Question

In an autocatalytic chemical reaction, the product formed is a catalyst for the reaction. If Qo is the amount of the original substance and x is the amount of catalyst formed, the rate of chemical reaction is Q'(x)= kx(Q0 − x)
For what value of x will the rate of chemical reaction be greatest?

Answer 1

ok, so I have no idea how to deal with this. i should find where Q''(x)=0, right? but how do you take the derivative with all those variables (still on my 1st semester).

Answer 2

Those variables that are constant are treated as constants.
Those variables that are variables are treated as functions of whatever you are taking the derivative to the respect to. Use the chain rule.
I'm pretty sure only the first sentence applies to you, however.

Answer 3

so is the derivative -2kx+Qo

Answer 4

I think so.

Answer 5

I've never taken a formal calculus course, so you probably should wait for another to answer, like @satellite73

Answer 6

or @UnkleRhaukus or @Zarkon ? any of you out there?

Answer 7

i know nothing about this, but you are given the derivative, it is
\[Q'(x)= kx(Q_0 − x)\]

Answer 8

you want the max, set it equal to zero and solve

Answer 9

ok that is wrong, you want the max of the rate
take the derivative of this mess, set it equal to zero and solve

Answer 10

you have
\[Q'(x)=Q_0kx-kx^2\] max is at the vertex, namely
\[-\frac{b}{2a}=-\frac{Q_0k}{2\times -k}=\frac{A_0}{2}\]

Answer 11

\[-\frac{b}{2a}=-\frac{Q_0k}{2\times -k}=\frac{Q_0}{2}\]

Answer 12

my typing is not so good

Answer 13

ok, that makes more sense. I'm with ya there

Answer 14

satellite, they are asking for the rate of the rate, which is a double derivative

Answer 15

so would I solve for Qo in 0=kx-kx^2 and then put it into Qo/2 to get the max?

Answer 16

Sorry, 0=Qo(kx)-kx^2

Answer 17

hick, they want when the rate is the greatest.

Answer 18

That means that the rate of the rate must be 0

Answer 19

0=-2kx+Qo

Answer 20

ok, so x=Qo/2k

Answer 21

yup.

Answer 22

but they're asking me for an x value

Answer 23

and they don't tell you Qo or k

Answer 24

that is the x value

Answer 25

and that is what you are asked for, so you are done

Answer 26

my homework won't accept that answer

Answer 27

am I missing something important here? I gave you everything they gave me for the question. I wonder what else they could be looking for?

Answer 28

The rate of chemical reaction will be greatest when \[x=\frac {Q_0}{2}\]

Answer 29

yay! it took that! i wonder where the k went?

Answer 30

Thanks a bunch!

Answer 31

I just read and re-read the post above made by satellite73,
i dont think they are asking for the rate of the rate