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Since in a first order reaction, the coefficient of the reactants is one, and "a" represents the coefficient of the reactant, thus "a" = 1. Thus, the equation becomes -d[A]/dt = k[A]. Normally on the right side of the equation [A] would be raised to the "n" power, but since the coefficient of A is one, then it simply becomes [A]. After that, we multiply around to get everything with A in it on one side of the equation and everything with "t" on it to the other side of the equation. We do this so that we can integrate. Thus, we obtain d[A]/[A] = -kdt (also multiply the negative over). We can now integrate to get ln[A] = -kt + C. Setting t = 0, C = ln[A initial] (consider this in terms of the reaction, at time = 0, there is the initial amount of the reactant left). Therefore, the final equation is ln[A] = -kt + ln[A initial].
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