Calculating Half-Life from Radioactive Decay

[sklotz]

Homework Statement

After 25 years, 60% of a radioactive material decays. What is the half-life?

Homework Equations

I used a ratio of 25/.60= x/.50

The Attempt at a Solution

I also tried this ratio as 25/.40= x/.50 I am not really sure what equation I should be using but this ratio set up isn't getting me the correct answer

 

[daveb]

Decay is an exponential dcay. That is:

N(t) = N(t₀)e^-k(t-t₀)

where N(t) is the amount at some time t, N(t₀) is the amount at time t₀, and k is the decay constant.

 

[sklotz]

Ok so I tried using this equation but I still got the problem wrong. I used these values:
N(t)= .6 N(t0)=1 t=25 and t0= 0. I then solved the equation for the decay constant and got: k=.020433025. From my book I found an equation that related the half-life and the decay constant. The equation I used was half-life= ln2/k. from this I got 33.9228861414. This is similar to the answer I got from doing the ratios, and was wrong. I only have one more attempt for full credit and I don't know exactly where I went wrong.

 

[TSny]

Note that N(t) represents the amount of radioactive substance that still remains at time t. So, if 60% has decayed, what % remains?

If you think about it, the half-life should be less than 25 years since more than half has decayed at 25 years.

 

[HallsofIvy]

Because you are talking about half life, in particular, I would use the formula (equivalent to the "e" formula TSny gives) $X= C(1/2)^{t/T}$ where C is the initial amount and T is the half life. (You can see that if t=0, $C(1/2)^{0/T}= C$ and if t= T, $C(1/2)^{T/T}= C/2$.)

If 60% has decayed then 40% is left so $.4C= C(1/2)^{25/T}$. The two "C"s will cancel leaving an equation to solve for T.