\documentclass[a4paper,10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{breqn}
\usepackage[colorlinks=false]{hyperref}
\newcommand{\pluseq}{\mathrel{{+}{=}}}
\newcommand{\minuseq}{\mathrel{{-}{=}}}
\newcommand{\ns}{N_{sp}}
\newcommand{\nr}{N_{reac}}
\newcommand{\Ru}{\mathcal{R}}
\begin{document}
\section{State Variables}
\begin{dmath} [C]_{k} = \frac{n_{k}}{V}\end{dmath}
\begin{dmath} \Phi = \left\{T,P,n[1],n[2]\ldots n[-1 + Ns()]\right\}\end{dmath}
\begin{dmath} \frac{\text{d} \Phi }{\text{d} t } = \left\{\frac{\text{d} T }{\text{d} t },\frac{\text{d} P }{\text{d} t },\frac{\text{d} n }{\text{d} t }[1],\frac{\text{d} n }{\text{d} t }[2]\ldots \frac{\text{d} n }{\text{d} t }[-1 + Ns()]\right\}\end{dmath}
\section{Source Terms}
\begin{dmath} \frac{\text{d} n }{\text{d} t }_{k} = V \dot{\omega}_{k}\end{dmath}
\begin{dmath} \frac{\text{d} T }{\text{d} t } = - \frac{\sum_{k=1}^{\ns} U_{k} \dot{\omega}_{k}}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}}\end{dmath}
From conservation of mass:
\begin{dmath} m = \sum_{k=1}^{\ns} W_{k} n_{k}\end{dmath}
\begin{dmath} 0 = \sum_{k=1}^{\ns} W_{k} \frac{\text{d} n }{\text{d} t }_{k}\end{dmath}
\begin{dmath} \frac{\text{d} n }{\text{d} t }_{\ns} = - \frac{1}{W_{\ns}} \sum_{k=1}^{-1 + \ns} W_{k} \frac{\text{d} n }{\text{d} t }_{k}\end{dmath}
\begin{dmath} n = \frac{P V}{T \Ru}\end{dmath}
Thus...\begin{dmath} \dot{\omega}_{\ns} = - \frac{1}{W_{\ns}} \sum_{k=1}^{-1 + \ns} W_{k} \dot{\omega}_{k}\end{dmath}
And...\begin{dmath} \frac{\text{d} T }{\text{d} t } = - \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k}\end{dmath}
\begin{dmath} \frac{\text{d} n }{\text{d} t } = \sum_{k=1}^{\ns} \frac{\text{d} n }{\text{d} t }_{k}\end{dmath}
\begin{dmath} \frac{\text{d} n }{\text{d} t } = \sum_{k=1}^{-1 + \ns} \left(1 - \frac{W_{k}}{W_{\ns}}\right) \frac{\text{d} n }{\text{d} t }_{k}\end{dmath}
From the ideal gas law:
\begin{dmath} \frac{\text{d} P }{\text{d} t } = \frac{\Ru}{V} \left(T \frac{\text{d} n }{\text{d} t } + \frac{\text{d} T }{\text{d} t } n\right)\end{dmath}
\begin{dmath} \frac{\text{d} P }{\text{d} t } = \frac{P}{T} \frac{\text{d} T }{\text{d} t } + T \Ru \sum_{k=1}^{-1 + \ns} \left(1 - \frac{W_{k}}{W_{\ns}}\right) \dot{\omega}_{k}\end{dmath}
\subsection{Other defns}
\begin{dmath} [C] = \frac{P}{T \Ru}\end{dmath}
\begin{dmath} \frac{\text{d} P }{\text{d} t } = \frac{P}{T} \frac{\text{d} T }{\text{d} t } + T \Ru \sum_{k=1}^{-1 + \ns} \left(1 - \frac{W_{k}}{W_{\ns}}\right) \dot{\omega}_{k}\end{dmath}
\begin{dmath} [C]_{\ns} = [C] - \sum_{k=1}^{-1 + \ns} [C]_{k}\end{dmath}
\begin{dmath} [C]_{\ns} = \frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} [C]_{k}\end{dmath}
\begin{dmath} W = \sum_{k=1}^{\ns} W_{k} X_{k}\end{dmath}
\begin{dmath} W = \frac{1}{[C]} \sum_{k=1}^{\ns} W_{k} [C]_{k}\end{dmath}
\begin{dmath} [C]_{\ns} = \frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} [C]_{k}\end{dmath}
\begin{dmath} W = \frac{1}{[C]} \left(\left([C] - \sum_{k=1}^{-1 + \ns} [C]_{k}\right) W_{\ns} + \sum_{k=1}^{-1 + \ns} W_{k} [C]_{k}\right)\end{dmath}
\begin{dmath} W = W_{\ns} + \frac{1}{[C]} \sum_{k=1}^{-1 + \ns} \left(- W_{\ns} + W_{k}\right) [C]_{k}\end{dmath}
\section{Thermo Definitions}
\begin{dmath} {C_{p,k}}^{\circ} = {C_p}_{k}\end{dmath}
\begin{dmath} {C_p}_{k} = \Ru \left(T \left(T \left(T \left(T a_{k,4} + a_{k,3}\right) + a_{k,2}\right) + a_{k,1}\right) + a_{k,0}\right)\end{dmath}
\begin{dmath} {C_p}_{k} = T^{4} \Ru a_{k,4} + T^{3} \Ru a_{k,3} + T^{2} \Ru a_{k,2} + T \Ru a_{k,1} + \Ru a_{k,0}\end{dmath}
\begin{dmath} \frac{\text{d} {C_p} }{\text{d} T }_{k} = \Ru \left(4 T^{3} a_{k,4} + 3 T^{2} a_{k,3} + 2 T a_{k,2} + a_{k,1}\right)\end{dmath}
\begin{dmath} \frac{\text{d} {C_p} }{\text{d} T }_{k} = \Ru \left(T \left(T \left(4 T a_{k,4} + 3 a_{k,3}\right) + 2 a_{k,2}\right) + a_{k,1}\right)\end{dmath}
\begin{dmath} \bar{c_p} = \sum_{k=1}^{\ns} \frac{n_{k} {C_p}_{k}}{n}\end{dmath}
\begin{dmath} {C_{v,k}}^{\circ} = {C_v}_{k}\end{dmath}
\begin{dmath} {C_v}_{k} = \Ru \left(T \left(T \left(T \left(T a_{k,4} + a_{k,3}\right) + a_{k,2}\right) + a_{k,1}\right) + a_{k,0} - 1\right)\end{dmath}
\begin{dmath} {C_v}_{k} = T^{4} \Ru a_{k,4} + T^{3} \Ru a_{k,3} + T^{2} \Ru a_{k,2} + T \Ru a_{k,1} + \Ru a_{k,0} - \Ru\end{dmath}
\begin{dmath} \frac{\text{d} {C_v} }{\text{d} T }_{k} = \Ru \left(4 T^{3} a_{k,4} + 3 T^{2} a_{k,3} + 2 T a_{k,2} + a_{k,1}\right)\end{dmath}
\begin{dmath} \frac{\text{d} {C_v} }{\text{d} T }_{k} = \Ru \left(T \left(T \left(4 T a_{k,4} + 3 a_{k,3}\right) + 2 a_{k,2}\right) + a_{k,1}\right)\end{dmath}
\begin{dmath} \bar{c_v} = \sum_{k=1}^{\ns} \frac{n_{k} {C_v}_{k}}{n}\end{dmath}
\begin{dmath} H_k^{\circ} = H_{k}\end{dmath}
\begin{dmath} H_{k} = \Ru \left(T \left(T \left(T \left(T \left(\frac{T a_{k,4}}{5} + \frac{a_{k,3}}{4}\right) + \frac{a_{k,2}}{3}\right) + \frac{a_{k,1}}{2}\right) + a_{k,0}\right) + a_{k,5}\right)\end{dmath}
\begin{dmath} H_{k} = \frac{T^{5} a_{k,4}}{5} \Ru + \frac{T^{4} a_{k,3}}{4} \Ru + \frac{T^{3} a_{k,2}}{3} \Ru + \frac{T^{2} a_{k,1}}{2} \Ru + T \Ru a_{k,0} + \Ru a_{k,5}\end{dmath}
\begin{dmath} \frac{\text{d} H }{\text{d} T }_{k} = \Ru \left(T \left(T \left(T \left(T a_{k,4} + a_{k,3}\right) + a_{k,2}\right) + a_{k,1}\right) + a_{k,0}\right)\end{dmath}
\begin{dmath} H_k = U_k + \frac{P V}{n}\end{dmath}
\begin{dmath} U_{k} = - T \Ru + H_{k}\end{dmath}
\begin{dmath} U_{k} = \Ru \left(T \left(T \left(T \left(T \left(\frac{T a_{k,4}}{5} + \frac{a_{k,3}}{4}\right) + \frac{a_{k,2}}{3}\right) + \frac{a_{k,1}}{2}\right) + a_{k,0}\right) - T + a_{k,5}\right)\end{dmath}
\begin{dmath} \frac{\text{d} U }{\text{d} T }_{k} = \Ru \left(T \left(T \left(T \left(T a_{k,4} + a_{k,3}\right) + a_{k,2}\right) + a_{k,1}\right) + a_{k,0} - 1\right)\end{dmath}
\begin{dmath} S_k^{\circ} = S_{k} = \Ru \left(T \left(T \left(T \left(\frac{T a_{k,4}}{4} + \frac{a_{k,3}}{3}\right) + \frac{a_{k,2}}{2}\right) + a_{k,1}\right) + \log{\left (T \right )} a_{k,0} + a_{k,6}\right)\end{dmath}
\section{Definitions}
\begin{dmath} \nu_{k,i} = \nu^{\prime\prime}_{k,i} - \nu^{\prime}_{k,i}\end{dmath}
\begin{dmath} \dot{\omega}_{k} = \sum_{i=1}^{\nr} \nu_{k,i} q_{i}\end{dmath}
\begin{dmath} q_{i} = R_{i} c_{i}\end{dmath}
\begin{dmath} \dot{\omega}_{k} = \sum_{i=1}^{\nr} \nu_{k,i} R_{i} c_{i}\end{dmath}
\section{Rate of Progress}
\begin{dmath} R_{i} = {R_f}_{i} - {R_r}_{i}\end{dmath}
\begin{dmath} {R_f}_{i} = {k_f}_{i} \prod_{k=1}^{\ns} [C]_{k}^{\nu^{\prime}_{k,i}}\end{dmath}
\begin{dmath} {R_r}_{i} = {k_r}_{i} \prod_{k=1}^{\ns} [C]_{k}^{\nu^{\prime\prime}_{k,i}}\end{dmath}
\section{Third-body effect}
\begin{dmath} c_{i}=1\text{\quad for elementary reactions}\end{dmath}
\begin{dmath} c_{i}=[X]_{i}\text{\quad for third-body enhanced reactions}\end{dmath}
\begin{dmath} c_{i}=\frac{F_{i} P_{r, i}}{P_{r, i} + 1}\text{\quad for unimolecular/recombination falloff reactions}\end{dmath}
\begin{dmath} c_{i}=\frac{F_{i}}{P_{r, i} + 1}\text{\quad for chemically-activated bimolecular reactions}\end{dmath}
\section{Forward Reaction Rate}
\begin{dmath} {k_f}_{i} = T^{\beta_{i}} \operatorname{exp}\left({- \frac{{E_{a}}_{i}}{T \Ru}}\right) A_{i}\end{dmath}
\section{Equilibrium Constants}
\begin{dmath} {K_c}_{i} = \left(\left(\frac{P_{atm}}{T \Ru}\right)^{\sum_{k=1}^{\ns} \nu_{k,i}}\right) {K_p}_{i}\end{dmath}
\begin{dmath} {K_p}_{i} = \text{exp}(\frac{\Delta S^{\circ}_k}{\Ru} - \frac{\Delta H^{\circ}_k}{\Ru T})\end{dmath}
\begin{dmath} {K_p}_{i} = \text{exp}\left(\sum_{k=1}^{\ns}\nu_{ki}\left(\frac{S^{\circ}_k}{\Ru} - \frac{H^{\circ}_k}{\Ru T}\right)\right)\end{dmath}
\begin{dmath} {K_c}_{i} = \left(\left(\frac{P_{atm}}{\Ru}\right)^{\sum_{k=1}^{\ns} \nu_{k,i}}\right) \operatorname{exp}\left({\sum_{k=1}^{\ns} \nu_{k,i} B_{k}}\right)\end{dmath}
\begin{dmath} B_{k}= \frac{S^{\circ}_k}{\Ru} - \frac{H^{\circ}_k}{\Ru T} - ln(T)\end{dmath}
\begin{dmath} B_{k} = T \left(T \left(T \left(\frac{T a_{k,4}}{20} + \frac{a_{k,3}}{12}\right) + \frac{a_{k,2}}{6}\right) + \frac{a_{k,1}}{2}\right) + \left(a_{k,0} - 1\right) \log{\left (T \right )} - a_{k,0} + a_{k,6} - \frac{a_{k,5}}{T}\end{dmath}
\section{Reverse Reaction Rate}
\begin{dmath} {k_r}_{i}=\frac{{k_f}_{i}}{{K_c}_{i}}\text{\quad if non-explicit}\end{dmath}
\begin{dmath} {R_r}_{i}=T^{{\beta_r}_{i}} \operatorname{exp}\left({- \frac{{E_{a,r}}_{i}}{T \Ru}}\right) {A_{r}}_{i} \prod_{k=1}^{\ns} [C]_{k}^{\nu^{\prime\prime}_{k,i}}\text{\quad if explicit}\end{dmath}
\section{Third-Body Efficiencies}
\begin{dmath} [X]_{i} = \sum_{k=1}^{\ns} \alpha_{k,i} [C]_{k}\end{dmath}
\begin{dmath} [X]_{i} = [C] + \sum_{k=1}^{\ns} \left(\alpha_{k,i} - 1\right) [C]_{k}\end{dmath}
\begin{dmath} [X]_{i} = [C] + \left(\frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} [C]_{k}\right) \left(\alpha_{\ns,i} - 1\right) + \sum_{k=1}^{-1 + \ns} \left(\alpha_{k,i} - 1\right) [C]_{k}\end{dmath}
\begin{dmath} [X]_{i}=[C] \alpha_{\ns,i} + \sum_{k=1}^{-1 + \ns} \left(- \alpha_{\ns,i} + \alpha_{k,i}\right) [C]_{k}\text{\quad for mixture as third-body}\end{dmath}
\begin{dmath} [X]_{i}=[C]\text{\quad for all $\alpha_{ki} = 1$}\end{dmath}
\begin{dmath} [X]_{i}=\left([C] - \sum_{k=1}^{-1 + \ns} [C]_{k}\right) \delta_{\ns m} + \left(- \delta_{\ns m} + 1\right) [C]_{m}\text{\quad for a single species third-body}\end{dmath}
\section{Falloff Reactions}
\begin{dmath} k_{0, i} = T^{\beta_0} A_{0} \operatorname{exp}\left({- \frac{E_{a, 0}}{T \Ru}}\right)\end{dmath}
\begin{dmath} k_{\infty, i} = T^{\beta_{\infty}} A_{\infty} \operatorname{exp}\left({- \frac{E_{a, \infty}}{T \Ru}}\right)\end{dmath}
\begin{dmath} P_{r, i}=\frac{[X]_{i} k_{0, i}}{k_{\infty, i}}\text{\quad for the mixture as the third-body}\end{dmath}
\begin{dmath} P_{r, i}=\frac{k_{0, i}}{k_{\infty, i}} \left(\left([C] - \sum_{k=1}^{-1 + \ns} [C]_{k}\right) \delta_{\ns m} + \left(- \delta_{\ns m} + 1\right) [C]_{m}\right)\text{\quad for species $m$ as the third-body}\end{dmath}
\begin{dmath} P_{r, i}=\frac{[C] k_{0, i}}{k_{\infty, i}}\text{\quad for for all $\alpha_{i, j} = 1$}\end{dmath}
\begin{dmath} F_{i}=1\text{\quad for Lindemann}\end{dmath}
\begin{dmath} F_{i}=F_{cent}^{\frac{1}{\frac{A_{Troe}^{2}}{B_{Troe}^{2}} + 1}}\text{\quad for Troe}\end{dmath}
\begin{dmath} F_{i}=T^{e} d \left(a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right)\right)^{X}\text{\quad for SRI}\end{dmath}
\begin{dmath} F_{cent} = a \operatorname{exp}\left({- \frac{T}{T^{*}}}\right) + \left(- a + 1\right) \operatorname{exp}\left({- \frac{T}{T^{***}}}\right) + \operatorname{exp}\left({- \frac{T^{**}}{T}}\right)\end{dmath}
\begin{dmath} A_{Troe} = - \frac{0.67 \log{\left (F_{cent} \right )}}{\log{\left (10 \right )}} + \frac{\log{\left (P_{r, i} \right )}}{\log{\left (10 \right )}} - 0.4\end{dmath}
\begin{dmath} B_{Troe} = - \frac{1.1762 \log{\left (F_{cent} \right )}}{\log{\left (10 \right )}} - \frac{0.14 \log{\left (P_{r, i} \right )}}{\log{\left (10 \right )}} + 0.806\end{dmath}
\begin{dmath} X = \frac{1}{\frac{\log^{2}{\left (P_{r, i} \right )}}{\log^{2}{\left (10 \right )}} + 1}\end{dmath}
\section{Pressure-Dependent Reactions}
For PLog reactions
\begin{dmath} k_{1}=T^{\beta_1} A_{1} \operatorname{exp}\left({\frac{E_{a_1}}{T \Ru}}\right)\text{\quad at $P_1$}\end{dmath}
\begin{dmath} k_{2}=T^{\beta_2} A_{2} \operatorname{exp}\left({\frac{E_{a_2}}{T \Ru}}\right)\text{\quad at $P_2$}\end{dmath}
\begin{dmath} \log{\left ({k_f}_{i} \right )} = \frac{\left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right) \left(- \log{\left (k_{1} \right )} + \log{\left (k_{2} \right )}\right)}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} + \log{\left (k_{1} \right )}\end{dmath}
For Chebyshev reactions
\begin{dmath} \frac{\log{\left ({k_f}_{i} \right )}}{\log{\left (10 \right )}} = \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} T_{j - 1}\left(\tilde{T}\right) T_{l - 1}\left(\tilde{P}\right) \eta_{l,j}\end{dmath}
\begin{dmath} \tilde{T} = \frac{- \frac{1}{T_{min}} - \frac{1}{T_{max}} + \frac{2}{T}}{- \frac{1}{T_{min}} + \frac{1}{T_{max}}}\end{dmath}
\begin{dmath} \tilde{P} = \frac{2 \log{\left (P \right )} - \log{\left (P_{max} \right )} - \log{\left (P_{min} \right )}}{\log{\left (P_{max} \right )} - \log{\left (P_{min} \right )}}\end{dmath}
\section{Derivatives}
\begin{dmath} \frac{\partial q }{\partial T }_{i} = R_{i} \frac{\partial c }{\partial T }_{i} + \frac{\partial R }{\partial T }_{i} c_{i}\end{dmath}
\begin{dmath} \frac{\partial \dot{\omega} }{\partial T }_{k} = \sum_{i=1}^{\nr} \left(\nu_{k,i} R_{i} \frac{\partial c }{\partial T }_{i} + \nu_{k,i} \frac{\partial R }{\partial T }_{i} c_{i}\right)\end{dmath}
\begin{dmath} \frac{\partial q }{\partial n[k] }_{i} = R_{i} \frac{\partial c }{\partial {n_j} }_{i} + \frac{\partial R }{\partial {n_j} }_{i} c_{i}\end{dmath}
\begin{dmath} \frac{\partial \dot{\omega} }{\partial {n_j} }_{k} = \sum_{i=1}^{\nr} \left(\nu_{k,i} R_{i} \frac{\partial c }{\partial {n_j} }_{i} + \nu_{k,i} \frac{\partial R }{\partial {n_j} }_{i} c_{i}\right)\end{dmath}
\begin{dmath} \frac{\partial q }{\partial P }_{i} = R_{i} \frac{\partial c }{\partial P }_{i} + \frac{\partial R }{\partial P }_{i} c_{i}\end{dmath}
\begin{dmath} \frac{\partial \dot{\omega} }{\partial P }_{k} = \sum_{i=1}^{\nr} \left(\nu_{k,i} R_{i} \frac{\partial c }{\partial P }_{i} + \nu_{k,i} \frac{\partial R }{\partial P }_{i} c_{i}\right)\end{dmath}
\section{Rate of Progress Derivatives}
\subsection{Molar Derivatives}
\begin{dmath} \frac{d}{d n_{k}} {R_f} = \left(\frac{\partial}{\partial n_{j}} \prod_{k=1}^{\ns} [C]_{k}^{\nu^{\prime}_{k,i}}\right) {k_f}_{i}\end{dmath}
\begin{dmath} \frac{\partial [C_k]}{\partial n_j} =\frac{\delta_{j k}}{V}\end{dmath}
\begin{dmath} \frac{\partial [C_{Ns}]}{\partial n_j} =- \frac{1}{V}\end{dmath}
\begin{dmath} \frac{\partial [C_{Ns}]^{\nu^{\prime}_{Ns, i}}}{\partial [n_j]} =- \frac{\left(\left(\frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} \frac{n_{k}}{V}\right)^{\nu^{\prime}_{\ns,i}}\right) \nu^{\prime}_{\ns,i} \sum_{k=1}^{-1 + \ns} \frac{\delta_{j k}}{V}}{\frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} \frac{n_{k}}{V}}\end{dmath}
\begin{dmath} \frac{\partial [C_{Ns}]^{\nu^{\prime}_{Ns, i}}}{\partial n_j} =- \frac{\nu^{\prime}_{\ns,i}}{V} [C]_{\ns}^{\nu^{\prime}_{\ns,i} - 1}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial {n_j} }_{i} = {k_f}_{i} \sum_{k=1}^{\ns} \left(- \frac{\delta_{\ns k}}{V} + \frac{\delta_{j k}}{V}\right) \nu^{\prime}_{k,i} [C]_{k}^{\nu^{\prime}_{k,i} - 1} \prod_{\substack{1 \leq l \leq k - 1\\k + 1 \leq l \leq \ns}} [C]_{l}^{\nu^{\prime}_{l,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial {n_j} }_{i} = \frac{{k_f}_{i}}{V} \left(- \nu^{\prime}_{\ns,i} [C]_{\ns}^{\nu^{\prime}_{\ns,i} - 1} \prod_{l=1}^{-1 + \ns} [C]_{l}^{\nu^{\prime}_{l,i}} + \nu^{\prime}_{j,i} [C]_{j}^{\nu^{\prime}_{j,i} - 1} \prod_{\substack{1 \leq l \leq j - 1\\j + 1 \leq l \leq \ns}} [C]_{l}^{\nu^{\prime}_{l,i}}\right)\end{dmath}
\begin{dmath} S^{\prime}_{l} = \nu^{\prime}_{l,i} [C]_{l}^{\nu^{\prime}_{l,i} - 1} \prod_{\substack{1 \leq l \leq l - 1\\l + 1 \leq l \leq \ns}} [C]_{l}^{\nu^{\prime}_{l,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial {n_j} }_{i} = \frac{{k_f}_{i}}{V} \left(- S^{\prime}_{\ns} + S^{\prime}_{j}\right)\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial {n_j} }_{i} = {k_r}_{i} \sum_{k=1}^{\ns} \left(- \frac{\delta_{\ns k}}{V} + \frac{\delta_{j k}}{V}\right) \nu^{\prime\prime}_{k,i} [C]_{k}^{\nu^{\prime\prime}_{k,i} - 1} \prod_{\substack{1 \leq l \leq k - 1\\k + 1 \leq l \leq \ns}} [C]_{l}^{\nu^{\prime\prime}_{l,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial {n_j} }_{i} = \frac{{k_r}_{i}}{V} \left(- \nu^{\prime\prime}_{\ns,i} [C]_{\ns}^{\nu^{\prime\prime}_{\ns,i} - 1} \prod_{l=1}^{-1 + \ns} [C]_{l}^{\nu^{\prime\prime}_{l,i}} + \nu^{\prime\prime}_{j,i} [C]_{j}^{\nu^{\prime\prime}_{j,i} - 1} \prod_{\substack{1 \leq l \leq j - 1\\j + 1 \leq l \leq \ns}} [C]_{l}^{\nu^{\prime\prime}_{l,i}}\right)\end{dmath}
\begin{dmath} S^{\prime\prime}_{l} = \nu^{\prime\prime}_{l,i} [C]_{l}^{\nu^{\prime\prime}_{l,i} - 1} \prod_{\substack{1 \leq l \leq l - 1\\l + 1 \leq l \leq \ns}} [C]_{l}^{\nu^{\prime\prime}_{l,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial {n_j} }_{i} = \frac{{k_r}_{i}}{V} \left(- S^{\prime\prime}_{\ns} + S^{\prime\prime}_{j}\right)\end{dmath}
For all reversible reactions
\begin{dmath} \frac{\partial R }{\partial {n_j} }_{i} = - \frac{{k_r}_{i}}{V} \left(- S^{\prime\prime}_{\ns} + S^{\prime\prime}_{j}\right) + \frac{{k_f}_{i}}{V} \left(- S^{\prime}_{\ns} + S^{\prime}_{j}\right)\end{dmath}
\subsection{Temperature Derivative}
\begin{dmath} {R_f} = {k_f}_{i} \prod_{k=1}^{\ns} [C]_{k}^{\nu^{\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\text{d} {k_f} }{\text{d} T }_{i} = \frac{{k_f}_{i}}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right)\end{dmath}
\begin{dmath} {R_f} = \left(\left(\frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} [C]_{k}\right)^{\nu^{\prime}_{\ns,i}}\right) {k_f}_{i} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial T }_{i} = - \frac{P \left(\left(\frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} [C]_{k}\right)^{\nu^{\prime}_{\ns,i}}\right) \nu^{\prime}_{\ns,i} {k_f}_{i} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime}_{k,i}}}{T^{2} \Ru \left(\frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} [C]_{k}\right)} + \left(\left(\frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} [C]_{k}\right)^{\nu^{\prime}_{\ns,i}}\right) \frac{\text{d} {k_f} }{\text{d} T }_{i} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial T }_{i} = \frac{\text{d} {k_f} }{\text{d} T }_{i} \prod_{k=1}^{\ns} [C]_{k}^{\nu^{\prime}_{k,i}} - \frac{[C] \nu^{\prime}_{\ns,i}}{T} [C]_{\ns}^{\nu^{\prime}_{\ns,i} - 1} {k_f}_{i} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial T }_{i} = - \frac{[C] S^{\prime}_{\ns}}{T} {k_f}_{i} + \frac{{R_f}_{i}}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right)\end{dmath}
For reactions with explicit reverse Arrhenius coefficients
\begin{dmath} \frac{\partial {R_r} }{\partial T }_{i} = - \frac{[C] S^{\prime\prime}_{\ns}}{T} {k_r}_{i} + \frac{{R_r}_{i}}{T} \left({\beta_r}_{i} + \frac{{E_{a,r}}_{i}}{T \Ru}\right)\end{dmath}
\begin{dmath} \frac{\partial R }{\partial T }_{i} = \frac{[C] S^{\prime\prime}_{\ns}}{T} {k_r}_{i} - \frac{[C] S^{\prime}_{\ns}}{T} {k_f}_{i} + \frac{{R_f}_{i}}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right) - \frac{{R_r}_{i}}{T} \left({\beta_r}_{i} + \frac{{E_{a,r}}_{i}}{T \Ru}\right)\end{dmath}
For non-explicit reversible reactions
\begin{dmath} \frac{\text{d} {k_r} }{\text{d} T }_{i} = - \frac{{k_f}_{i}}{{K_c}_{i}^{2}} \frac{\text{d} {K_c} }{\text{d} T }_{i} + \frac{1}{{K_c}_{i}} \frac{\text{d} {k_f} }{\text{d} T }_{i}\end{dmath}
\begin{dmath} \frac{\text{d} {k_r} }{\text{d} T }_{i} = \left(- \frac{1}{{K_c}_{i}} \frac{\text{d} {K_c} }{\text{d} T }_{i} + \frac{1}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right)\right) {k_r}_{i}\end{dmath}
\begin{dmath} \frac{\text{d} {K_c} }{\text{d} T }_{i} = {K_c}_{i} \sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k}\end{dmath}
\begin{dmath} \frac{\text{d} {k_r} }{\text{d} T }_{i} = \left(- \sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{1}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right)\right) {k_r}_{i}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial T }_{i} = \left(- \sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{1}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right)\right) {R_r}_{i} - \frac{[C] S^{\prime\prime}_{\ns}}{T} {k_r}_{i}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial T }_{i} = \left(- \sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{1}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right)\right) {R_r}_{i} - \frac{[C] S^{\prime\prime}_{\ns}}{T} {k_r}_{i}\end{dmath}
\begin{dmath} \frac{\partial R }{\partial T }_{i} = - \left(- \sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{1}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right)\right) {R_r}_{i} + \frac{[C] S^{\prime\prime}_{\ns}}{T} {k_r}_{i} - \frac{[C] S^{\prime}_{\ns}}{T} {k_f}_{i} + \frac{{R_f}_{i}}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right)\end{dmath}
\begin{dmath} \frac{\text{d} B }{\text{d} T }_{k} = T \left(T \left(\frac{T a_{k,4}}{5} + \frac{a_{k,3}}{4}\right) + \frac{a_{k,2}}{3}\right) + \frac{a_{k,1}}{2} + \frac{1}{T} \left(a_{k,0} - 1 + \frac{a_{k,5}}{T}\right)\end{dmath}
\subsection{Pressure derivatives}
\begin{dmath} \frac{\partial [C]_{k} }{\partial P } = 0\end{dmath}
\begin{dmath} \frac{\partial [C]_{\ns} }{\partial P } = \frac{1}{T \Ru}\end{dmath}
\begin{dmath} \frac{\partial [C_{Ns}]^{\nu^{\prime}_{Ns, i}}}{\partial P } = \frac{\nu^{\prime}_{k,i} [C]_{\ns}^{\nu^{\prime}_{k,i} - 1}}{T \Ru}\end{dmath}
\begin{dmath} \frac{\partial [C] }{\partial P } = \frac{1}{T \Ru}\end{dmath}
\begin{dmath} {R_f}_{i} = \left(\left(\frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} [C]_{k}\right)^{\nu^{\prime}_{\ns,i}}\right) {k_f}_{i} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial P }_{i} = \frac{\nu^{\prime}_{\ns,i} [C]_{\ns}^{\nu^{\prime}_{\ns,i}} {k_f}_{i}}{T \Ru [C]_{\ns}} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial P }_{i} = \frac{\nu^{\prime}_{\ns,i} [C]_{\ns}^{\nu^{\prime}_{\ns,i}} {k_f}_{i}}{T \Ru [C]_{\ns}} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial P }_{i} = \frac{\nu^{\prime}_{\ns,i} [C]_{\ns}^{\nu^{\prime}_{\ns,i}} {k_f}_{i}}{T \Ru [C]_{\ns}} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial P }_{i} = \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial P }_{i} = \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru}\end{dmath}
\begin{dmath} {R_r}_{i} = \left(\left(\frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} [C]_{k}\right)^{\nu^{\prime\prime}_{\ns,i}}\right) {k_r}_{i} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial P }_{i} = \frac{\nu^{\prime\prime}_{\ns,i} [C]_{\ns}^{\nu^{\prime\prime}_{\ns,i}} {k_r}_{i}}{T \Ru [C]_{\ns}} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial P }_{i} = \frac{\nu^{\prime\prime}_{\ns,i} [C]_{\ns}^{\nu^{\prime\prime}_{\ns,i}} {k_r}_{i}}{T \Ru [C]_{\ns}} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial P }_{i} = \frac{\nu^{\prime\prime}_{\ns,i} [C]_{\ns}^{\nu^{\prime\prime}_{\ns,i}} {k_r}_{i}}{T \Ru [C]_{\ns}} \prod_{k=1}^{-1 + \ns} [C]_{k}^{\nu^{\prime\prime}_{k,i}}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial P }_{i} = \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial P }_{i} = \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru}\end{dmath}
\section{Third-Body\slash Falloff Derivatives}
\subsection{Elementary reactions
}
\begin{dmath} \frac{\partial c }{\partial T }_{i} = 0\end{dmath}
\begin{dmath} \frac{\partial c }{\partial {n_j} }_{i} = 0\end{dmath}
\begin{dmath} \frac{\partial c }{\partial P }_{i} = 0\end{dmath}
\subsection{Third-body enhanced reactions}
\begin{dmath} \frac{\partial [X]_i }{\partial T } = - \frac{[C] \alpha_{\ns,i}}{T}\end{dmath}
\begin{dmath} \frac{\partial [X]_i }{\partial {n_j} } = \frac{1}{V} \left(- \alpha_{\ns,i} + \alpha_{j,i}\right)\end{dmath}
\begin{dmath} \frac{\partial [X]_i }{\partial P } = \frac{\alpha_{\ns,i}}{T \Ru}\end{dmath}
For species $m$ as the third-body
\begin{dmath} \frac{\partial c }{\partial T }_{i} = - \frac{\delta_{\ns m}}{T} [C]\end{dmath}
\begin{dmath} \frac{\partial c }{\partial {n_j} }_{i} = \frac{1}{V} \left(- \delta_{\ns m} \delta_{j m} - \delta_{\ns m} + \delta_{j m}\right)\end{dmath}
\begin{dmath} \frac{\partial c }{\partial {n_j} }_{i} = \frac{1}{V} \left(- \delta_{\ns m} + \delta_{j m}\right)\end{dmath}
\begin{dmath} \frac{\partial c }{\partial P }_{i} = \frac{\delta_{\ns m}}{T \Ru}\end{dmath}
If all $\alpha_{j, i} = 1$ for all species j
\begin{dmath} \frac{\partial c }{\partial T }_{i} = - \frac{[C]}{T}\end{dmath}
\begin{dmath} \frac{\partial c }{\partial {n_j} }_{i} = 0\end{dmath}
\begin{dmath} \frac{\partial c }{\partial P }_{i} = \frac{1}{T \Ru}\end{dmath}
\subsection{Unimolecular/recombination fall-off reactions}
\begin{dmath} \frac{\partial c }{\partial T }_{i} = \frac{1}{P_{r, i} + 1} \left(P_{r, i} \frac{\partial F_{i} }{\partial T } + \frac{\partial P_{r, i} }{\partial T } \left(F_{i} - c_{i}\right)\right)\end{dmath}
\begin{dmath} \frac{\partial c }{\partial {n_j} }_{i} = \frac{1}{P_{r, i} + 1} \left(P_{r, i} \frac{\partial F_{i} }{\partial {n_j} } + \frac{\partial P_{r, i} }{\partial {n_j} } \left(F_{i} - c_{i}\right)\right)\end{dmath}
\begin{dmath} \frac{\partial c }{\partial P }_{i} = \frac{1}{P_{r, i} + 1} \left(P_{r, i} \frac{\partial F_{i} }{\partial P } + \frac{\partial P_{r, i} }{\partial P } \left(F_{i} - c_{i}\right)\right)\end{dmath}
\subsection{Chemically-activated bimolecular reactions}
\begin{dmath} \frac{\partial c }{\partial T }_{i} = \frac{1}{P_{r, i} + 1} \left(\frac{\partial F_{i} }{\partial T } - \frac{\partial P_{r, i} }{\partial T } c_{i}\right)\end{dmath}
\begin{dmath} \frac{\partial c }{\partial {n_j} }_{i} = \frac{1}{P_{r, i} + 1} \left(\frac{\partial F_{i} }{\partial {n_j} } - \frac{\partial P_{r, i} }{\partial {n_j} } c_{i}\right)\end{dmath}
\begin{dmath} \frac{\partial c }{\partial P }_{i} = \frac{1}{P_{r, i} + 1} \left(\frac{\partial F_{i} }{\partial P } - \frac{\partial P_{r, i} }{\partial P } c_{i}\right)\end{dmath}
\subsection{Reduced Pressure derivatives}

