The half-life of radioactive potassium is 1.3 billion years. If 10 grams is present now?

Question

The half-life of radioactive potassium is 1.3 billion years. If 10 grams is present now, how much will be present in 100 years? 1000 years?

Answer 1

This is a classic example of an exponential growth and decay problem. These are typically discussed in a first semester differential equations course or in the second quarter of the calculus sequence when discussing logarithmic and exponential functions.

In this case, let y(t) be the amount present at time t. Because we are talking about radioactive decay, we assume that

y(t)=y(0)e^(k t)

Because y(0)=10, we have
y(t)=10e^(k t).

Noting that e^(kt)=(e^k)^t, we now find e^k. Because
the half life is 1300000000 years, when t=1300000000, y(t)=5:

5=10 (e^k)^(1300000000)
1/2=(e^k)^(1300000000)
(1/2)^(1/1300000000)=e^k

This gives us

y(t)=10 (1/2)^(t/1300000000)

To find how much is present in 100 and 1000 years, we evaluate y(100) and y(1000):

y(100)=10 (1/2)^(1/13000000) approx 9.999999467

y(1000)=10 (1/2)^(1/1300000) approx 9.999994668

Because there is such a huge difference between 1.3 billion and 100 and 1000, you see that it takes a long time for their to be a signifcant change.

But, eventually it does decay!

After 3.9 billion years, we have

y(3900000000)=10 (1/2)^(3) approx 1.25

Answer 2

Radioactive Potassium

Answer 3

With a half-life in the billions (1.3x10^9) of years, the amount of decay will be exceedingly small -- 7.7x10^-8 half-lives in 100 years, 7.7x10^-7 half-lives in 1000 years, so in both cases you'd still have effectively 10 grams of radioactive potassium.

Answer 4

The amount of a weight w (g) that decays in a time t (years) is given by

w * ln(2) * t / (half life)

so for t=100 years and w=10g it is about 533 ng

Answer 5

10 grams, 10 grams. Its going to take 2,539,062 years to dacay to 9.99023475 grams. You have some time.

Answer 6

http://www.1728.com/halflife.htm

for 1000 yrs 9.999994668 gms
for 100 yrs 9.9999994670 gms

Answer 7

less than 10 grams, lol