# How do we know that rates depend on the product and not the sum of reactant concentrations [closed]

I've been struggling to understand the mathematics behind the calculation in enzyme kinetics within systems biology at an intuitive level.

Every research article and textbook says it is based on the mass-action law, and I never came across how this law was invented.

for e.g in the chemical reaction $$\ce{\alpha A + \beta B <=> \gamma C + \delta D}.$$

The law of mass action gives the following kinetics

$$V^- = {-k[C]^\gamma [D]^\delta}$$ $$V^+ = {k[A]^\alpha [B]^\beta}$$

What was the reason behind taking the product, instead of taking addition.

Why is the following equation incorrect?

$$V^- = {-k[C]^\gamma + [D]^\delta}$$ $$V^+ = {k[A]^\alpha + [B]^\beta}$$

Was the law of mass-action experimentally derived or it was mathematically derived? What derivation rule was used to take the product instead of addition.

I want to understand this concept of taking the product of concentrations rather than cramming it.

• chemistry.stackexchange.com/questions/68195/… Aug 22, 2022 at 12:08
• chemistry.stackexchange.com/questions/6924/… Aug 22, 2022 at 12:10
• @Dendrobium - OK, what about them (and other linked answers) not answer? I might have chosen chemistry.stackexchange.com/questions/14836/… instead. Just saying it didn't answer you question doesn't really help the rest of us figure out what you are having difficulty with. Aug 22, 2022 at 12:55
• babel.hathitrust.org/cgi/… If anyone can read german here's the paper that proposed it Aug 22, 2022 at 13:01
• Note this site is not a forum for extended discussions and dialogue, although there are chat rooms available as need be. The idea is to compile "canonical" answers, the best answers that address clear questions. If there are answers that already address your problem then yours might be closed as a duplicate. OTOH if it seems unclear what you are confused about (requiring you to explain yourself in more detail - in this case justify the equation you wrote), it might also get closed. Aug 23, 2022 at 8:32

What was the reason behind taking the product, instead of taking addition.

There are at least three reasons:

1. Experiment: Empirical data on reactions that were experimentally studied.
2. Logic: If one reactant is missing, an "addition formula" would still give a reaction even though the reaction is impossible
3. Theory: The combined evidence led to the concept of collision theory, where reactants collide to give a reaction. The probability of collision is given by an expression that includes the product of the concentrations of particles colliding.

The relationship between kinetics and law of mass action can be more complicated for reactions that do not consist of an elementary step, and the answers to the linked questions give some information about that topic.

Was the law of mass-action experimentally derived or it was mathematically derived? What derivation rule was used to take the product instead of addition.

The law of mass action was experimentally derived but now has a rigorous derivation as part of equilibrium thermodynamics.