For the mixture as the third body
\begin{dmath} \frac{\partial P_{r, i} }{\partial T } = \frac{P_{r, i}}{T} \left(\beta_{0} - \beta_{\infty} + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right) - \frac{[C] k_{0, i} \alpha_{\ns,i}}{T k_{\infty, i}}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial {n_j} } = \frac{k_{0, i} \left(- \alpha_{\ns,i} + \alpha_{j,i}\right)}{k_{\infty, i} V}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial P } = \frac{k_{0, i} \alpha_{\ns,i}}{T k_{\infty, i} \Ru}\end{dmath}
Simplifying:
\begin{dmath} \frac{\partial P_{r, i} }{\partial T } = P_{r, i} \Theta_{P_{r,i}, \partial T, mix} + \bar{\theta}_{P_{r, i}, \partial T, mix}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial {n_j} } = \frac{\bar{\theta}_{P_{r, i}, \partial n_j, mix}}{k_{\infty, i} V} k_{0, i}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial P } = \bar{\theta}_{P_{r, i}, \partial P, mix}\end{dmath}
\begin{dmath} \Theta_{P_{r,i}, \partial T, mix} = \frac{1}{T} \left(\beta_{0} - \beta_{\infty} + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right)\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial T, mix} = - \frac{[C] k_{0, i} \alpha_{\ns,i}}{T k_{\infty, i}}\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial n_j, mix} = - \alpha_{\ns,i} + \alpha_{j,i}\end{dmath}
\begin{dmath} \Theta_{P_{r,i}, \partial P, mix} = 0\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial P, mix} = \frac{k_{0, i} \alpha_{\ns,i}}{T k_{\infty, i} \Ru}\end{dmath}
For species $m$ as the third-body
\begin{dmath} \frac{\partial P_{r, i} }{\partial T } = \frac{P_{r, i}}{T} \left(\beta_{0} - \beta_{\infty} + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right) - \frac{[C] k_{0, i} \delta_{\ns m}}{T k_{\infty, i}}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial {n_j} } = \frac{k_{0, i}}{k_{\infty, i} V} \left(- \delta_{\ns m} + \delta_{j m}\right)\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial P } = \frac{k_{0, i} \delta_{\ns m}}{T k_{\infty, i} \Ru}\end{dmath}
Simplifying:
\begin{dmath} \frac{\partial P_{r, i} }{\partial T } = P_{r, i} \Theta_{P_{r,i}, \partial T, spec} + \bar{\theta}_{P_{r, i}, \partial T, spec}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial {n_j} } = \frac{\bar{\theta}_{P_{r, i}, \partial n_j, spec}}{k_{\infty, i} V} k_{0, i}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial P } = \bar{\theta}_{P_{r, i}, \partial P, spec}\end{dmath}
\begin{dmath} \Theta_{P_{r,i}, \partial T, spec} = \frac{1}{T} \left(\beta_{0} - \beta_{\infty} + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right)\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial T, spec} = - \frac{[C] k_{0, i} \delta_{\ns m}}{T k_{\infty, i}}\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial n_j, spec} = - \delta_{\ns m} + \delta_{j m}\end{dmath}
\begin{dmath} \Theta_{P_{r,i}, \partial P, spec} = 0\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial P, spec} = \frac{k_{0, i} \delta_{\ns m}}{T k_{\infty, i} \Ru}\end{dmath}
If all $\alpha_{j, i} = 1$ for all species j
\begin{dmath} \frac{\partial P_{r, i} }{\partial T } = \frac{P_{r, i}}{T} \left(\beta_{0} - \beta_{\infty} - 1 + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right)\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial {n_j} } = 0\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial {n_j} } = \frac{k_{0, i}}{k_{\infty, i}} \frac{\partial [C] }{\partial P }\end{dmath}
Simplifying:
\begin{dmath} \frac{\partial P_{r, i} }{\partial T } = P_{r, i} \Theta_{P_{r,i}, \partial T, unity}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial {n_j} } = \bar{\theta}_{P_{r, i}, \partial n_j, unity}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial P } = \bar{\theta}_{P_{r, i}, \partial P, unity}\end{dmath}
\begin{dmath} \Theta_{P_{r,i}, \partial T, unity} = \frac{1}{T} \left(\beta_{0} - \beta_{\infty} - 1 + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right)\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial T, unity} = 0\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial n_j, unity} = 0\end{dmath}
\begin{dmath} \Theta_{P_{r,i}, \partial P, unity} = 0\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial P, unity} = \frac{k_{0, i}}{k_{\infty, i}} \frac{\partial [C] }{\partial P }\end{dmath}
Thus we write:
\begin{dmath} \frac{\partial P_{r, i} }{\partial T } = P_{r, i} \Theta_{P_{r,i}, \partial T} + \bar{\theta}_{P_{r, i}, \partial T}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial {n_j} } = \frac{k_{0, i} \bar{\theta}_{P_{r, i}, \partial n_j}}{k_{\infty, i} V}\end{dmath}
\begin{dmath} \frac{\partial P_{r, i} }{\partial P } = \bar{\theta}_{P_{r, i}, \partial P}\end{dmath}
For
\begin{dgroup}
\begin{dmath} \Theta_{P_{r,i}, \partial T} = \frac{1}{T} \left(\beta_{0} - \beta_{\infty} + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right)\text{\quad if mix}\end{dmath}
\begin{dmath} \Theta_{P_{r,i}, \partial T} = \frac{1}{T} \left(\beta_{0} - \beta_{\infty} + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right)\text{\quad if species}\end{dmath}
\begin{dmath} \Theta_{P_{r,i}, \partial T} = \frac{1}{T} \left(\beta_{0} - \beta_{\infty} - 1 + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right)\text{\quad if unity}\end{dmath}
\end{dgroup}
\begin{dgroup}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial T} = - \frac{[C] k_{0, i} \alpha_{\ns,i}}{T k_{\infty, i}}\text{\quad if mix}\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial T} = - \frac{[C] k_{0, i} \delta_{\ns m}}{T k_{\infty, i}}\text{\quad if species}\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial T} = 0\text{\quad if unity}\end{dmath}
\end{dgroup}
\begin{dgroup}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial n_j} = - \alpha_{\ns,i} + \alpha_{j,i}\text{\quad if mix}\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial n_j} = - \delta_{\ns m} + \delta_{j m}\text{\quad if species}\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial n_j} = 0\text{\quad if unity}\end{dmath}
\end{dgroup}
\begin{dgroup}
\begin{dmath} \Theta_{P_{r,i}, \partial P} = 0\text{\quad if mix}\end{dmath}
\begin{dmath} \Theta_{P_{r,i}, \partial P} = 0\text{\quad if species}\end{dmath}
\begin{dmath} \Theta_{P_{r,i}, \partial P} = 0\text{\quad if unity}\end{dmath}
\end{dgroup}
\begin{dgroup}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial P} = \frac{k_{0, i} \alpha_{\ns,i}}{T k_{\infty, i} \Ru}\text{\quad if mix}\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial P} = \frac{k_{0, i} \delta_{\ns m}}{T k_{\infty, i} \Ru}\text{\quad if species}\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial P} = \frac{k_{0, i}}{k_{\infty, i}} \frac{\partial [C] }{\partial P }\text{\quad if unity}\end{dmath}
\end{dgroup}
\subsection{Falloff Blending Factor derivatives}

