[maakii]

I am wondering how one would find a rate equation that would work for all circumstances, without any assumptions such as steady-state?

for example,

X + 2Y -> XY₂

For this reaction, the mechanism is

Y + Y -> Y₂        (fast)
X + Y₂ -> XY₂     (slow)

So for this equation, the rate would be

rate = k[X][Y₂]

but how would we be able to find out the concentration of Y₂? thanks in advance!

[vhpk]

Remember that in the equation you can't use the concentration of the intermediate subtance like Y₂ and you must use the reactants of the first reaction like X and Y

[maakii]

Hmm yes, but how do you find [Y₂] in terms of [Y] and [X]?

[charco]

in your example the Y₂ is produced from a previous step and is consequently an intermediate or transition state.
Y + Y --> Y2

This step must be an equilibrium as otherwise any pure Y must exist in the form of Y₂

The rate equation in such a case is given by

Rate = k[X][Y]²

[Yggdrasil]

As charco said, in most cases you will assume that the first step is in equilibrium and [Y₂] = K₁[Y]², where K₁ is the equilibrium constant for reaction 1.

To find the concentrations without making any assumptions is a much trickier task.  First you have to set up a system of coupled, non-linear differential equations, each corresponding to the rate of change of each chemical species involved in the reaction.  Then, you have to solve this system of equations (a very non-trivial task).  Since analytical solutions do not exist for most non-linear differential equations, usually you'll have to use some numerical methods to model the evolution of the system.

[charco]

yes, and if the forward reaction of the step:

Y + Y --> Y₂ is very fast, then the equilibrium constant k1 tends to infinity (as near as dammit)

and the sample of Y will, in fact, be mainly Y₂ anyway

[maakii]

so in short i guess there isn't any easy way to determine the rate equation without assumptions lol

thanks for all the replies!