Question

Complete the table for the radioactive isotope. (Round your answer to 2 decimal places .
Isotope : 226Ra
Half Life(years) : 1599
Initial quantity : ? g
Amount after 3000 years: 1.3g

Answer 1

Do you know what an exponential decay formula looks like?

Answer 2

I think it is y=ae^-bx , b>0 ?

Answer 3

There are several ways to solve this sort of problem, but a pretty straight-forward way is to start with a generic exponential function, say,
\[A=Pe^{rt}\]
and use the given information to solve for r.

Answer 4

sorry don't have it typed properly

Answer 5

No, that's fine. Same thing, different names for the variables.

Answer 6

I'll use your variables,
\[y=ae^{-bx}\]
At half-life, y/a = 1/2, so
\[\large \frac{1}{2}=e^{-b(1599)}\]
Take the natural log, LN, of both sides to solve for b.

Answer 7

Once you have b, you can put in 1.3 for y, 3000 for x, and solve for a.

Answer 8

I see, so I would want to start off with --> \[\ln \frac{ 1 }{ 2 }\] = \[\ln e ^{b(1599)}\]

Answer 9

Yep, but don't forget that you're calling b negative.

Answer 10

oh yes, lol.. don't want to miss that do I :)

Answer 11

It'll still work out either way, you'll either get a positive number for b that you'll put a '-' on or you'll get a negative value for b, which you'll use as-is.

Answer 12

trying to figure out how I would get "b" .. not sure why I am drawing a blank

Answer 13

\[\ln(1/2)=\ln(e^{1599(-b)}) \rightarrow -\ln(2)=-1599b\]

Answer 14

not sure if I did my math right, I came up with -.00125 (rounded to 5 decimals) Does that sound right to you?

Answer 15

for b? no, that is way too big for a half-life that long.

Answer 16

hmm, that is what I was hoping for "b"

Answer 17

From this equation,
−ln(2)=−1599b
the negative on both sides cancels out (told you it wouldn't matter), and
b= ln(2) ÷ 1599

Answer 18

ahh I see what you mean ... does 4.229 E-4 sound right? for "b"

Answer 19

I got 4.3349E-4, but that's probably close enough.

Answer 20

(sorry for delay, was eating dinner)

Answer 21

No at all. Thx again for steps and help