 For Lindemann reactions
\begin{dmath} \frac{\partial F_{i} }{\partial T } = 0\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial {n_j} } = 0\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial P } = 0\end{dmath}
For Troe reactions
\begin{dmath} \frac{\partial F_{i} }{\partial T } = \frac{\partial F_{i} }{\partial F_{cent} } \frac{\text{d} F_{cent} }{\text{d} T } + \frac{\partial F_{i} }{\partial P_{r, i} } \frac{\partial P_{r, i} }{\partial T }\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial {n_j} } = \frac{\partial F_{i} }{\partial P_{r, i} } \frac{\partial P_{r, i} }{\partial {n_j} }\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial P } = \frac{\partial F_{i} }{\partial P_{r, i} } \frac{\partial P_{r, i} }{\partial P }\end{dmath}
where
\begin{dmath} \frac{\partial F_{i} }{\partial F_{cent} } = \frac{F_{i}}{\frac{A_{Troe}^{2}}{B_{Troe}^{2}} + 1} \left(\frac{2 A_{Troe} \log{\left (F_{cent} \right )}}{B_{Troe}^{2} \left(\frac{A_{Troe}^{2}}{B_{Troe}^{2}} + 1\right)} \left(\frac{A_{Troe}}{B_{Troe}} \frac{\partial B_{Troe} }{\partial F_{cent} } - \frac{\partial A_{Troe} }{\partial F_{cent} }\right) + \frac{1}{F_{cent}}\right)\end{dmath}
\begin{dmath} \frac{\text{d} F_{cent} }{\text{d} T } = - \frac{a}{T^{*}} \operatorname{exp}\left({- \frac{T}{T^{*}}}\right) - \frac{\operatorname{exp}\left({- \frac{T}{T^{***}}}\right)}{T^{***}} \left(- a + 1\right) + \frac{T^{**}}{T^{2}} \operatorname{exp}\left({- \frac{T^{**}}{T}}\right)\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial P_{r, i} } = \frac{2 F_{i} A_{Troe} \log{\left (F_{cent} \right )}}{B_{Troe}^{2} \left(\frac{A_{Troe}^{2}}{B_{Troe}^{2}} + 1\right)^{2}} \left(\frac{A_{Troe}}{B_{Troe}} \frac{\partial B_{Troe} }{\partial P_{r, i} } - \frac{\partial A_{Troe} }{\partial P_{r, i} }\right)\end{dmath}
And
\begin{dmath} \frac{\partial A_{Troe} }{\partial F_{cent} } = - \frac{0.67}{F_{cent} \log{\left (10 \right )}}\end{dmath}
\begin{dmath} \frac{\partial B_{Troe} }{\partial F_{cent} } = - \frac{1.1762}{F_{cent} \log{\left (10 \right )}}\end{dmath}
\begin{dmath} \frac{\partial A_{Troe} }{\partial P_{r, i} } = \frac{1}{P_{r, i} \log{\left (10 \right )}}\end{dmath}
\begin{dmath} \frac{\partial B_{Troe} }{\partial P_{r, i} } = - \frac{0.14}{P_{r, i} \log{\left (10 \right )}}\end{dmath}
Thus
\begin{dmath} \frac{\partial F_{i} }{\partial F_{cent} } = - \frac{F_{i} B_{Troe}}{F_{cent} \left(A_{Troe}^{2} + B_{Troe}^{2}\right)^{2} \log{\left (10 \right )}} \left(2 A_{Troe} \left(1.1762 A_{Troe} - 0.67 B_{Troe}\right) \log{\left (F_{cent} \right )} - B_{Troe} \left(A_{Troe}^{2} + B_{Troe}^{2}\right) \log{\left (10 \right )}\right)\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial P_{r, i} } = - \frac{2 F_{i} A_{Troe} \left(\frac{0.14 A_{Troe}}{B_{Troe}} + 1\right) \log{\left (F_{cent} \right )}}{B_{Troe}^{2} P_{r, i} \left(\frac{A_{Troe}^{2}}{B_{Troe}^{2}} + 1\right)^{2} \log{\left (10 \right )}}\end{dmath}
And
\begin{dmath} \frac{\partial F_{i} }{\partial T } = F_{i} \Theta_{F_i, \partial T}\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial {n_j} } = \frac{F_{i} k_{0, i} \Theta_{F_i, \partial n_j}}{k_{\infty, i} V} \bar{\theta}_{P_{r, i}, \partial n_j}\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial P } = F_{i} \Theta_{F_i, \partial P}\end{dmath}
Where
\begin{dmath} \Theta_{F_i, \partial T} = - \frac{B_{Troe}}{F_{cent} P_{r, i} \left(A_{Troe}^{2} + B_{Troe}^{2}\right)^{2} \log{\left (10 \right )}} \left(2 A_{Troe} F_{cent} \left(0.14 A_{Troe} + B_{Troe}\right) \left(P_{r, i} \Theta_{P_{r,i}, \partial T} + \bar{\theta}_{P_{r, i}, \partial T}\right) \log{\left (F_{cent} \right )} + P_{r, i} \frac{\text{d} F_{cent} }{\text{d} T } \left(2 A_{Troe} \left(1.1762 A_{Troe} - 0.67 B_{Troe}\right) \log{\left (F_{cent} \right )} - B_{Troe} \left(A_{Troe}^{2} + B_{Troe}^{2}\right) \log{\left (10 \right )}\right)\right)\end{dmath}
\begin{dmath} \Theta_{F_i, \partial n_j} = - \frac{2 A_{Troe} B_{Troe} \left(0.14 A_{Troe} + B_{Troe}\right) \log{\left (F_{cent} \right )}}{P_{r, i} \left(A_{Troe}^{2} + B_{Troe}^{2}\right)^{2} \log{\left (10 \right )}}\end{dmath}
\begin{dmath} \Theta_{F_i, \partial P} = - \frac{2 A_{Troe} B_{Troe} \bar{\theta}_{P_{r, i}, \partial P} \left(0.14 A_{Troe} + B_{Troe}\right) \log{\left (F_{cent} \right )}}{P_{r, i} \left(A_{Troe}^{2} + B_{Troe}^{2}\right)^{2} \log{\left (10 \right )}}\end{dmath}
For SRI reactions
\begin{dmath} \frac{\partial F_{i} }{\partial T } = F_{i} \left(\frac{X \left(- \frac{\operatorname{exp}\left({- \frac{T}{c}}\right)}{c} + \frac{a b}{T^{2}} \operatorname{exp}\left({- \frac{b}{T}}\right)\right)}{a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right)} + \frac{\partial P_{r, i} }{\partial T } \frac{\text{d} X }{\text{d} P_{r, i} } \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )} + \frac{e}{T}\right)\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial {n_j} } = F_{i} \frac{\partial P_{r, i} }{\partial {n_j} } \frac{\text{d} X }{\text{d} P_{r, i} } \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial P } = F_{i} \frac{\partial P_{r, i} }{\partial P } \frac{\text{d} X }{\text{d} P_{r, i} } \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}\end{dmath}
Where
\begin{dmath} \frac{\text{d} X }{\text{d} P_{r, i} } = - \frac{2 X^{2} \log{\left (P_{r, i} \right )}}{P_{r, i} \log^{2}{\left (10 \right )}}\end{dmath}
\begin{dmath} \frac{\partial X}{\partial n_j} = \frac{\partial P_{r, i} }{\partial {n_j} } \frac{\text{d} X }{\text{d} P_{r, i} }\end{dmath}
And
\begin{dmath} \frac{\partial F_{i} }{\partial T } = F_{i} \Theta_{F_i, \partial T}\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial {n_j} } = \frac{F_{i} k_{0, i} \Theta_{F_i, \partial n_j}}{k_{\infty, i} V} \bar{\theta}_{P_{r, i}, \partial n_j}\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial P } = F_{i} \Theta_{F_i, \partial P}\end{dmath}
Where
\begin{dmath} \Theta_{F_i, \partial T} = - \frac{X \left(\frac{\operatorname{exp}\left({- \frac{T}{c}}\right)}{c} - \frac{a b}{T^{2}} \operatorname{exp}\left({- \frac{b}{T}}\right)\right)}{a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right)} + \frac{e}{T} - \frac{2 X^{2} \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}}{P_{r, i} \log^{2}{\left (10 \right )}} \left(P_{r, i} \Theta_{P_{r,i}, \partial T} + \bar{\theta}_{P_{r, i}, \partial T}\right) \log{\left (P_{r, i} \right )}\end{dmath}
\begin{dmath} \Theta_{F_i, \partial n_j} = - \frac{2 X^{2} \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}}{P_{r, i} \log^{2}{\left (10 \right )}} \log{\left (P_{r, i} \right )}\end{dmath}
\begin{dmath} \Theta_{F_i, \partial P} = - \frac{2 X^{2} \bar{\theta}_{P_{r, i}, \partial P} \log{\left (P_{r, i} \right )}}{P_{r, i} \log^{2}{\left (10 \right )}} \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}\end{dmath}
Simplifying:
\begin{dmath} \frac{\partial F_{i} }{\partial T } = F_{i} \Theta_{F_i, \partial T}\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial {n_j} } = \frac{F_{i} k_{0, i} \Theta_{F_i, \partial n_j}}{k_{\infty, i} V} \bar{\theta}_{P_{r, i}, \partial n_j}\end{dmath}
\begin{dmath} \frac{\partial F_{i} }{\partial P } = F_{i} \Theta_{F_i, \partial P}\end{dmath}
Where:
\begin{dgroup}
\begin{dmath} \Theta_{F_i, \partial T} = 0\text{\quad if Lindemann}\end{dmath}
\begin{dmath} \Theta_{F_i, \partial T} = - \frac{B_{Troe}}{F_{cent} P_{r, i} \left(A_{Troe}^{2} + B_{Troe}^{2}\right)^{2} \log{\left (10 \right )}} \left(2 A_{Troe} F_{cent} \left(0.14 A_{Troe} + B_{Troe}\right) \left(P_{r, i} \Theta_{P_{r,i}, \partial T} + \bar{\theta}_{P_{r, i}, \partial T}\right) \log{\left (F_{cent} \right )} + P_{r, i} \frac{\text{d} F_{cent} }{\text{d} T } \left(2 A_{Troe} \left(1.1762 A_{Troe} - 0.67 B_{Troe}\right) \log{\left (F_{cent} \right )} - B_{Troe} \left(A_{Troe}^{2} + B_{Troe}^{2}\right) \log{\left (10 \right )}\right)\right)\text{\quad if Troe}\end{dmath}
\begin{dmath} \Theta_{F_i, \partial T} = - \frac{X \left(\frac{\operatorname{exp}\left({- \frac{T}{c}}\right)}{c} - \frac{a b}{T^{2}} \operatorname{exp}\left({- \frac{b}{T}}\right)\right)}{a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right)} + \frac{e}{T} - \frac{2 X^{2} \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}}{P_{r, i} \log^{2}{\left (10 \right )}} \left(P_{r, i} \Theta_{P_{r,i}, \partial T} + \bar{\theta}_{P_{r, i}, \partial T}\right) \log{\left (P_{r, i} \right )}\text{\quad if SRI}\end{dmath}
\end{dgroup}
\begin{dgroup}
\begin{dmath} \Theta_{F_i, \partial n_j} = 0\text{\quad if Lindemann}\end{dmath}
\begin{dmath} \Theta_{F_i, \partial n_j} = - \frac{2 A_{Troe} B_{Troe} \left(0.14 A_{Troe} + B_{Troe}\right) \log{\left (F_{cent} \right )}}{P_{r, i} \left(A_{Troe}^{2} + B_{Troe}^{2}\right)^{2} \log{\left (10 \right )}}\text{\quad if Troe}\end{dmath}
\begin{dmath} \Theta_{F_i, \partial n_j} = - \frac{2 X^{2} \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}}{P_{r, i} \log^{2}{\left (10 \right )}} \log{\left (P_{r, i} \right )}\text{\quad if SRI}\end{dmath}
\end{dgroup}
\begin{dgroup}
\begin{dmath} \Theta_{F_i, \partial P} = 0\text{\quad if Lindemann}\end{dmath}
\begin{dmath} \Theta_{F_i, \partial P} = - \frac{2 A_{Troe} B_{Troe} \bar{\theta}_{P_{r, i}, \partial P} \left(0.14 A_{Troe} + B_{Troe}\right) \log{\left (F_{cent} \right )}}{P_{r, i} \left(A_{Troe}^{2} + B_{Troe}^{2}\right)^{2} \log{\left (10 \right )}}\text{\quad if Troe}\end{dmath}
\begin{dmath} \Theta_{F_i, \partial P} = - \frac{2 X^{2} \bar{\theta}_{P_{r, i}, \partial P} \log{\left (P_{r, i} \right )}}{P_{r, i} \log^{2}{\left (10 \right )}} \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}\text{\quad if SRI}\end{dmath}
\end{dgroup}
\subsection{Unimolecular/recombination fall-off reactions (complete)}
\begin{dmath} \frac{\partial c }{\partial T }_{i} = \frac{F_{i} \bar{\theta}_{P_{r, i}, \partial T}}{P_{r, i} + 1} + \left(- \frac{P_{r, i} \Theta_{P_{r,i}, \partial T}}{P_{r, i} + 1} + \Theta_{F_i, \partial T} + \Theta_{P_{r,i}, \partial T} - \frac{\bar{\theta}_{P_{r, i}, \partial T}}{P_{r, i} + 1}\right) c_{i}\end{dmath}
\begin{dmath} \frac{\partial c }{\partial {n_j} }_{i} = \frac{k_{0, i} \bar{\theta}_{P_{r, i}, \partial n_j}}{k_{\infty, i} V \left(P_{r, i} + 1\right)} \left(F_{i} \left(P_{r, i} \Theta_{F_i, \partial n_j} + 1\right) - c_{i}\right)\end{dmath}
\begin{dmath} \frac{\partial c }{\partial P }_{i} = \frac{F_{i} \bar{\theta}_{P_{r, i}, \partial P}}{P_{r, i} + 1} + \left(\Theta_{F_i, \partial P} - \frac{\bar{\theta}_{P_{r, i}, \partial P}}{P_{r, i} + 1}\right) c_{i}\end{dmath}
\subsection{Chemically-activated bimolecular reactions (complete)}
\begin{dmath} \frac{\partial c }{\partial T }_{i} = \left(- \frac{P_{r, i} \Theta_{P_{r,i}, \partial T}}{P_{r, i} + 1} + \Theta_{F_i, \partial T} - \frac{\bar{\theta}_{P_{r, i}, \partial T}}{P_{r, i} + 1}\right) c_{i}\end{dmath}
\begin{dmath} \frac{\partial c }{\partial {n_j} }_{i} = \frac{k_{0, i} \bar{\theta}_{P_{r, i}, \partial n_j} \left(F_{i} \Theta_{F_i, \partial n_j} - c_{i}\right)}{k_{\infty, i} V \left(P_{r, i} + 1\right)}\end{dmath}
\begin{dmath} \frac{\partial c }{\partial P }_{i} = \left(\Theta_{F_i, \partial P} - \frac{\bar{\theta}_{P_{r, i}, \partial P}}{P_{r, i} + 1}\right) c_{i}\end{dmath}
\section{Pressure-dependent reaction derivatives}
For PLog reactions
\begin{dmath} \frac{\text{d} {k_f} }{\text{d} T }_{i} = \left(\frac{1}{k_{1}} \frac{\text{d} k_1 }{\text{d} T } + \frac{1}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} \left(- \frac{1}{k_{1}} \frac{\text{d} k_1 }{\text{d} T } + \frac{1}{k_{2}} \frac{\text{d} k_2 }{\text{d} T }\right) \left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right)\right) {k_f}_{i}\end{dmath}
\begin{dmath} \frac{\text{d} {k_f} }{\text{d} T }_{i} = \left(\frac{1}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} \left(- \frac{1}{T} \left(\beta_1 + \frac{E_{a_1}}{T \Ru}\right) + \frac{1}{T} \left(\beta_2 + \frac{E_{a_2}}{T \Ru}\right)\right) \left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right) + \frac{1}{T} \left(\beta_1 + \frac{E_{a_1}}{T \Ru}\right)\right) {k_f}_{i}\end{dmath}
\begin{dmath} \frac{\text{d} {k_f} }{\text{d} T }_{i} = \frac{{k_f}_{i}}{T} \left(\beta_1 + \frac{\left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right) \left(- \beta_1 + \beta_2 - \frac{E_{a_1}}{T \Ru} + \frac{E_{a_2}}{T \Ru}\right)}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} + \frac{E_{a_1}}{T \Ru}\right)\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial T }_{i} = - \frac{[C] S^{\prime}_{\ns}}{T} {k_f}_{i} + \frac{{R_f}_{i}}{T} \left(\beta_1 + \frac{\left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right) \left(- \beta_1 + \beta_2 - \frac{E_{a_1}}{T \Ru} + \frac{E_{a_2}}{T \Ru}\right)}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} + \frac{E_{a_1}}{T \Ru}\right)\end{dmath}
\begin{dmath} \frac{\text{d} {k_r} }{\text{d} T }_{i} = \left(- \sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{1}{T} \left(\beta_1 + \frac{\left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right) \left(- \beta_1 + \beta_2 - \frac{E_{a_1}}{T \Ru} + \frac{E_{a_2}}{T \Ru}\right)}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} + \frac{E_{a_1}}{T \Ru}\right)\right) {k_r}_{i}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial T }_{i} = \left(- \sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{1}{T} \left(\beta_1 + \frac{\left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right) \left(- \beta_1 + \beta_2 - \frac{E_{a_1}}{T \Ru} + \frac{E_{a_2}}{T \Ru}\right)}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} + \frac{E_{a_1}}{T \Ru}\right)\right) {R_r}_{i} - \frac{[C] S^{\prime\prime}_{\ns}}{T} {k_r}_{i}\end{dmath}
\begin{dmath} \frac{\partial R }{\partial T }_{i} = - \left(- \sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{1}{T} \left(\beta_1 + \frac{\left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right) \left(- \beta_1 + \beta_2 - \frac{E_{a_1}}{T \Ru} + \frac{E_{a_2}}{T \Ru}\right)}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} + \frac{E_{a_1}}{T \Ru}\right)\right) {R_r}_{i} + \frac{[C]}{T} \left(S^{\prime\prime}_{\ns} {k_r}_{i} - S^{\prime}_{\ns} {k_f}_{i}\right) + \frac{{R_f}_{i}}{T} \left(\beta_1 + \frac{\left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right) \left(- \beta_1 + \beta_2 - \frac{E_{a_1}}{T \Ru} + \frac{E_{a_2}}{T \Ru}\right)}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} + \frac{E_{a_1}}{T \Ru}\right)\end{dmath}
\begin{dmath} \frac{\partial {k_f} }{\partial P }_{i} = \frac{\left(- \log{\left (k_{1} \right )} + \log{\left (k_{2} \right )}\right) {k_f}_{i}}{P \left(- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}\right)}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial P }_{i} = \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru} + \frac{\left(- \log{\left (k_{1} \right )} + \log{\left (k_{2} \right )}\right) {R_f}_{i}}{P \left(- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}\right)}\end{dmath}
\begin{dmath} \frac{\partial {k_r} }{\partial P }_{i} = \frac{1}{{K_c}_{i}} \frac{\partial {k_f} }{\partial P }_{i}\end{dmath}
\begin{dmath} \frac{\partial {k_r} }{\partial P }_{i} = \frac{\left(- \log{\left (k_{1} \right )} + \log{\left (k_{2} \right )}\right) {k_r}_{i}}{P \left(- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}\right)}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial P }_{i} = \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru} + \frac{\left(- \log{\left (k_{1} \right )} + \log{\left (k_{2} \right )}\right) {R_r}_{i}}{P \left(- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}\right)}\end{dmath}
\begin{dmath} \frac{\partial R }{\partial P }_{i} = \frac{1}{T \Ru} \left(- S^{\prime\prime}_{\ns} {k_r}_{i} + S^{\prime}_{\ns} {k_f}_{i}\right) + \frac{\left(- \log{\left (k_{1} \right )} + \log{\left (k_{2} \right )}\right) \left({R_f}_{i} - {R_r}_{i}\right)}{P \left(- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}\right)}\end{dmath}
For Chebyshev reactions
