Question

Will some please check my work?

1. RE: Dr. Brown has a radioisotope source containing 1.2 x 106 (also written as 1.2E+06) atoms of Plutonium-238 in 1985, how many atoms will remain in 352 years?
Answer Details
The equation to model exponential decay is
P(t)=P0ekt, where P(t) is the amount left after t years, P0 is the original amount, and k is the rate of decay. The half life of Plutonium-238 is 87.7 years. Solving for the rate of decay, k yields
1/2=e87.7k
ln(1/2)=87.7k
k=ln(1/2)/87.7=-.007895
Now,...
The equation to model exponential decay is
P(t)=P0ekt, where P(t) is the am

Answer 1

The equation for exponential decay is: P(t)=Po*e^(-kt)

The half life is the amount of time it takes for half of the sample to decay.
So we can say P(t)=(1/2)*Po (Since Po is our initial amount and P(t) is our final amount) and we can set t=H (H meaning Half-life) This changes our equation to:

(1/2)*Po=Po*e^(-k*H)

From there you can divide Po from both sides and take the natural logarithm to both sides to get:

ln(1/2)=-kH

with properties of logarithms we can separate ln(1/2) into ln1-ln2, and since ln1=0 our equation further simplifies to:

ln2=kH

You now have been shown how the natural log of 2 is related to the rate and half-life completely irrespective of what the initial amount is you've started with. Interestingly it follows that this is the same exact relationship between doubling time since the reverse is also true.

I hope this helps.