The "Ostwald Isolation Method": How does this work?

Question

Recently started reading up on Chemical Kinetics, and I came across a certain "Ostwald Isolation Method" that figures in my school-issued workbook. Apparently, it's used to determine (approximately) the order of a given reaction.

However, the painfully short passage that constitutes that section of the book, coupled with my teacher's reluctance to take up the said section in class, means I still don't know how this "method" works. Heck, I don't even know if the said "method" even exists (at least not under that name anyways...the book is notorious for its numerous errors that have gone unchecked over the years. We fondly view it as the Mater Omnium Errorum)

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My book, on the "Ostwald Isolation Method":

This method is particularly useful in case of reactions where two or more reactants are involved.

Consider the reaction between $a$ moles of species $A$ and $b$ moles of species $B$ to give product(s) $P$: $$\ce{aA + bB -> P}$$

The reaction is first carried out with $B$ present in great excess. So the reaction must only depend on the concentration of $A$: $$\ce{r = k[A]^{x}}$$

Similarly, when the reaction is now carried out in a large excess of $A$, the rate of reaction must only depend on the concentration of $B$: $$\ce{r = j[B]^{y}}$$

The order, $n$, of the overall reaction is therefore: $$\ce{n = x + y}$$

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Okay, I understand how the first two bits work (i.e- Concentrations of whatever species present in excess can safely be considered constant/invariant).

But I don't get how directly adding $x$ and $y$, both obtained from different rate equations, having different rate constants , gives us the overall order of the original (without excess) reaction.

Could someone explain the logic behind this to me? That is, if there is any logic behind it...

Answer 1

The powers $x$ and $y$ are intrinsic to the reaction (at least if the reaction pathway remains the same) and will not vary with reaction parameters, so that adding them then poses no difficulty.

More specifically, it is presumed that the reaction rate is given by an equation of the form $$r = c\ce{[A]}^x\ce{[B]}^y;$$ the Ostwald isolation method finds the powers $x$ and $y$ by flooding the system with $\ce{B}$ and $\ce{A}$ respectively in excess in two separate experiments, so that in the former case we have $$r = (c\ce{[B]}^y)\ce{[A]}^x \approx k\ce{[A]}^x$$ and in the latter $$r = (c\ce{[A]}^x)\ce{[B]}^y \approx j\ce{[B]}^y.$$

Adding the powers $x$ and $y$ gives the order of reaction by definition.

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Incidentally, Wikipedia lists this method as the 'method of flooding', here.