\begin{dmath} \frac{\text{d} {k_f} }{\text{d} T }_{i} = \log{\left (10 \right )} {k_f}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \frac{\text{d} \tilde{T} }{\text{d} T } \left(j - 1\right) T_{l - 1}\left(\tilde{P}\right) U_{j - 2}\left(\tilde{T}\right) \eta_{l,j}\end{dmath}
\begin{dmath} \frac{\text{d} {k_f} }{\text{d} T }_{i} = \log{\left (10 \right )} {k_f}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} - \frac{2 T_{l - 1}\left(\tilde{P}\right) U_{j - 2}\left(\tilde{T}\right) \eta_{l,j}}{T^{2} \left(- \frac{1}{T_{min}} + \frac{1}{T_{max}}\right)} \left(j - 1\right)\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial T }_{i} = \log{\left (10 \right )} {R_f}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} - \frac{2 T_{l - 1}\left(\tilde{P}\right) U_{j - 2}\left(\tilde{T}\right) \eta_{l,j}}{T^{2} \left(- \frac{1}{T_{min}} + \frac{1}{T_{max}}\right)} \left(j - 1\right) - \frac{[C] S^{\prime}_{\ns}}{T} {k_f}_{i}\end{dmath}
\begin{dmath} \frac{\text{d} {k_r} }{\text{d} T }_{i} = - \left(\sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{2 \log{\left (10 \right )}}{T^{2} \left(- \frac{1}{T_{min}} + \frac{1}{T_{max}}\right)} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \left(j - 1\right) T_{l - 1}\left(\tilde{P}\right) U_{j - 2}\left(\tilde{T}\right) \eta_{l,j}\right) {k_r}_{i}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial T }_{i} = - \left(\sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{2 \log{\left (10 \right )}}{T^{2} \left(- \frac{1}{T_{min}} + \frac{1}{T_{max}}\right)} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \left(j - 1\right) T_{l - 1}\left(\tilde{P}\right) U_{j - 2}\left(\tilde{T}\right) \eta_{l,j}\right) {R_r}_{i} - \frac{[C] S^{\prime\prime}_{\ns}}{T} {k_r}_{i}\end{dmath}
\begin{dmath} \frac{\partial R }{\partial T }_{i} = \left(\sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{2 \log{\left (10 \right )}}{T^{2} \left(- \frac{1}{T_{min}} + \frac{1}{T_{max}}\right)} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \left(j - 1\right) T_{l - 1}\left(\tilde{P}\right) U_{j - 2}\left(\tilde{T}\right) \eta_{l,j}\right) {R_r}_{i} + \log{\left (10 \right )} {R_f}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} - \frac{2 T_{l - 1}\left(\tilde{P}\right) U_{j - 2}\left(\tilde{T}\right) \eta_{l,j}}{T^{2} \left(- \frac{1}{T_{min}} + \frac{1}{T_{max}}\right)} \left(j - 1\right) + \frac{[C]}{T} \left(S^{\prime\prime}_{\ns} {k_r}_{i} - S^{\prime}_{\ns} {k_f}_{i}\right)\end{dmath}
\begin{dmath} \frac{\partial {k_f} }{\partial P }_{i} = \log{\left (10 \right )} {k_f}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \frac{\text{d} \tilde{P} }{\text{d} P } \left(l - 1\right) T_{j - 1}\left(\tilde{T}\right) U_{l - 2}\left(\tilde{P}\right) \eta_{l,j}\end{dmath}
\begin{dmath} \frac{\partial {k_f} }{\partial P }_{i} = \log{\left (10 \right )} {k_f}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \frac{2 \left(l - 1\right) T_{j - 1}\left(\tilde{T}\right) U_{l - 2}\left(\tilde{P}\right) \eta_{l,j}}{P \left(\log{\left (P_{max} \right )} - \log{\left (P_{min} \right )}\right)}\end{dmath}
\begin{dmath} \frac{\partial {R_f} }{\partial P }_{i} = \log{\left (10 \right )} {R_f}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \frac{2 \left(l - 1\right) T_{j - 1}\left(\tilde{T}\right) U_{l - 2}\left(\tilde{P}\right) \eta_{l,j}}{P \left(\log{\left (P_{max} \right )} - \log{\left (P_{min} \right )}\right)} + \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru}\end{dmath}
\begin{dmath} \frac{\partial {k_r} }{\partial P }_{i} = \log{\left (10 \right )} {k_r}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \frac{2 \left(l - 1\right) T_{j - 1}\left(\tilde{T}\right) U_{l - 2}\left(\tilde{P}\right) \eta_{l,j}}{P \left(\log{\left (P_{max} \right )} - \log{\left (P_{min} \right )}\right)}\end{dmath}
\begin{dmath} \frac{\partial {R_r} }{\partial P }_{i} = \log{\left (10 \right )} {R_r}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \frac{2 \left(l - 1\right) T_{j - 1}\left(\tilde{T}\right) U_{l - 2}\left(\tilde{P}\right) \eta_{l,j}}{P \left(\log{\left (P_{max} \right )} - \log{\left (P_{min} \right )}\right)} + \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru}\end{dmath}
\begin{dmath} \frac{\partial R }{\partial P }_{i} = \log{\left (10 \right )} {R_f}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \frac{2 \left(l - 1\right) T_{j - 1}\left(\tilde{T}\right) U_{l - 2}\left(\tilde{P}\right) \eta_{l,j}}{P \left(\log{\left (P_{max} \right )} - \log{\left (P_{min} \right )}\right)} - \log{\left (10 \right )} {R_r}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \frac{2 \left(l - 1\right) T_{j - 1}\left(\tilde{T}\right) U_{l - 2}\left(\tilde{P}\right) \eta_{l,j}}{P \left(\log{\left (P_{max} \right )} - \log{\left (P_{min} \right )}\right)} + \frac{1}{T \Ru} \left(- S^{\prime\prime}_{\ns} {k_r}_{i} + S^{\prime}_{\ns} {k_f}_{i}\right)\end{dmath}
\section{Jacobian entries}
\subsection{Energy Equation}
\begin{dmath} \frac{\text{d} T }{\text{d} t } = - \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k}\end{dmath}
\begin{dmath} \frac{\text{d} T }{\text{d} t } = - \frac{\sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k}}{\left(\frac{P}{T \Ru} - \sum_{k=1}^{-1 + \ns} [C]_{k}\right) {C_v}_{\ns} + \sum_{k=1}^{-1 + \ns} [C]_{k} {C_v}_{k}}\end{dmath}
\begin{dmath} \frac{\text{d} T }{\text{d} t } = - \frac{\sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k}}{[C] {C_v}_{\ns} + \sum_{k=1}^{-1 + \ns} \left(- {C_v}_{\ns} + {C_v}_{k}\right) [C]_{k}}\end{dmath}
\subsection{\texorpdfstring{$\dot{T}$}{dTdt} Derivatives}
Molar derivative
\begin{dmath} \frac{\partial\dot{T}}{\partial{n_j}} = - \frac{\sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial {n_j} }_{k}}{[C] {C_v}_{\ns} + \sum_{k=1}^{-1 + \ns} - \left({C_v}_{\ns} - {C_v}_{k}\right) [C]_{k}} + \frac{\left(\sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k}\right) \sum_{k=1}^{-1 + \ns} - \frac{\delta_{j k}}{V} \left({C_v}_{\ns} - {C_v}_{k}\right)}{\left([C] {C_v}_{\ns} + \sum_{k=1}^{-1 + \ns} - \left({C_v}_{\ns} - {C_v}_{k}\right) [C]_{k}\right)^{2}}\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{n_j}} = \frac{1}{\left(\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}\right)^{2}} \left(\sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k}\right) \sum_{k=1}^{-1 + \ns} - \frac{\delta_{j k}}{V} \left({C_v}_{\ns} - {C_v}_{k}\right) - \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial {n_j} }_{k}\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{n_j}} = \frac{1}{V \left(\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}\right)^{2}} \left(- {C_v}_{\ns} + {C_v}_{j}\right) \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k} - \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial {n_j} }_{k}\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{n_j}} = \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(- \frac{1}{V} \frac{\text{d} T }{\text{d} t } \left(- {C_v}_{\ns} + {C_v}_{j}\right) - \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial {n_j} }_{k}\right)\end{dmath}
Temperature derivative
\begin{dmath} \frac{\text{d} T }{\text{d} t } = - \frac{\sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k}}{\frac{P {C_v}_{\ns}}{T \Ru} + \sum_{k=1}^{-1 + \ns} \left(- {C_v}_{\ns} + {C_v}_{k}\right) [C]_{k}}\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{T}} = - \frac{1}{\frac{P {C_v}_{\ns}}{T \Ru} + \sum_{k=1}^{-1 + \ns} \left(- {C_v}_{\ns} + {C_v}_{k}\right) [C]_{k}} \sum_{k=1}^{-1 + \ns} \left(\left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial T }_{k} + \left(\frac{\text{d} U }{\text{d} T }_{k} - \frac{W_{k}}{W_{\ns}} \frac{\text{d} U }{\text{d} T }_{\ns}\right) \dot{\omega}_{k}\right) - \frac{1}{\left(\frac{P {C_v}_{\ns}}{T \Ru} + \sum_{k=1}^{-1 + \ns} \left(- {C_v}_{\ns} + {C_v}_{k}\right) [C]_{k}\right)^{2}} \left(- \frac{P}{T \Ru} \frac{\text{d} {C_v} }{\text{d} T }_{\ns} + \frac{P {C_v}_{\ns}}{T^{2} \Ru} - \sum_{k=1}^{-1 + \ns} \left(- \frac{\text{d} {C_v} }{\text{d} T }_{\ns} + \frac{\text{d} {C_v} }{\text{d} T }_{k}\right) [C]_{k}\right) \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k}\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{T}} = - \frac{1}{[C] {C_v}_{\ns} + \sum_{k=1}^{-1 + \ns} \left(- {C_v}_{\ns} + {C_v}_{k}\right) [C]_{k}} \sum_{k=1}^{-1 + \ns} \left(\left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial T }_{k} + \left(\frac{\text{d} U }{\text{d} T }_{k} - \frac{W_{k}}{W_{\ns}} \frac{\text{d} U }{\text{d} T }_{\ns}\right) \dot{\omega}_{k}\right) - \frac{1}{\left([C] {C_v}_{\ns} + \sum_{k=1}^{-1 + \ns} \left(- {C_v}_{\ns} + {C_v}_{k}\right) [C]_{k}\right)^{2}} \left(- [C] \frac{\text{d} {C_v} }{\text{d} T }_{\ns} - \sum_{k=1}^{-1 + \ns} \left(- \frac{\text{d} {C_v} }{\text{d} T }_{\ns} + \frac{\text{d} {C_v} }{\text{d} T }_{k}\right) [C]_{k} + \frac{[C] {C_v}_{\ns}}{T}\right) \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k}\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{T}} = - \frac{1}{\left(\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}\right)^{2}} \left(- [C] \frac{\text{d} {C_v} }{\text{d} T }_{\ns} - \sum_{k=1}^{-1 + \ns} \left(- \frac{\text{d} {C_v} }{\text{d} T }_{\ns} + \frac{\text{d} {C_v} }{\text{d} T }_{k}\right) [C]_{k} + \frac{[C] {C_v}_{\ns}}{T}\right) \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k} - \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \sum_{k=1}^{-1 + \ns} \left(\left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial T }_{k} + \left(\frac{\text{d} U }{\text{d} T }_{k} - \frac{W_{k}}{W_{\ns}} \frac{\text{d} U }{\text{d} T }_{\ns}\right) \dot{\omega}_{k}\right)\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{T}} = - \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(\frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(- [C] \frac{\text{d} {C_v} }{\text{d} T }_{\ns} - \sum_{k=1}^{-1 + \ns} \left(- \frac{\text{d} {C_v} }{\text{d} T }_{\ns} + \frac{\text{d} {C_v} }{\text{d} T }_{k}\right) [C]_{k} + \frac{[C] {C_v}_{\ns}}{T}\right) \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \dot{\omega}_{k} + \sum_{k=1}^{-1 + \ns} \left(\left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial T }_{k} + \left(\frac{\text{d} U }{\text{d} T }_{k} - \frac{W_{k}}{W_{\ns}} \frac{\text{d} U }{\text{d} T }_{\ns}\right) \dot{\omega}_{k}\right)\right)\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{T}} = \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(\frac{\text{d} T }{\text{d} t } \left(- [C] \frac{\text{d} {C_v} }{\text{d} T }_{\ns} - \sum_{k=1}^{-1 + \ns} \left(- \frac{\text{d} {C_v} }{\text{d} T }_{\ns} + \frac{\text{d} {C_v} }{\text{d} T }_{k}\right) [C]_{k} + \frac{[C] {C_v}_{\ns}}{T}\right) - \sum_{k=1}^{-1 + \ns} \left(\left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial T }_{k} + \left(\frac{\text{d} U }{\text{d} T }_{k} - \frac{W_{k}}{W_{\ns}} \frac{\text{d} U }{\text{d} T }_{\ns}\right) \dot{\omega}_{k}\right)\right)\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{T}} = \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(\frac{\text{d} T }{\text{d} t } \left(- \frac{\text{d} {C_v} }{\text{d} T }_{\ns} \sum_{k=1}^{\ns} [C]_{k} - \sum_{k=1}^{-1 + \ns} \left(- \frac{\text{d} {C_v} }{\text{d} T }_{\ns} + \frac{\text{d} {C_v} }{\text{d} T }_{k}\right) [C]_{k} + \frac{{C_v}_{\ns}}{T} \sum_{k=1}^{\ns} [C]_{k}\right) - \sum_{k=1}^{-1 + \ns} \left(\left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial T }_{k} + \left(\frac{\text{d} U }{\text{d} T }_{k} - \frac{W_{k}}{W_{\ns}} \frac{\text{d} U }{\text{d} T }_{\ns}\right) \dot{\omega}_{k}\right)\right)\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{T}} = \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(\frac{\text{d} T }{\text{d} t } \sum_{k=1}^{\ns} \left(- \frac{\text{d} {C_v} }{\text{d} T }_{k} + \frac{{C_v}_{\ns}}{T}\right) [C]_{k} + \sum_{k=1}^{-1 + \ns} \left(\left(- U_{k} + \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial T }_{k} + \left(- \frac{\text{d} U }{\text{d} T }_{k} + \frac{W_{k}}{W_{\ns}} \frac{\text{d} U }{\text{d} T }_{\ns}\right) \dot{\omega}_{k}\right)\right)\end{dmath}
\begin{dmath} \frac{\partial\dot{T}}{\partial{T}} = \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(\frac{\text{d} T }{\text{d} t } \sum_{k=1}^{\ns} \left(- \frac{\text{d} {C_v} }{\text{d} T }_{k} + \frac{{C_v}_{\ns}}{T}\right) [C]_{k} + \sum_{k=1}^{-1 + \ns} \left(\left(- U_{k} + \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial T }_{k} + \left(- {C_v}_{k} + \frac{W_{k}}{W_{\ns}} \frac{\text{d} U }{\text{d} T }_{\ns}\right) \dot{\omega}_{k}\right)\right)\end{dmath}
Pressure Derivative
\begin{dmath} \frac{\partial\dot{T}}{\partial{P}} = \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(- \sum_{k=1}^{-1 + \ns} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial P }_{k} - \frac{{C_v}_{\ns}}{T \Ru} \frac{\text{d} T }{\text{d} t }\right)\end{dmath}
\subsection{\texorpdfstring{$\dot{P}$}{dPdt} Derivatives}
Temperature Derivative
\begin{dmath} \frac{\partial \dot{ P } }{\partial T } = \frac{P}{T} \left(\frac{\text{d} \dot{T} }{\text{d} T } - \frac{1}{T} \frac{\text{d} T }{\text{d} t }\right) + \Ru \sum_{k=1}^{-1 + \ns} \left(1 - \frac{W_{k}}{W_{\ns}}\right) \left(T \frac{\partial \dot{\omega} }{\partial T }_{k} + \dot{\omega}_{k}\right)\end{dmath}
Molar Derivative
\begin{dmath} \frac{\partial \dot{ P } }{\partial {n_j} } = \frac{P}{T} \frac{\text{d} \dot{T} }{\text{d} {n_j} } + T \Ru \sum_{k=1}^{-1 + \ns} \left(1 - \frac{W_{k}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial {n_j} }_{k}\end{dmath}
Pressure Derivative
\begin{dmath} \frac{\partial \dot{ P } }{\partial P } = T \Ru \sum_{k=1}^{-1 + \ns} \left(1 - \frac{W_{k}}{W_{\ns}}\right) \frac{\partial \dot{\omega} }{\partial P }_{k} + \frac{1}{T} \left(P \frac{\text{d} \dot{T} }{\text{d} P } + \dot{T}\right)\end{dmath}
\subsection{\texorpdfstring{$\dot{n_k}$}{dnkdt} Derivatives}
\begin{dmath} \frac{\partial \dot{n} }{\partial {n_j} }_{k} = V \frac{\partial \dot{\omega} }{\partial {n_j} }_{k}\end{dmath}
\begin{dmath} \frac{\partial \dot{n} }{\partial T }_{k} = V \frac{\partial \dot{\omega} }{\partial T }_{k}\end{dmath}
\begin{dmath} \frac{\partial \dot{n} }{\partial P }_{k} = V \frac{\partial \dot{\omega} }{\partial P }_{k}\end{dmath}
\section{Jacobian Update Form}
\subsection{Temperature Derivatives}
\begin{dmath} \mathcal{J}_{1,1} = \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(\frac{\text{d} T }{\text{d} t } \sum_{k=1}^{\ns} \left(- \frac{\text{d} {C_v} }{\text{d} T }_{k} + \frac{{C_v}_{\ns}}{T}\right) [C]_{k} + \sum_{k=1}^{-1 + \ns} \left(\frac{1}{V} \left(- U_{k} + \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{n} }{\partial T }_{k} + \left(- {C_v}_{k} + \frac{W_{k}}{W_{\ns}} \frac{\text{d} U }{\text{d} T }_{\ns}\right) \dot{\omega}_{k}\right)\right)\end{dmath}
\begin{dmath} \mathcal{J}_{2,1} = \frac{P}{T} \left(\frac{\text{d} \dot{T} }{\text{d} T } - \frac{1}{T} \frac{\text{d} T }{\text{d} t }\right) + \Ru \sum_{k=1}^{-1 + \ns} \left(1 - \frac{W_{k}}{W_{\ns}}\right) \left(\frac{T}{V} \frac{\partial \dot{n} }{\partial T }_{k} + \dot{\omega}_{k}\right)\end{dmath}
\begin{dmath} \mathcal{J}_{k + 2,1} = V \sum_{i=1}^{\nr} \nu_{k,i} \frac{\partial q }{\partial T }_{i}\end{dmath}
Converting to update form:
\begin{dmath} \mathcal{J}_{k + 2,1}\pluseq V \nu_{k,i} \frac{\partial q }{\partial T }_{i}\text{\quad} k = 1, \ldots, N_{sp} - 1\end{dmath}
\subsubsection{Explicit reversible reactions}
\begin{dmath} \frac{\partial q }{\partial T }_{i} = \Theta_{\partial T, i} c_{i} + R_{i} \frac{\partial c }{\partial T }_{i}\end{dmath}
\begin{dmath} \Theta_{\partial T, i} = \frac{[C] S^{\prime\prime}_{\ns}}{T} {k_r}_{i} - \frac{[C] S^{\prime}_{\ns}}{T} {k_f}_{i} + \frac{{R_f}_{i}}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right) - \frac{{R_r}_{i}}{T} \left({\beta_r}_{i} + \frac{{E_{a,r}}_{i}}{T \Ru}\right)\end{dmath}
\subsubsection{Non-explicit reversible reactions}
\begin{dmath} \frac{\partial q }{\partial T }_{i} = \Theta_{\partial T, i} c_{i} + R_{i} \frac{\partial c }{\partial T }_{i}\end{dmath}
