500 grams of a radioactive substance is expected to decay exponentially to 125 grams in the course of 20 year

Question

500 grams of a radioactive substance is expected to decay exponentially to 125 grams in the course of 20 years.


(a) Write down the half-life, and the rate of decay of the radioactive substance.









(b) Write down the function in time, N(t), where t is in years, representing the exponential decay of the radioactive substance.











(c) How much of the substance is expected to remain after 15, 25, and 40 years, respectively?. How long would it take for the radioactive substance to decay to under 1 gram?.

Answer 1

(A)
Half life... Amount of Sample
0 .................500g
1 ................ 250g
2 .................125g

In twenty years there are 2 half lives, therefore each half-life is 10 years.

To find the rate we use this equation:
k = 0.693/T

where k is the rate and T is the half-life.

k=0.693/10= 6.93 x 10^-2

(B) N(t) = No * e^(-kt) where N= final amount, No= initial amount, t= time, and k is the rate



(C) Use N=No* e^(-kt) equation
After 15 years:
N= 500g* e^(-6.93 x 10^-2*15)=176 grams

I'll let you calculate the 25 and 40 years.

How long would it take for the radioactive substance to decay to under 1 gram?

Use this equation t= T * [log(N/No)/ log(1/2)]

We'll call under 1 gram 0.99 g.

t= 10* [log(0.99 g/500 g)/ log (1/2)]
t= 89 years

Lets check to make sure that makes sense:

Half life (years) Amount of Sample
0 .................... 500g
10 ..................250g
20...................125g
30.....................62.5g
40.....................31.25
50.....................15.6g
60.....................7.8g
70.....................3.9g
80.....................1.9g
90.....................0.98 g

So, our answer above of 89 years makes sense.
I hope that this helps!

Answer 2

(b)
Use this one:
Final Mass = Initial Mass (1/2)^{time / half-life}

(c)
Final mass = 500grams (1/2)^{15years/10years}
Final mass = 500grams (0.5)^(1.5) = 176.78 grams

Final mass = 500grams (1/2)^{25years/10years}
Final mass = 500grams (0.5)^(2.5) = 88.39 grams

Final mass = 500grams (1/2)^{40years/10years}
Final mass = 500grams (0.5)^(4) = 31.25 grams

Answer 3

a) Half life = 10years
k = 6.93 x 10^-2

b) N(t) = No * e^(-kt)

c) Use above equation.