Conceptual question about rate laws

[minivanhighwa]

So anytime I've seen textbooks explain integrated rate laws, they usually start with a reaction of the form A -> B and then from there say, if we know the reaction is first order with respect to [A] then:

v = -d[A]/dt = k[A]

And then subsequently integrate this to find ln[A]/[A]i = -kt

I get that. My issue is this. From what I always thought, the integrated rate laws are universal, and now that I have this equation I can use it any time I know that a reaction is first order with respect to A. But what if I have an equation of the form:

aA -> bB

where there is a stoichiometric coefficient in front of A. I was always under the impression that v = - (1/a) d[A]/dt = (1/b) d/dt... If experimental data still tells us that the rate law is first order with respect to A, can I just use the same integrated rate law that I found above? or would I have to say the following:

v = - (1/a) d[A]/dt = k[A]

in which case you would get:
ln[A]/[A]i = -(a)kt

So for a reaction 2A -> B

You would find ln[A]/[A]i = -2kt

Assuming the rate was always first order with respect to [A], wouldn't you get a different integrated rate law for every instance that you have a different stoichiometric factor (a) in front of your reactant? If so why isn't the general form of the equation given as

ln[A]/[A]i = -akt

Hope this makes sense - not sure where my reasoning is off

 

[Mike H]

I suspect this is a matter of definitions and pedagogy.

When I think of a "different rate law," I view the zero-order rate law

[A] = [A]₀ - kt

to be different than the first-order rate law

[A] = [A]₀*exp(-kt).

Adjusting a rate law by a constant to correct for stoichiometry does not substantively change the form of the rate law.

I suspect the reason that a more general rate law is not introduced for a few reasons, most of which has to do with avoiding confusion between order of a reaction and stoichiometry (I used to notice this happening quite a bit while a teaching assistant for introductory general chemistry) and since - if feasible - one can pick what to measure in an experiment to make your subsequent analysis of the reaction kinetics & mechanism easier.