\begin{dmath} \Theta_{\partial T, i} = - \left(- \sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{1}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right)\right) {R_r}_{i} + \frac{[C] S^{\prime\prime}_{\ns}}{T} {k_r}_{i} - \frac{[C] S^{\prime}_{\ns}}{T} {k_f}_{i} + \frac{{R_f}_{i}}{T} \left(\beta_{i} + \frac{{E_{a}}_{i}}{T \Ru}\right)\end{dmath}
\subsubsection{Pressure-dependent reactions}
\begin{dmath} \frac{\partial q }{\partial T }_{i} = \Theta_{\partial T, i}\end{dmath}
For PLog reactions:
\begin{dmath} \Theta_{\partial T, i} = - \left(- \sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{1}{T} \left(\beta_1 + \frac{\left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right) \left(- \beta_1 + \beta_2 - \frac{E_{a_1}}{T \Ru} + \frac{E_{a_2}}{T \Ru}\right)}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} + \frac{E_{a_1}}{T \Ru}\right)\right) {R_r}_{i} + \frac{[C]}{T} \left(S^{\prime\prime}_{\ns} {k_r}_{i} - S^{\prime}_{\ns} {k_f}_{i}\right) + \frac{{R_f}_{i}}{T} \left(\beta_1 + \frac{\left(\log{\left (P \right )} - \log{\left (P_{1} \right )}\right) \left(- \beta_1 + \beta_2 - \frac{E_{a_1}}{T \Ru} + \frac{E_{a_2}}{T \Ru}\right)}{- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}} + \frac{E_{a_1}}{T \Ru}\right)\end{dmath}
For Chebyshev reactions:
\begin{dmath} \Theta_{\partial T, i} = \left(\sum_{k=1}^{\ns} \nu_{k,i} \frac{\text{d} B }{\text{d} T }_{k} + \frac{2 \log{\left (10 \right )}}{T^{2} \left(- \frac{1}{T_{min}} + \frac{1}{T_{max}}\right)} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \left(j - 1\right) T_{l - 1}\left(\tilde{P}\right) U_{j - 2}\left(\tilde{T}\right) \eta_{l,j}\right) {R_r}_{i} + \log{\left (10 \right )} {R_f}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} - \frac{2 T_{l - 1}\left(\tilde{P}\right) U_{j - 2}\left(\tilde{T}\right) \eta_{l,j}}{T^{2} \left(- \frac{1}{T_{min}} + \frac{1}{T_{max}}\right)} \left(j - 1\right) + \frac{[C]}{T} \left(S^{\prime\prime}_{\ns} {k_r}_{i} - S^{\prime}_{\ns} {k_f}_{i}\right)\end{dmath}
\subsubsection{Pressure independent reactions}
\begin{dmath} \frac{\partial q }{\partial T }_{i} = \Theta_{\partial T, i}\end{dmath}
\subsubsection{Third-body enhanced reactions}
For mixture as third-body:
\begin{dmath} \frac{\partial q }{\partial T }_{i} = [X]_{i} \Theta_{\partial T, i} - \frac{[C] \alpha_{\ns,i}}{T} R_{i}\end{dmath}
For species $m$ as third-body:
\begin{dmath} \frac{\partial q }{\partial T }_{i} = \Theta_{\partial T, i} \left(\left(- \delta_{\ns m} + 1\right) [C]_{m} + \delta_{\ns m} [C]_{\ns}\right) - \frac{\delta_{\ns m}}{T} [C] R_{i}\end{dmath}
If all $\alpha_{j,i} = 1$ for all species j:
\begin{dmath} \frac{\partial q }{\partial T }_{i} = [C] \left(\Theta_{\partial T, i} - \frac{R_{i}}{T}\right)\end{dmath}
\subsubsection{Unimolecular\slash recombination fall-off reactions}
\begin{dmath} \frac{\partial q }{\partial T }_{i} = \Theta_{\partial T, i} c_{i} + \left(\frac{F_{i} \bar{\theta}_{P_{r, i}, \partial T}}{P_{r, i} + 1} + \left(- \frac{P_{r, i} \Theta_{P_{r,i}, \partial T}}{P_{r, i} + 1} + \Theta_{F_i, \partial T} + \Theta_{P_{r,i}, \partial T} - \frac{\bar{\theta}_{P_{r, i}, \partial T}}{P_{r, i} + 1}\right) c_{i}\right) R_{i}\end{dmath}
\subsubsection{Chemically-activated bimolecular reactions}
\begin{dmath} \frac{\partial q }{\partial T }_{i} = \left(\Theta_{\partial T, i} + \left(- \frac{P_{r, i} \Theta_{P_{r,i}, \partial T}}{P_{r, i} + 1} + \Theta_{F_i, \partial T} - \frac{\bar{\theta}_{P_{r, i}, \partial T}}{P_{r, i} + 1}\right) R_{i}\right) c_{i}\end{dmath}
\subsubsection{Reduced Pressure Derivatives}
For mixture as third-body:
\begin{dmath} \Theta_{P_{r,i}, \partial T} = \frac{1}{T} \left(\beta_{0} - \beta_{\infty} + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right)\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial T} = - \frac{[C] k_{0, i} \alpha_{\ns,i}}{T k_{\infty, i}}\end{dmath}
For species $m$ as third-body:
\begin{dmath} \Theta_{P_{r,i}, \partial T} = \frac{1}{T} \left(\beta_{0} - \beta_{\infty} + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right)\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial T} = - \frac{[C] k_{0, i} \delta_{\ns m}}{T k_{\infty, i}}\end{dmath}
If all $\alpha_{j,i} = 1$ for all species j:
\begin{dmath} \Theta_{P_{r,i}, \partial T} = \frac{1}{T} \left(\beta_{0} - \beta_{\infty} - 1 + \frac{E_{a, 0}}{T \Ru} - \frac{E_{a, \infty}}{T \Ru}\right)\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial T} = 0\end{dmath}
\subsubsection{Falloff Blending Function Forms}
For Lindemann
\begin{dmath} \Theta_{F_i, \partial T} = 0\end{dmath}
For Troe
\begin{dmath} \Theta_{F_i, \partial T} = - \frac{B_{Troe}}{F_{cent} P_{r, i} \left(A_{Troe}^{2} + B_{Troe}^{2}\right)^{2} \log{\left (10 \right )}} \left(2 A_{Troe} F_{cent} \left(0.14 A_{Troe} + B_{Troe}\right) \left(P_{r, i} \Theta_{P_{r,i}, \partial T} + \bar{\theta}_{P_{r, i}, \partial T}\right) \log{\left (F_{cent} \right )} + P_{r, i} \frac{\text{d} F_{cent} }{\text{d} T } \left(2 A_{Troe} \left(1.1762 A_{Troe} - 0.67 B_{Troe}\right) \log{\left (F_{cent} \right )} - B_{Troe} \left(A_{Troe}^{2} + B_{Troe}^{2}\right) \log{\left (10 \right )}\right)\right)\end{dmath}
For SRI
\begin{dmath} \Theta_{F_i, \partial T} = - \frac{X \left(\frac{\operatorname{exp}\left({- \frac{T}{c}}\right)}{c} - \frac{a b}{T^{2}} \operatorname{exp}\left({- \frac{b}{T}}\right)\right)}{a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right)} + \frac{e}{T} - \frac{2 X^{2} \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}}{P_{r, i} \log^{2}{\left (10 \right )}} \left(P_{r, i} \Theta_{P_{r,i}, \partial T} + \bar{\theta}_{P_{r, i}, \partial T}\right) \log{\left (P_{r, i} \right )}\end{dmath}
\subsection{Molar Derivatives}
\begin{dmath} \mathcal{J}_{1,j + 2} = \frac{\partial\dot{T}}{\partial{n_j}} = \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(- \frac{1}{V} \frac{\text{d} T }{\text{d} t } \left(- {C_v}_{\ns} + {C_v}_{j}\right) - \sum_{k=1}^{-1 + \ns} \frac{1}{V} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{n} }{\partial {n_j} }_{k}\right)\end{dmath}
\begin{dmath} \mathcal{J}_{2,j + 2} = \frac{\partial \dot{ P } }{\partial {n_j} } = \frac{P}{T} \frac{\text{d} \dot{T} }{\text{d} {n_j} } + T \Ru \sum_{k=1}^{-1 + \ns} \frac{1}{V} \left(1 - \frac{W_{k}}{W_{\ns}}\right) \frac{\partial \dot{n} }{\partial {n_j} }_{k}\end{dmath}
\begin{dmath} \mathcal{J}_{k + 2,j + 2} = \frac{\partial \dot{n_k} }{\partial n_{j} } = V \sum_{i=1}^{\nr} \nu_{k,i} \frac{\partial q }{\partial {n_j} }_{i}\end{dmath}
Converting to Jacobian Update form:
\begin{dmath} \mathcal{J}_{k + 2,j + 2}\pluseq V \nu_{k,i} \frac{\partial q }{\partial {n_j} }_{i}\end{dmath}
\begin{dmath} V \frac{\partial q }{\partial {n_j} }_{k} = \left(\left(S^{\prime\prime}_{\ns} - S^{\prime\prime}_{j}\right) {k_r}_{i} - \left(S^{\prime}_{\ns} - S^{\prime}_{j}\right) {k_f}_{i}\right) c_{i} + V R_{i} \frac{\partial c }{\partial {n_j} }_{i}\end{dmath}
\subsubsection{Pressure-dependent reactions}
\begin{dmath} V \frac{\partial q }{\partial {n_j} }_{k} = \left(S^{\prime\prime}_{\ns} - S^{\prime\prime}_{j}\right) {k_r}_{i} - \left(S^{\prime}_{\ns} - S^{\prime}_{j}\right) {k_f}_{i}\end{dmath}
\subsubsection{Pressure independent reactions}
\begin{dmath} V \frac{\partial q }{\partial {n_j} }_{k} = \left(S^{\prime\prime}_{\ns} - S^{\prime\prime}_{j}\right) {k_r}_{i} - \left(S^{\prime}_{\ns} - S^{\prime}_{j}\right) {k_f}_{i}\end{dmath}
\subsubsection{Third-body enhanced reactions}
\textbf{For mixture as third-body}:
\begin{dmath} V \frac{\partial q }{\partial {n_j} }_{k} = [X]_{i} \left(\left(S^{\prime\prime}_{\ns} - S^{\prime\prime}_{j}\right) {k_r}_{i} - \left(S^{\prime}_{\ns} - S^{\prime}_{j}\right) {k_f}_{i}\right) + \left(- \alpha_{\ns,i} + \alpha_{j,i}\right) R_{i}\end{dmath}
\textbf{For species $m$ as third-body}:
\begin{dmath} V \frac{\partial q }{\partial {n_j} }_{k} = \left(\left(- \delta_{\ns m} + 1\right) [C]_{m} + \delta_{\ns m} [C]_{\ns}\right) \left(\left(S^{\prime\prime}_{\ns} - S^{\prime\prime}_{j}\right) {k_r}_{i} - \left(S^{\prime}_{\ns} - S^{\prime}_{j}\right) {k_f}_{i}\right) + \left(- \delta_{\ns m} + \delta_{j m}\right) R_{i}\end{dmath}
\textbf{If all $\alpha_{j,i} = 1$}:
\begin{dmath} V \frac{\partial q }{\partial {n_j} }_{k} = [C] \left(\left(S^{\prime\prime}_{\ns} - S^{\prime\prime}_{j}\right) {k_r}_{i} - \left(S^{\prime}_{\ns} - S^{\prime}_{j}\right) {k_f}_{i}\right)\end{dmath}
\subsubsection{Falloff Reactions}
\textbf{Unimolecular\slash recombination fall-off reactions}:
\begin{dmath} V \frac{\partial q }{\partial {n_j} }_{i} = \frac{k_{0, i} \bar{\theta}_{P_{r, i}, \partial n_j} R_{i}}{k_{\infty, i} \left(P_{r, i} + 1\right)} \left(F_{i} P_{r, i} \Theta_{F_i, \partial n_j} + F_{i} - c_{i}\right) + \left(\left(S^{\prime\prime}_{\ns} - S^{\prime\prime}_{j}\right) {k_r}_{i} - \left(S^{\prime}_{\ns} - S^{\prime}_{j}\right) {k_f}_{i}\right) c_{i}\end{dmath}
\subsubsection{Chemically-activated bimolecular reactions}
\begin{dmath} V \frac{\partial q }{\partial {n_j} }_{i} = \frac{k_{0, i} \bar{\theta}_{P_{r, i}, \partial n_j} R_{i}}{k_{\infty, i} \left(P_{r, i} + 1\right)} \left(F_{i} \Theta_{F_i, \partial n_j} - c_{i}\right) + \left(\left(S^{\prime\prime}_{\ns} - S^{\prime\prime}_{j}\right) {k_r}_{i} - \left(S^{\prime}_{\ns} - S^{\prime}_{j}\right) {k_f}_{i}\right) c_{i}\end{dmath}
\subsubsection{Reduced Pressure Derivatives}
\textbf{For mixture as third-body}:
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial n_j} = - \alpha_{\ns,i} + \alpha_{j,i}\end{dmath}
\textbf{For species $m$ as third-body}:
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial n_j} = - \delta_{\ns m} + \delta_{j m}\end{dmath}
\textbf{If all $\alpha_{j,i} = 1$}:
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial n_j} = 0\end{dmath}
\subsubsection{Falloff Blending Function Forms}
For Lindemann
\begin{dmath} \Theta_{F_i, \partial n_j} = 0\end{dmath}
For Troe
\begin{dmath} \Theta_{F_i, \partial n_j} = - \frac{2 A_{Troe} B_{Troe} \left(0.14 A_{Troe} + B_{Troe}\right) \log{\left (F_{cent} \right )}}{P_{r, i} \left(A_{Troe}^{2} + B_{Troe}^{2}\right)^{2} \log{\left (10 \right )}}\end{dmath}
For SRI
\begin{dmath} \Theta_{F_i, \partial n_j} = - \frac{2 X^{2} \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}}{P_{r, i} \log^{2}{\left (10 \right )}} \log{\left (P_{r, i} \right )}\end{dmath}
\subsection{Pressure Derivatives}
\begin{dmath} \mathcal{J}_{1,2} = \frac{\partial\dot{T}}{\partial{P}} = \frac{1}{\sum_{k=1}^{\ns} [C]_{k} {C_v}_{k}} \left(- \sum_{k=1}^{-1 + \ns} \frac{1}{V} \left(U_{k} - \frac{W_{k} U_{\ns}}{W_{\ns}}\right) \frac{\partial \dot{n} }{\partial P }_{k} - \frac{{C_v}_{\ns}}{T \Ru} \frac{\text{d} T }{\text{d} t }\right)\end{dmath}
\begin{dmath} \mathcal{J}_{2,2} = \frac{\partial \dot{ P } }{\partial P } = T \Ru \sum_{k=1}^{-1 + \ns} \frac{1}{V} \left(1 - \frac{W_{k}}{W_{\ns}}\right) \frac{\partial \dot{n} }{\partial P }_{k} + \frac{1}{T} \left(P \frac{\text{d} \dot{T} }{\text{d} P } + \dot{T}\right)\end{dmath}
\begin{dmath} \mathcal{J}_{k + 2,2} = \frac{\partial \dot{n_k} }{\partial P } = V \sum_{i=1}^{\nr} \nu_{k,i} \frac{\partial q }{\partial P }_{i}\end{dmath}
Converting to Jacobian Update form:
\begin{dmath} \mathcal{J}_{k + 2,2}\pluseq V \nu_{k,i} \frac{\partial q }{\partial P }_{i}\end{dmath}
\begin{dmath} \frac{\partial q }{\partial P }_{k} = \left(- \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru} + \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru}\right) c_{i} + R_{i} \frac{\partial c }{\partial P }_{i}\end{dmath}
\subsubsection{Pressure-dependent reactions}
For PLOG:
\begin{dmath} \frac{\partial q }{\partial P }_{k} = \frac{1}{T \Ru} \left(- S^{\prime\prime}_{\ns} {k_r}_{i} + S^{\prime}_{\ns} {k_f}_{i}\right) + \frac{\left(- \log{\left (k_{1} \right )} + \log{\left (k_{2} \right )}\right) \left({R_f}_{i} - {R_r}_{i}\right)}{P \left(- \log{\left (P_{1} \right )} + \log{\left (P_{2} \right )}\right)}\end{dmath}
For Chebyshev:
\begin{dmath} \frac{\partial q }{\partial P }_{k} = \log{\left (10 \right )} {R_f}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \frac{2 \left(l - 1\right) T_{j - 1}\left(\tilde{T}\right) U_{l - 2}\left(\tilde{P}\right) \eta_{l,j}}{P \left(\log{\left (P_{max} \right )} - \log{\left (P_{min} \right )}\right)} - \log{\left (10 \right )} {R_r}_{i} \sum_{\substack{1 \leq l \leq N_{P}\\1 \leq j \leq N_{T}}} \frac{2 \left(l - 1\right) T_{j - 1}\left(\tilde{T}\right) U_{l - 2}\left(\tilde{P}\right) \eta_{l,j}}{P \left(\log{\left (P_{max} \right )} - \log{\left (P_{min} \right )}\right)} + \frac{1}{T \Ru} \left(- S^{\prime\prime}_{\ns} {k_r}_{i} + S^{\prime}_{\ns} {k_f}_{i}\right)\end{dmath}
\subsubsection{Pressure independent reactions}
\begin{dmath} \frac{\partial q }{\partial P }_{k} = - \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru} + \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru}\end{dmath}
\subsubsection{Third-body enhanced reactions}
\textbf{For mixture as third-body}:
\begin{dmath} \frac{\partial q }{\partial P }_{k} = [X]_{i} \left(- \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru} + \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru}\right) + \frac{\alpha_{\ns,i} R_{i}}{T \Ru}\end{dmath}
\textbf{For species $m$ as third-body}:
\begin{dmath} \frac{\partial q }{\partial P }_{k} = \left(\left(- \delta_{\ns m} + 1\right) [C]_{m} + \delta_{\ns m} [C]_{\ns}\right) \left(- \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru} + \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru}\right) + \frac{\delta_{\ns m} R_{i}}{T \Ru}\end{dmath}
\textbf{If all $\alpha_{j,i} = 1$}:
\begin{dmath} \frac{\partial q }{\partial P }_{k} = [C] \left(- \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru} + \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru}\right) + \frac{R_{i}}{T \Ru}\end{dmath}
\subsubsection{Unimolecular/recombination fall-off reactions}
\begin{dmath} \frac{\partial q }{\partial P }_{i} = \left(\frac{F_{i} \bar{\theta}_{P_{r, i}, \partial P}}{P_{r, i} + 1} + \left(\Theta_{F_i, \partial P} - \frac{\bar{\theta}_{P_{r, i}, \partial P}}{P_{r, i} + 1}\right) c_{i}\right) R_{i} + \left(- \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru} + \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru}\right) c_{i}\end{dmath}
\subsubsection{Chemically-activated bimolecular reactions}
\begin{dmath} \frac{\partial q }{\partial P }_{i} = \left(\left(\Theta_{F_i, \partial P} - \frac{\bar{\theta}_{P_{r, i}, \partial P}}{P_{r, i} + 1}\right) R_{i} - \frac{S^{\prime\prime}_{\ns} {k_r}_{i}}{T \Ru} + \frac{S^{\prime}_{\ns} {k_f}_{i}}{T \Ru}\right) c_{i}\end{dmath}
\subsubsection{Reduced Pressure Derivatives}
\textbf{For mixture as third-body}:
\begin{dmath} \Theta_{P_{r,i}, \partial P} = 0\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial P} = \frac{k_{0, i} \alpha_{\ns,i}}{T k_{\infty, i} \Ru}\end{dmath}
\textbf{For species $m$ as third-body}:
\begin{dmath} \Theta_{P_{r,i}, \partial P} = 0\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial P} = \frac{k_{0, i} \delta_{\ns m}}{T k_{\infty, i} \Ru}\end{dmath}
\textbf{If all $\alpha_{j,i} = 1$}:
\begin{dmath} \Theta_{P_{r,i}, \partial P} = 0\end{dmath}
\begin{dmath} \bar{\theta}_{P_{r, i}, \partial P} = \frac{k_{0, i}}{k_{\infty, i}} \frac{\partial [C] }{\partial P }\end{dmath}
\subsubsection{Falloff Blending Function Forms}
For Lindemann
\begin{dmath} \Theta_{F_i, \partial P} = 0\end{dmath}
For Troe
\begin{dmath} \Theta_{F_i, \partial P} = - \frac{2 A_{Troe} B_{Troe} \bar{\theta}_{P_{r, i}, \partial P} \left(0.14 A_{Troe} + B_{Troe}\right) \log{\left (F_{cent} \right )}}{P_{r, i} \left(A_{Troe}^{2} + B_{Troe}^{2}\right)^{2} \log{\left (10 \right )}}\end{dmath}
For SRI
\begin{dmath} \Theta_{F_i, \partial P} = - \frac{2 X^{2} \bar{\theta}_{P_{r, i}, \partial P} \log{\left (P_{r, i} \right )}}{P_{r, i} \log^{2}{\left (10 \right )}} \log{\left (a \operatorname{exp}\left({- \frac{b}{T}}\right) + \operatorname{exp}\left({- \frac{T}{c}}\right) \right )}\end{dmath}
\end{document}


