# Is it possible to derive integrated rate law algebraically?

Here is the calculus way: I have no prior knowledge of calculus. But my teacher asks me to derive the integrated rate law for any order between one to ten.

It would be fine for me to memorize the derivations for zero to third orders. But I can't do all ten for sure.

• What is your question exactly ... ? And what have you tried ? – Babounet Feb 12 '15 at 21:09
• My question is how to derive integrated rate law algebraically. I found what is image. I can googlr all the formulas out. But It all seemed to me that the only way is by calculus. – most venerable sir Feb 12 '15 at 21:11
• For a function $f(x) = x^n$, the primitive integral is $F(x) = \vert\ln(x)\vert$ if $n=-1$, otherwise $F(x) = \frac{1}{n+1}x^{n+1}$. – Klaus-Dieter Warzecha Feb 12 '15 at 21:24
• We are fortunate to have a very nice $\LaTeX$ capability on the site. Please consider writing out the equations rather than pasting in a picture. – jonsca Feb 13 '15 at 0:07

Calculus is pretty much just a shorthand for algebra (plus the whole limits concept), so the answer is technically yes.

Your teacher might be intending for you to use the pattern @Klaus Warzecha mentioned:

For a function $f(x) = x^n$, the primitive integral is $F(x) = \vert\ln(x)\vert$ if $n=−1$, otherwise $F(x) = \frac{1}{n+1}x^{n+1}$

So in that sense, you can use algebra if you let

$$\int_{A_0}^{A_\tau} \! \frac{1}{[A]^n} \, \mathrm{d}[A] = \frac{1}{n - 1} \left[ \frac{1}{[A]^{n-1}_{\tau}} - \frac{1}{[A]^{n-1}_{0}} \right]$$

for $n \neq 1$

and

$$\int_{A_0}^{A_\tau} \! \frac{1}{[A]} \, \mathrm{d}[A] = \ln \frac{[A]_\tau}{[A]_0}$$

for $n = 1$

Note that this only works for single-species rate equations - if you have multiple species the integration result changes. Note that this also isn't really using algebra - it's memorizing and applying one of many general integration results. Which is what you do in integral calculus.

In any case, it doesn't seem fair to me to expect you to know how to derive integrated rate laws if integral calculus isn't a prerequisite for your course. If it is a prereq, then now you are seeing why. If it isn't, then what your teacher is asking you to do is, in my opinion, unrealistic.