tag{3} $$

I'm wondering if this was an original formula for solving rate equations, and if the formula is obsolete as there must be different equations for rate of reaction dependent on temperature and pressure, which are just reduced to the constant $k$ in this equation.

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    $\begingroup$ No, these are $r_i$. $\endgroup$
    – Ivan Neretin
    Nov 7 at 1:03
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    $\begingroup$ Chem+Math Expression formatting reference: MathJax Basics / Chem+Math expressions/formulas/equations / Upright vs italic / Math SE Mathjax tutorial // MathJax is preferred not to be used in CH SE Q titles. $\endgroup$
    – Poutnik
    Nov 7 at 2:50
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    $\begingroup$ OK what are the $r_i$? You define the version without subscript, but not the one with. Also showing your derivation would help $\endgroup$
    – Ian Bush
    Nov 7 at 7:56
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    $\begingroup$ I tried my best to convert text from the image, but please do feel free to check notations as well as define them. $\endgroup$
    – andselisk ♦
    Nov 7 at 8:06
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    $\begingroup$ Also $ln_n$, what is meant by that? $\endgroup$
    – Ian Bush
    Nov 7 at 16:45

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