The half-life for the radioactive decay of calcium-47 is 4.5days. If a sample has an activity of 3.0Ci after 9.0days, what was the initial activity of the sample?

Question

The half-life for the radioactive decay of calcium-47 is 4.5days. If a sample has an activity of 3.0Ci  after 9.0days, what was the initial activity of the sample?

xishem · Answer

Well, think about it logically. If it's been decaying for 9.0 days and its half-life is 4.5 days, how many half-lifes has it gone through?

xishem · Answer

In a more analytic sense...

This situation can be modeled by this equation...$$N(t)=N_0\frac{1}{2}^{\frac{t}{t_{1/2}}}$$Where...
N(t) is the amount (or activity in this case) at time t as a function of time
N_0 is the initial amount (activity)
t is the time
t_{1/2} is the half-life

So...$$3.0Ci=N_0*(\frac{1}{2})^\frac{9.0}{4.5}$$$$3.0Ci=N_0*(1/2)^2 \rightarrow 3.0Ci = N_0*0.25$$$$N_0=12.0Ci $$