# Using calculus to find the rate of a bivariate reaction

Recently, I've been learning about rates and have found the concept fairly easy to understand in the case of a reaction with a single reactant, which is essentially solving $$f'(t) = kf(t)^n$$ where $$f(t)$$ gives the concentration of the reactant at time $$t$$. Now, for a reaction with two reactants, I'm not exactly sure how to represent the rate in terms of a function. My best guess was defining a function $$r(t) = \left(x(t), y(t) \right)$$ where $$x(t)$$ and $$y(t)$$ represent the concentration of both reactants respectively at time $$t$$, then solving $$\mid \nabla r(t) \mid=kx(t)^ay(t)^b$$ This boils down to mostly a math problem but I don't know much about multivariable calculus. I'd appreciate if anyone could provide an answer or a link to a source. Thanks.

• Reactants don't react independently of each other, they are linked by stoichiometry. It is still a single variable problem. – Karsten Theis Dec 5 '19 at 16:01
• Agree with Karsten. Also, this is first of all chemistry, not maths, so there is no 'bivariate' reaction. In fact, regardless of how many different reagents are present, the rate of the reaction depends on its mechanism. A reaction with one reagent can be second order. Some reactions are pseudo-zero order because the rate-determining step depends on species whose concentrations don't change much. So my main advice here is: look at this as a chemistry problem first, understand what is actually happening in the system, and only then apply the appropriate maths to solve it. – user6376297 Dec 5 '19 at 17:39
• Do you mean something like this? – Rodrigo de Azevedo Dec 6 '19 at 20:03
• @Rodrigo: look at the comment by Ivan Neretin to the post you linked. Far too complicated for its own good. And again, while maths is necessary to solve some problems related to chemical systems, one must first understand what is happening chemically, then find the simplest mathematical model that addresses it. IMO, the main issue with the OP is that the 'two reactants' are treated as independent variables in some abstract mathematical world, whereas they are actual molecules colliding with each other, and their concentrations are linked by mass balances. It can't be right. – user6376297 Dec 7 '19 at 13:34
• @Rodrigo - actually I see now that your own answer to that post does take into account the concentrations mass balance, so it's correct, and basically the same as orthocresol suggested. Maybe you should have linked your own answer chemistry.stackexchange.com/a/84059/41751 rather than the OP. – user6376297 Dec 7 '19 at 13:53