Question

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

Answer 1

1620 years

Explanation:

Given : Radium-226 is a radioactive element, and its decay rate is modeled by the equation R = R0e-0.000428t

Solution:

We will find total number of particles in 100 gram sample :

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

Now we will find total number of particles when the 100 gram sample is reduced to half its mass:

R = Ro/2 = 0.4425/2 = 0.2212

On substituting values of Ro and R to the decay rate equation, we get

R = Ro e^-0.000428t

0.2212 = 0.4425 e^-0.000428t

`(0.2212)/(0.4425)` = e^-0.000428t

Now, take natural logarithm on both sides of the equation in order to find value of t .

ln (0.2212/0.4425) = -0.000428t

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

t = 1620 years

Answer 2

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