Does anyone know the analytical solutions to the following rate laws:
1) $\frac{\mathrm{d}\rho_\text{W}}{\mathrm{d}t} = -\rho_\text{W} \cdot (K_1+K_2+K_3)$
2) $\frac{\mathrm{d}\rho_\text{T}}{\mathrm{d}t} = \rho_\text{W} \cdot K_2 - (K_4+K_5) \cdot \rho_\text{T}$
3) $\frac{\mathrm{d}\rho_\text{C}}{\mathrm{d}t} = \rho_\text{W} \cdot K_3 + \rho_\text{T} \cdot K_5$
As an example, I have determined the following analytical solution for this rate law:
$\frac{\mathrm{d}\rho_\text{W}}{\mathrm{d}t} = -\rho_\text{W} \cdot (K_1+K_2+K_3) \qquad \Rightarrow \qquad \rho_\text{W} = \rho_{\text{W}0} \cdot \operatorname{e}^{-(K_1+K_2+K_3) \cdot t}$
where $\rho_{\text{W}0} = $ initial density, $K = $ reaction rate constant, $t =$ time
The scheme for the reactions are:
$\ce{wood} \xrightarrow{K_{1}}{} \ce{gas}$
$\ce{wood} \xrightarrow{K_{2}}{} \ce{tar}$
$\ce{wood} \xrightarrow{K_{3}}{} \ce{char}$
$\ce{tar} \xrightarrow{K_{4}}{} \ce{gas}$
$\ce{tar} \xrightarrow{K_{5}}{} \ce{char}$
Any help with the three rate laws above would be greatly appreciated.