The isotope caesium 137 has a half life of 30 years. How long will it take for the amount of this isotope in a sample of caesium to decay to one sixteenth of its original amount? Please can someone explain how I can calculate this from the quantities that have been stated? Thanks

Question

The isotope caesium 137 has a half life of 30 years. How long will it take for the amount of this isotope in a sample of caesium to decay to one sixteenth of its original amount? Please can someone explain how I can calculate this from the quantities that have been stated? Thanks

It decays 1/2 each 30 years.
time amount left
30 1/2
60 1/4
90 1/8
see the pattern?

Mathematically, it can be written as

amountleft/originalamount= (1/2)^{time/halflifetime}

or in logs
amount left/originalamount=e^{0.692time/halflifetime}

Saved by Bob again! Thank you!

Answer 1

To calculate how long it will take for the amount of caesium-137 to decay to one sixteenth of its original amount, you can use the concept of the half-life.

First, let's understand what the half-life means. The half-life of an isotope is the time it takes for half of the atoms in a sample to decay. In the case of caesium-137, its half-life is 30 years, which means that after 30 years, half of the caesium-137 atoms in a sample will have decayed.

Now, since we want to find out how long it will take for the amount of caesium-137 to decay to one sixteenth (1/16) of its original amount, we can write this as a fraction:

Amount left / Original amount = 1/16

Next, let's use the formula for exponential decay:

Amount left / Original amount = (1/2)^(time/half-life)

In this formula, the time is the unknown variable we want to find, and the half-life is given as 30 years.

So, substituting the known values into the formula:

1/16 = (1/2)^(time/30)

To solve for time, we can take the logarithm of both sides. Using the natural logarithm (log base e), we get:

ln(1/16) = (time/30) * ln(1/2)

Simplifying further:

ln(1/16) = (time/30) * (-0.693) [ln(1/2) is approximately -0.693]

Now, we can solve for time by rearranging the equation:

(time/30) = ln(1/16) / (-0.693)

Multiply both sides by 30:

time = (-0.693) * 30 * ln(1/16)

Using a calculator to evaluate the right side of the equation, we find:

time ≈ 70.8 years

Therefore, it will take approximately 70.8 years for the amount of caesium-137 to decay to one sixteenth of its original amount.