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I have the equation $\frac{dN}{dt}= - Nk$ where $k$ is the decay constant.

When $time = 0$,

we get $N(t) = N(0) e^{-kt}$.

How would I rearrange this to the $y = mx + c$ format? How would I find the decay constant? Thanks in advance.

RedRose
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Matt
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1 Answers1

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You can transform the equation into a linear equation, but then you have to take logs of $N(t)$ and $N(0)$

$\ln(N(t))=\ln \left[(N(0))\cdot e^{k\cdot t}\right]$

$\ln(N(t))=\ln (N(0))+\ln\left[ e^{k\cdot t}\right]$

$\ln(N(t))=\ln (N(0))+k\cdot t$

This is equivalent to $y=c+mx$

Let´s say the origin values are

t      0  1  2  3  4

N(t)  16  8  4  2  1

Now you calculate the logs of $N(t)$. The values of $t$ musn´t be transformed. Two value pairs are sufficient to evaluate the value of $k$.

Remark

I have $+k$ at the function, not $-k$. But it doesn´t matter. If you have a decay then the value of k is negative.

callculus42
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  • I was wondering how you gained those values of t and N(t)? – Matt Mar 03 '17 at 19:28
  • @Matt It´s just the function $N(t)=16\cdot 0.5^t$ . This is equal to $N(t)=16\cdot e^{\ln(0.5)\cdot t}$, where $N(0)=16$ and $k=\ln(0.5)=-0.693147...$ – callculus42 Mar 03 '17 at 20:47
  • I do not think that value of k is right because I am working through these questions on a site (I am a GCSE student attempting to teach myself some A level) and it marked that answer of k as wrong. – Matt Mar 04 '17 at 11:34
  • @Matt Please post the question. Without out it I can´t say something if I´m wrong or not. – callculus42 Mar 04 '17 at 13:51
  • It is the exact same except N(t)=N(0)e^(-kt). The N(0) is actually N (little zero done in the style of of chemistry e.g. O2 as in oxygen molecules) I do not know how to actually make it little though. – Matt Mar 04 '17 at 22:22
  • @Matt I had a second, third... look on you question. Isnt´t that simple that you have to calculate k from N(t)=N(0)e^(-kt). Since you have the value of N(0) you need one data point to calculate the value of k. Was it your idea with the linear function y = mx + c ? Another thing: I would appreciate it if you would post all information you have-but not in the comments. Make an edit of the quesstion for that purpose. Your thoughts and your work are very, very, very welcomed, but seperate it from the question. – callculus42 Mar 04 '17 at 22:41
  • The question asked for y = mx + c and then the next question asked for the decay constant – Matt Mar 05 '17 at 09:46
  • @matt The decay constant can be calculated by using the equation $N(t)=N(0)e^{-kt}$. Insert the given values and solve for k. To get the form y = mx + c see my answer. – callculus42 Mar 05 '17 at 11:06
  • The values you gave? Because the question did not give any? – Matt Mar 06 '17 at 17:50
  • @Matt If you have no values, no function no other information then you must solve the equation $\ln(N(t))=\ln (N(0))+k\cdot t$ in general for $k$. – callculus42 Mar 06 '17 at 18:00