Question

You have 2000 grams of substance A which has decay constant 0.03, and you have 3000 grams of substance B, which has decay constant 0.05. After how many years will the amounts of the two substances be equal?

Answer 1

If you have a quantity X of a substance, with a decay constant r, then the equation that tells you the amount of substance that you have, at a time t, is:

C(t) = X*e^(-r*t)

Now, we know that:

We have 2000g of substance A, and it has a decay constant of 0.03 (i assume that is in 1/year because the question asks in years)

And we have 3000 grams of substance B, with a decay constant of 0.05.

Then the equations for both of them will be:

Ca = 2000g*e^(-0.03*t)

Cb = 3000g*e^(-0.05*t)

Where t is in years.

We want to find the value of t such that Ca = Cb.

So we need to solve:

2000g*e^(-0.03*t) = 3000g*e^(-0.05*t)

e^(-0.03*t) = (3/2)e^(-0.05*t)

e^(-0.03*t)/e^(-0.05*t) = 3/2

e^(t*(0.05 - 0.03)) = 3/2

e^(t*0.02) = 3/2

Now we can apply Ln(x) to both sides, and get:

Ln(e^(t*0.02)) = Ln(3/2)

t*0.02 = Ln(3/2)

t = Ln(3/2)/0.02 = 20.3

Then after 20.3 years, both substances will have the same mass.