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Numerade Educator

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Problem 7 Hard Difficulty

Experiments show that if the chemical reaction
$ N_2 O_5 \to 2NO_2 + \frac {1}{2}O_2 $
takes place at $ 45^oC, $ the rate of reaction of dinitrogen pent-oxide is proportional to its concentration as follows:
$ -\frac {d [N_2 O_5]}{d} = 0.0005[N_2 O_5] $
(See Example 3.7.4.)
(a) Find an expression for the concentration $ [N_2 O_5] $ after $ t $ seconds if the initial concentration is C.
(b) How long will the reaction take to reduce the concentration of $ (N_2 O_5), $ to $ 90\% $ of its original value?

Answer

a) $C e^{-0.0005 t}$
b) $T=211$ seconds

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Video Transcript

So we start out with this pretty intimidating looking equation. It's a differential equation because it has a derivative involved in it, and what we need to do is simplify things for ourselves. So let's say that why is equal to the concentration And then let's substitute that into our equation. And what we have is the opposite of de y t t equals 0.5 One more zero in there times. Why now? Hopefully we remember from the reading in the textbook that we were told that the solution to the equation de y d T equals K T, which is really very similar to what we have here, is why of tea equals Why not you to the Katie? In other words, the solution to this differential equation is this exponential growth model. And so what we need to do with our differential equation is just divide both sides by negative one so that we can get it to look more like this. Groups more like this. Okay, so divide both sides by negative one, and we have d y d t equals negative 0.0 y five way and the number negative. 0.5 were calling that k. So then our model is going to be based on this right here is going to be Why equals the initial amount times e to that value of K times time. Now we were told that the initial amount, the initial concentration is capital C. So let's replace our Why not with a capital C and here we have our model. Now we can also remember that we substituted and why is equal to the concentration so we can replace the why with that concentration. And we have what we're looking for for this part of the problem, that is our model. Okay, so we can use that model to solve part B. So for part B, we're told that the concentration is going to be 90% of the original. So 90% of C and 90% of C could be expressed as 900.9 seat. So we're going to put 0.9 c and for the concentration into our equation and sell for the time, how long does it take to reach a 90% of the concentration? So let's divide both sides by sea, and we get 0.9 equals e to the negative 0.5 t. Then we take the natural log of both sides. And then we divide both sides by negative 0.5 So we get time equals natural log a 0.9 divided by negative 0.5 And we put that into a calculator and it is approximately 211. And what were the units of time here? They were seconds, I believe so. It's 211 seconds.