I’m the AP Chemistry teacher, btw. I spent 20 minutes with the Calc teacher, and we can’t find the units in this calculation. Where are they?
We discover that our chemical reaction is a first order reaction, so rate = k[A] (This happens before the math shown.) A is the reactant. k is the constant that makes the 2 sides actually equal. t is time after the reaction starts. Brackets shows that the value of A is the concentration of A in water, in moles/L, which is molarity, or M. The integrated rate law is ln[Ao]=ln[At]+kt. This lets us find molarity at a certain time if we know the concentration at another time.
My contention is that the brackets on the left side only indicate that we should find a M unit when we solve for it, not that it has M as its unit. It is a concept, not a measurement, and I should find a number with the correct unit on the right, to which the unknown is equal. A chem teacher I emailed about this seems to be saying that there is an M unit on the ln[At] value, but then I couldn’t add it to the other number on the right side, because those units have canceled out.
*I just now talked this through with my top math/science student (10th grade), and his idea is that taking ln of the molarity on the right side removes the unit, allowing me to add it to the unitless number, but then taking the inverse ln of the right side restores the unit. It’s the best idea I’ve heard on this, but if true, it’s news to me.