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How do we obtain the expressions for integrated rate laws ?!? What are the limits we integrate it under and with respect to what do we integrate it?

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Remember first that the integrated rate laws that we are familiar with for zeroth, first, and second order reactions are based on systems involving a single reactant in their rate determining step (or those that have been manipulated to give very large excesses of one reactant such that its concentration remains essentially fixed during the reaction process.)

With that in mind, there are only two variables involved, the concentration of our reactant of interest, $[A]$, and time, $t$. We can't directly integrate the differential rate law; it's a differential equation and must be solved as such.

For an example, consider the first order differential rate law: $rate = k[A]$

The meaning of $rate$ here is the rate of change of the concentration of reactant A with respect to time. Since we want the rate to be positive, we use the opposite of that quantity.

$-\frac{d[A]}{dt} = k[A]$

This is a differential equation, specifically one that is separable (and thus fairly easy to solve.) Multiply both sides by $-dt$ and we get:

$\frac{d[A]}{[A]} = -k\ dt$

$\int{\frac{1}{[A]}d[A]} = \int{-k\ dt}$

Integrating both sides and applying initial condition to find the constant gives the integrated first order rate law:

$ln[A]=-kt+ln[A_0]$

The other integrated rate laws are found in similar ways.

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