\section{History of the Algorithms} \label{sec:history}
Back when I was a young naive programmer, I made a thing. Now a few years down the line I made the thing again, but infinitely better. So I have no use for the old thing anymore. But fear not,
old algorithms (used by CLAuDE) will be collected here. This is just for historical purposes.

\subsection{Radiation}
\subsubsection{Adding Layers}
Remember \autoref{eq:atmos change}? We need this equation for every layer in the atmosphere. This also means that we have to adjust the main calculation of the code, which is described  in
\autoref{alg:temperature with density}. The $T_a$ needs to change, we need to either add a dimension (to indicate which layer of the atmosphere we are talking about) or we need to add different
matrices for each atmosphere layer. We opt for adding a dimension as that costs less memory than defining new arrays
\footnote{This has to do with pointers, creating a new object always costs a bit more space than adding a dimension as we need a pointer to the object and what type of object it is whereas with
adding a dimension we do not need this additional information as it has already been defined}. So $T_a$, and all other matrices that have to do with the atmosphere (so not $T_p$ for instance)
are no longer indexed by $lat, lon$ but are indexed by $lat, lon, layer$. We need to account for one more thing, the absorbtion of energy from another layer. The new equation is shown in
\autoref{eq:atmos change layer}. Here $k$ is the layer of the atmosphere, $k = -1$ means that you use $T_p$ and $k = nlevels$ means that $T_{a_{nlevels}} = 0$ as that is space. Also, let us
rewrite the equation a bit such that the variables that are repeated are only written once and stuff that is divided out is removed, which is done in \autoref{eq:atmos change layer improved}.
Let us also clean up the equation for the change in the surface temperature (\autoref{eq:surface change}) in \autoref{eq:surface change improved}.

