Stuck on a Question about Isotopes?

Question

I think the question is pretty easy but I'm really bad at questions with isotopes and activities. I've been trying for like 30 minutes. It says: A freshly prepared sample of a certain radioactive isotope has an activity of 10 mCi. After 4 h, its activity is 8 mCi. The half-life is 12.4 hours and the decay constant is 0.0558 h^(-1). How many atoms of the isotope were contained in the freshly prepared sample? Thanks a lot!

Answer 1

It's been a while since I've done something like this, but the first step would be to calculate the number of decays per second.

1 Ci = 3.7x10^10 decays per second, so 10 mCi = 3.7x10^8 decays per second.

You've given a lot of figures which tell me exactly the same thing (the half life, how much is left after 4 hours, the decay constant, these are all different ways to express the same thing). Just working with the decay constant is probably easiest... convert it to decays per second (divide it by 3600 - number of seconds in an hour) and call it k.

Therefore Y = X*exp(-k*t), where X is the starting number of atoms, X is the final number of atoms after time t.

The number of decays per second is therefore -dY/dt:

dY/dt = (-k)*X*exp(-k*t)

At t=0, dY/dt = (-k)*X

Therefore 3.7x10^8 = k*X

X = 3.7x10^8*3600/0.0558

=2.4x10^13 (I hope, check my working...)