Question

Suppose that you measure the intensity of radiation from carbon-14 in an ancient piece of wood to be 6% of what it would be in a freshly cut piece of wood. Show that the age of this artifact is 23,000 years old.

Answer 1

Answer:

t = 23136 years old

Explanation:

The intensity of radiation obeys the exponential decay law.

$I(t)=I_{0}e^{-\lambda t}$ (1)

Here:

λ is the radiation decay constant.

I is the intensity of radiation after t time.

I₀ is the initial  intensity of radiation.

t is the time

Now, we know that the intensity of radiation is a 6% of what it would be in a freshly cut piece of wood, in other words:

$I(t)=0.06I_{0}$ (2)

Combining (2) with (1), we have:

$0.06I_{0}=I_{0}e^{-\lambda t}$

$0.06=e^{-\lambda t}$ (3)

Now, we just need to solve (3) for t.

$t=\frac{-ln(0.06)}{\lambda}$ (4)

and $\lambda = \frac{ln(2)}{t_{1/2}}$ (5)

$t_{1/2}$ is the half-life.

In our case, we have ¹⁴C, so the $t_{1/2} = 5700 y$

Finally, we can find t putting (5) in (4):

$t=\frac{-t_{1/2}ln(0.06)}{ln(2)} = 23136 y$

I hope it helps you!