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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?

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Respuesta :

zoexoe
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

Answer:

1620 years

Step-by-step 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

       [tex]\frac{0.2212}{0.4425}[/tex] = 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

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