How do you arrive at the second form of the 1st order integrated rate law? [closed]








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As I understand it, the first-order integrated rate law is:


$$\ln[\ce{A}] = \ln[\ce{A}]\_0 - k t$$


However, I'm also told that this can be expressed as a ratio of $[\ce{A}]\_0$ and $[\ce{A}]$, as follows


$$\ln \left(\frac{[\ce{A}]\_0}{[\ce{A}]\,\,}\right) = kt$$


How did they arrive at this expression of the integrated rate law? I don't see a way to arrive at it algebraically, unless I'm greatly missing something.




To get to ln([A0]/[A])=kt from ln[A]=−kt+ln[A]0 just requires some understandings of log properties and a few log rules.


First rearrange ln[A]=−kt+ln[A]0


to


kt = ln[A]0 - ln[A] 


From there, the log rule
log(a/b)=log(a)−log(b) means that you can state this as


kt = ln([A]0/[A])


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