Half life problem

cmerickson21 · #1 $Old$ September 4th, 2009, 01:05 PM

Here is the question:
An unknown radioactive element decays into non-radioactive substances. In

days the radioactivity of a sample decreases by

percent.

(a) What is the half-life of the element?
(b)How long will it take for a sample of

mg to decay to

mg?

I've tried it several times and I keep getting the same answers which are wrong

I used $dy/dx=-ky$ I integrate this and got $y=ce^{-kt}$ .

I used $y(0)=y_0$ which gave me the equation $y=y_0e^{-kt}$ .

I then plugged in $0.37y_0$ for y and 400 for t from the question. this gave me k= 0.0025. giving me the equation $y(t)=y_0e^{-.0025t}$ .

I then set $y(t)= 0.5y_0$ and solved for t to get the half life. Which gave me the answer 277 days.

What am I doing wrong?

TKHunny · #2 $Old$ September 4th, 2009, 06:12 PM

1) Why do you think it is wrong?

2) Check your notation. How do you get from dy/dx = -ky to y = f(t)? It's okay, since everyone knows what you mean, but you should write it as clearly as you can.

3) You rounded 'k' rather severely. Use more decimal places.

CaptainBlack · #3 $Old$ September 4th, 2009, 11:57 PM

Quote:

Originally Posted by cmerickson21 $View Post$

Here is the question:
An unknown radioactive element decays into non-radioactive substances. In

days the radioactivity of a sample decreases by

percent.

(a) What is the half-life of the element?
(b)How long will it take for a sample of

mg to decay to

mg?

I've tried it several times and I keep getting the same answers which are wrong

I used $dy/dx=-ky$ I integrate this and got $y=ce^{-kt}$ .

I used $y(0)=y_0$ which gave me the equation $y=y_0e^{-kt}$ .

I then plugged in $0.37y_0$ for y and 400 for t from the question. this gave me k= 0.0025. giving me the equation $y(t)=y_0e^{-.0025t}$ .

I then set $y(t)= 0.5y_0$ and solved for t to get the half life. Which gave me the answer 277 days.

What am I doing wrong?

I always find this approach to half life problems bizarrely over complex.

If half-life means anything it tells you that if you start at $t=0$ with an initial quantity $m_0$ which decays (exponentially) with half-life $t_{1/2}$ , then the quantity remaining at $t$ is:

$m=m_0 2^{-\frac{t}{t_{1/2}}}$

CB