Radium-226 is a radioactive element, and its decay rate is modeled by the equation R = R0e-0.000428t. How many years will it take for 100 grams of radium-226 to reduce to half its mass?

810

1,620

2,690

5,380

It will take 1620 years.

Solution:

We calculate for the total number of particles in the 100 gram sample:

Ro = 100 grams * 1 mol / 226 g = 0.4425 mol

We also calculate for the total number of particles when the 100 gram sample is reduced to half its mass:

R = 100 grams/2 * 1 mol / 226 g = 0.2212 mol

We substitute the values to the decay rate equation

R = Ro e^-0.000428t0.2212

= 0.4425 e^-0.000428t0.2212/0.4425

= e^-0.000428t

Taking the natural logarithm of both sides of our equation, we can compute now for the years t:

ln (0.2212/0.4425) = - 0.000428t

t = ln (0.2212/0.4425) / (-0.000428)

t = 1620 years