\begin{subequations}
    \begin{equation}
        \label{eq:atmos change layer}
        \Delta T_{a_k} = \frac{\delta t (\sigma \epsilon_{k - 1}T_{a_{k - 1}}^4 + \sigma \epsilon_{k + 1}T_{a_{k + 1}}^4 - 2\epsilon_k\sigma T_{a_k}^4)}{C_a}
    \end{equation}
    \begin{equation}
        \label{eq:atmos change layer improved}
        \Delta T_{a_k} = \frac{\delta t \sigma (\epsilon_{k - 1}T_{a_{k - 1}}^4 + \epsilon_{k + 1}T_{a_{k + 1}}^4 - 2\epsilon_kT_{a_k}^4)}{C_a}
    \end{equation}
    \begin{equation}
        \label{eq:surface change improved}
        \Delta T_p = \frac{\delta t (S + \sigma(4\epsilon_pT_a^4 - 4T_p^4))}{4C_p}
    \end{equation}
\end{subequations}

With the changes made to the equation, we need to make those changes in the code as well. We need to add the new dimension to all matrices except $T_p$ and $a$ as they are unaffected (with
regards to the storage of the values) by the addition of multiple atmospheric layers. Every other matrix is affected. The new code can be found in \autoref{alg:temperature layer}. $\delta z$

