The radioactive decay of Sm-151 can be modeled by the differential equation
$$\ \frac{dy}{dt} = -0.0077y $$ where t is measured in years. Find the half-life of Sm-151.

Question

The radioactive decay of Sm-151 can be modeled by the differential equation 
$$\ \frac{dy}{dt} = -0.0077y $$ where t is measured in years. Find the half-life of Sm-151.

shamil98 · Answer

$$\int\limits_{}^{} \frac{ 1 }{ y } dy  = \int\limits_{}^{} -0.0077 ~dt$$
$$\ln|y| = -0.0077t + C$$
$$y = Ce^{-0.0077t}$$
Not sure what to do from here.

anonymous · Answer

set e^(-0.0077) = (1/2)^(1/h), and solve for h

anonymous · Answer

Just an initial note, you should say C2; since e^C is a constant, but it's a different constant. Break that habit before getting further in.

kainui · Answer

The key thing here to know is, WHAT'S A HALF LIFE? That will make your life simpler. All it is, thankfully, is a simple little relation.

It's the amount of time it takes for the amount you have to decay by a half. $$y=Ce^{-at}$$ you have this much at a time t. Now some time later you have: $$\frac{ y }{ 2}=Ce^{-a(t+t_{1/2})}$$

Now a little algebra lets us rearrange this:

$$\frac{ 1 }{ 2 }=e^{-at_{1/2}}$$

And we can solve for our half life, which is:

$$t_{1/2}=\frac{ \ln2 }{ a }$$

And of course, a=.0077 so plug that back in. Notice I didn't include the negative sign when I picked a, so don't confuse yourself. Half lives are always positive numbers, even though they could easily be negative since you could view that as the "doubling" time backwards if that doesn't overcomplicate it... haha.