# How to rewrite kinetic mass balance to get to proper SI units?

Suppose from kinetic mass balance I get the following differential equation of the molarity substances $$A$$,$$B$$, and $$C$$ in a reactor, with molar in- or outflow rate $$\phi$$, and rate constant $$k$$:

$$\frac{\mathrm{d}}{\mathrm{d}t}[C] = k[A]^2[B]^3 + \phi{C_{\mathrm{in}}} - \phi{C_{\mathrm{out}}}$$

Where the dimensions convert to:

$$[\pu{mol L-1 s-1}] = [\pu{s-1}] [\pu{mol5 L-5}] + [\pu{mol s-1}] - [\pu{mol s-1}]$$

Which can be simplified by adding the flow rate terms to:

$$[\pu{mol L-1 s-1}] = [\pu{s-1}] [\pu{mol5 L-5}] \pm [\pu{mol s-1}]$$

By cancelling similar terms mole,seconds on both sides to:

$$[\pu{L-1}] = [\pu{mol4 L-5}] \pm 1$$

This seems that something is off about the rate constant, but also that there is no $$\pu{L-1}$$ term in the molar flow rate.

I don't expect exact answers to this, but could someone point me the right way, or send some clear explanation on either reaction rate constants or on kinetic mass balances? It seems I cannot find clear explanations, not even in textbooks.

• Consider Formatting of chem expressions by using mhchem MathJax package, available in CH SE by default, involving MJ syntax \ce{} and \pu{} E.g. write $\ce{H2SO4}$ or $\ce{a A <=> p P}$ or $\pu{6.022E23 mol-1}$ to get $\ce{H2SO4}$ or $\ce{a A <=> p P}$ or $\pu{6.022E23 mol-1}$ (all eventually with double dollars in the display mode like $$\ce{H2SO4}$$. // E.g. typographic rules say unit symbols are never in italic. Jul 10, 2022 at 11:22
• Okay will do, but do you have an answer to my question? Jul 10, 2022 at 14:57
• I know nothing about this, but looking at the dimensions you firstly seemed to have used for concentration $mol L^{-1}$ on the LHS and simply $mol$ on the RHS, fixing that solves your second problem, and secondly by dimensionality arguments your assertion that the dimensions for k are $s^{-1}$ must be wrong. But don't ask me how to justify what the correct dimensions must be ($mol^{-4} L^4$). Jul 10, 2022 at 15:08
• Yes the units of k should be mol$^{-4}$L$^4$ Jul 10, 2022 at 15:52
• @IanBush I do not understand what you mean, the concentration is the same on both sides, except that on the LHS it is the derivative, so it is over time. Jul 10, 2022 at 16:22