\begin{algorithm}[hbt]
    \caption{The main function for the temperature calculations}
    \label{alg:temperature layer}
    \SetAlgoLined
    \SetKwInput{Input}{Input}
    \SetKwInOut{Output}{Output}
    \Input{amount of energy that hits the planet $S$}
    \Output{Temperature of the planet $T_p$, temperature of the atmosphere $T_a$}
    \For{$lat \leftarrow -nlat$ \KwTo $nlat$}{
        \For{$lon \leftarrow 0$ \KwTo $nlot$}{
            \For{$layer \leftarrow 0$ \KwTo $nlevels$}{
                $T_p[lat, lon] \leftarrow T_p[lat, lon] + \frac{\delta t ((1 - a[lat, lon])S + \sigma(4\epsilon[0](T_a[lat, lon, 0])^4 - 4(T_p[lat, lon])^4))}
                {4C_p[lat, lon]}$ \;
                \uIf{$layer = 0$}{
                    $T_a[lat, lon, layer] \leftarrow T_a[lat, lon, layer] + \frac{\delta t \sigma((T_p[lat, lon])^4 - 2\epsilon[layer](T_a[lat, lon, layer])^4)}
                    {\rho[lat, lon, layer]C_a\delta z[layer]}$ \;
                }\uElseIf{$layer = nlevels - 1$}{
                    $T_a[lat, lon, layer] \leftarrow T_a[lat, lon, layer] + \frac{\delta t \sigma(\epsilon[layer - 1](T_a[lat, lon, layer - 1])^4 - 2\epsilon[layer](T_a[lat, lon, layer])^4)}
                    {\rho[lat, lon, layer]C_a\delta z[layer]}$ \;
                }\uElse{
                    $T_a[lat, lon, layer] \leftarrow T_a[lat, lon, layer] + \frac{\delta t \sigma(\epsilon[layer - 1](T_a[lat, lon, layer - 1])^4 + \epsilon[layer + 1]T_a[lat, lon, layer + 1]
                    - 2\epsilon[layer](T_a[lat, lon, layer])^4)}{\rho[lat, lon, layer]C_a\delta z[layer]}$ \;
                }
            }
        }
    }
\end{algorithm}

We also need to initialise the $\epsilon$ value for each layer. We do that in \autoref{alg:epsilon}.

\begin{algorithm}
    \caption{Intialisation of the insulation of each layer (also known as $\epsilon$)}
    \label{alg:epsilon}
    $\epsilon[0] \leftarrow 0.75$ \;
    \For{$i \leftarrow 1$ \KwTo $nlevels$}{
        $\epsilon[i] \leftarrow 0.5\epsilon[i - 1]$
    }
\end{algorithm}

\subsection{Velocity}
\subsubsection{The Primitive Equations and Geostrophy} \label{sec:primitive}
The primitive equations (also known as the momentum equations) is what makes the air move. It is actually kind of an injoke between physicists as they are called the primitive equations but
actually look quite complicated (and it says $fu$ at the end! \cite{simon}). The primitive equations are a set of equations dictating the direction in the $u$ and $v$ directions as shown in
\autoref{eq:primitive u} and \autoref{eq:primitive v}. We can make the equations simpler by using and approximation called geostrophy which means that we have no vertical motion, such that the
terms with $\omega$ in \autoref{eq:primitive u} and \autoref{eq:primitive v} become 0. We also assume that we are in a steady state, i.e. there is no acceleration which in turn means that the
whole middle part of the equations are $0$. Hence we are left with \autoref{eq:primitive u final} and \autoref{eq:primitive v final}.

\begin{subequations}
    \begin{equation}
        \label{eq:primitive u}
        \frac{du}{dt} = \frac{\delta u}{\delta t} + u\frac{\delta u}{ \delta x} + v\frac{\delta u}{\delta v} + \omega\frac{\delta u}{\delta p} = -\frac{\delta \Phi}{\delta x} + fv
    \end{equation}
    \begin{equation}
        \label{eq:primitive v}
        \frac{dv}{dt} = \frac{\delta v}{\delta t} + u\frac{\delta v}{ \delta x} + v\frac{\delta v}{\delta v} + \omega\frac{\delta v}{\delta p} = -\frac{\delta \Phi}{\delta y} - fu
    \end{equation}
    \begin{equation}
        \label{eq:primitive u final}
        0 = -\frac{\delta \Phi}{\delta x} + fv
    \end{equation}
    \begin{equation}
        \label{eq:primitive v final}
        0 = -\frac{\delta \Phi}{\delta y} - fu
    \end{equation}
\end{subequations}

\autoref{eq:primitive u final} can be split up into to parts, the $\frac{\delta \Phi}{\delta x}$ part (the gradient force) and the $fv$ part (the coriolis force). The same applies to
\autoref{eq:primitive v final}. Effectively we have a balance between the gradient and the coriolis force as shown in \autoref{eq:pu simple} and \autoref{eq:pv simple}. The symbols in both of
these equations are:

\begin{itemize}
    \item $\Phi$: The geopotential, potential (more explanation in \autoref{sec:potential}) of the planet's gravity field ($Jkg^{-1}$).
    \item $x$: The change in the East direction along the planet surface ($m$).
    \item $y$: The change in the North direction along the planet surface ($m$).
    \item $f$: The coriolis parameter as described by \autoref{eq:coriolis}, where $\Omega$ is the rotation rate of the planet (for Earth $7.2921 \cdot 10^{-5}$) ($rad \ s^{-1}$) and $\theta$ is the
    latitude \cite{coriolis}.
    \item $u$: The velocity in the latitude ($ms^{-1}$).
    \item $v$: The velocity in the longitude ($ms^{-1}$).
\end{itemize}

\begin{subequations}
    \begin{equation}
        \label{eq:coriolis}
        f = 2\Omega\sin(\theta)
    \end{equation}
    \begin{equation}
        \label{eq:pu simple}
        \frac{\delta \Phi}{\delta x} = fv
    \end{equation}
    \begin{equation}
        \label{eq:pv simple}
        \frac{\delta \Phi}{\delta y} = -fu
    \end{equation}
    \begin{equation}
        \label{eq:pu simple final}
        \frac{\delta p}{\rho \delta x} = fv
    \end{equation}
    \begin{equation}
        \label{eq:pv simple final}
        \frac{\delta p}{\rho \delta y} = -fu
    \end{equation}
\end{subequations}

Since we want to know how the atmosphere moves, we want to get the v and u components of the velocity vector (since $v$ and $u$ are the veolicites in longitude and latitude, if we combine them
in a vector we get the direction of the overall velocity). So it is time to start coding and calculating! If we look back at \autoref{alg:stream1v2}, we can see that we already have a double
for loop. In computer science, having multiple loops is generally considered a bad coding practice as you usually can just reuse the indices of the already existing loop, so you do not need to
create a new one. However this is a special case, since we are calculating new temperatures in the double for loop. If we then also would start to calculate the velocities then we would use new
information and old information at the same time. Since at index $i - 1$ the new temperature has already been calculated, but at the index $i + 1$ the old one is still there. So in order to fix
that we need a second double for loop to ensure that we always use the new temperatures. We display this specific loop in \autoref{alg:stream2}. Do note that everything in \autoref{alg:stream1v2}
is still defined and can still be used, but since we want to focus on the new code, we leave out the old code to keep it concise and to prevent clutter.

\begin{algorithm}[hbt]
    \caption{The main loop of the velocity of the atmosphere calculations}
    \label{alg:stream2}
    \SetAlgoLined
    \For{$lat \in [-nlat, nlat]$}{
        \For{$lon \in [0, nlon]$}{
            $u[lat, lon] \leftarrow -\frac{p[lat + 1, lon] - p[lat - 1, lon]}{\delta y} \cdot \frac{1}{f[lat]\rho}$ \;
            $v[lat, lon] \leftarrow \frac{p[lat, lon + 1] - p[lat, lon - 1]}{\delta x[lat]} \cdot \frac{1}{f[lat]\rho}$ \;
        }
    }
\end{algorithm}

The gradient calculation is done in \autoref{alg:gradient}. For this to work, we need the circumference of the planet. Herefore we need to assume that the planet is a sphere. While that is not
technically true, it makes little difference in practice and is good enough for our model. The equation for the circumference can be found in \autoref{eq:circumference} \cite{circumference},
where $r$ is the radius of the planet. Here we also use the f-plane approximation, where the coriolis paramter has one value for the northern hemisphere and one value for the southern hemisphere
\cite{fplane}.

\begin{equation}
    \label{eq:circumference}
    2 \pi r
\end{equation}

\begin{algorithm}
    \caption{Calculating the gradient $\delta x$ (note that this algorithm is obsolete)}
    \label{alg:gradient}
    \SetAlgoLined
    $C \leftarrow 2\pi R$ \;
    $\delta y \leftarrow \frac{C}{nlat}$ \;

    \For{$lat \in [-nlat, nlat]$}{
        $\delta x[lat] \leftarrow \delta y \cos(lat \cdot \frac{\pi}{180})$ \;

        \eIf{$lat < 0$}{
            $f[lat] \leftarrow -10^{-4}$ \;
        }{
            $f[lat] \leftarrow 10^{-4}$ \;
        }
    }
\end{algorithm}

Because of the geometry of the planet and the construction of the longitude latitude grid, we run into some problems when calculating the gradient. Since the planet is not flat ("controversial
I know"\cite{simon}) whenever we reach the end of the longitude we need to loop around to get to the right spot to calculate the gradients (as the planet does not stop at the end of the
longitude line but loops around). So to fix that we use the modulus (mod) function which does the looping for us if we exceed the grid's boundaries. We do haveanother problem though, the poles.
As the latitude grows closer to the poles, they are converging on the center point of the pole. Looping around there is much more difficult so to fix it, we just do not consider that center
point in the main loop. The changed algorithm can be found in \autoref{alg:stream2v2}

\begin{algorithm}[hbt]
    \caption{The main loop of the velocity of the atmosphere calculations}
    \label{alg:stream2v2}
    \SetAlgoLined
    \For{$lat \in [-nlat + 1, nlat - 1]$}{
        \For{$lon \in [0, nlon]$}{
            $u[lat, lon] \leftarrow -\frac{p[(lat + 1) \text{ mod } nlat, lon] - p[(lat -1) \text{ mod } nlat, lon]}{\delta y} \cdot \frac{1}{f[lat]\rho}$ \;
            $v[lat, lon] \leftarrow \frac{p[lat, (lon + 1) \text{ mod } nlon] - p[lat, (lon -1) \text{ mod } nlon]}{\delta x[lat]} \cdot \frac{1}{f[lat]\rho}$ \;
        }
    }
\end{algorithm}

Do note that the pressure calculation is done between the temperature calculation in \autoref{alg:stream1v2} and the $u, v$ calculations in \autoref{alg:stream2v2}. At this point our model shows
a symmetric vortex around the sun that moves with the sun. This is not very realistic as you usually have convection and air flowing from warm to cold, but we do not have that complexity yet
(due to our single layer atmosphere).


%!TEX root=report.tex
\subsection{k-means clustering}
\label{section:result-kmeans}

\subsubsection{Indentifing the amount of clusters}
Due to the size of the dataset (64800, 341), and the fact multiple simulated datasets of the same size would be needed to calculate the gap-statistic it was calculated on the HPC cluster that DTU offers for students and faculty.
It should be noted that if such a setup had not been available, one could have used smaller subsamples of the data.
20 simulation samples of size (64800,341) were made using a multivariate uniform distribution. The following is the plot of the resulting gap statistic with its standard deviation.
\begin{figure}[H]
	\center
	\includegraphics[width=\textwidth]{figures/kmeans-gap}
	\caption{Gap-statistics with standard deviation. Note the standard deviation is very small.}
	\label{fig:kmeans-gap}
\end{figure}

From Figure \ref{fig:kmeans-gap} it is seen that according to the gap-statistics the optimal amount of cluster, is more than 20. Unfortunately such a high number of clusters are not suitable for visualization. Instead 7 clusters have been chosen; this is based on the large slope change which can be observed in the gap-statistics graph. This is also seems like a suitable number of colors for visualization.

\begin{figure}[H]
	\center
	\includegraphics[width=\textwidth]{figures/kmeans-world}
	\caption{Each position is an point with belongs to the cluster, with the closest centroid.}
	\label{fig:kmeans-world}
\end{figure}
\begin{figure}[H]
	\center
	\includegraphics[width=\textwidth]{figures/kmeans-centroids}
	\caption{Cluster centroids. The colors correspond to those in Figure \ref{fig:kmeans-world}}
	\label{fig:kmeans-centroids}
\end{figure}

Comparing the two plots above a few interesting insights are gained:
\begin{itemize}
	\item Orange areas have a slight mass increase. At the south pole it appears that some of the mass loss at the edge actually moves inward towards the South Pole. This may be caused by post glacial rebound as no GIA has been performed.
	\item  Green and blue correlates with extreme and regular seasonality (i.e. rain season in Amazon Basin).
	\item Light green and pink correspond with trending mass loss. The most significant locations appear to be located around the tip of the Western Antarctica along with Greenland's east coast.
\end{itemize}