How Do You Calculate the Decay Rate of C14 with a Given Half-Life?

[KevinMWHM]

Doing some self prep for Diff EQ starting next week.

Determine the decay rate of C14 which has a 1/2 life of 5230. Using e^kt as a function,
I solve using k5230=ln.5 which gives the obvious answer of negative what I want. How do I know to use the reciprocal (ln2) other than to "just know" I want the positive k value? What if I'm looking for something where I may want a negative but don't know it? Will there be such an instance?

-thx

 

[BruceW]

trust your answer bro. if your function is e^kt then you do want k to be negative. A nice way to check is to think about the graph of e^kt and what it looks like for k negative and k positive. And keep in mind the question is about decay, so then which graph makes most sense?

 

[Ray Vickson]

Doing some self prep for Diff EQ starting next week.

Determine the decay rate of C14 which has a 1/2 life of 5230. Using e^kt as a function,
I solve using k5230=ln.5 which gives the obvious answer of negative what I want. How do I know to use the reciprocal (ln2) other than to "just know" I want the positive k value? What if I'm looking for something where I may want a negative but don't know it? Will there be such an instance?

-thx

Your intuition is off. You either want $e^{kt}$ with $k < 0$ or $e^{-kt}$ with $k > 0$.The quantity $Q = Q(t)$ of C14 must be a decreasing function of t, since the substance is decaying. If you had $Q = e^{kt}$ with $k > 0$ then Q would be increasing without bound, so the amount of C14 would grow so much over a long time as to swallow up the whole earth---not what you want.