91影视

Given that the rate of decay of a radioactive substance is directly proportional to the amount of the substance present. Let the present amount of the radioactive substance be N.

Therefore,$\frac{\mathbf{d}\mathbf{N}}{\mathbf{d}\mathbf{t}}\mathbf{鈭�}\mathit{N}$

Given that there is only 2% of the original amount of carbon-14 remains in the burnt wood of the campfire. We have to estimate theage of the skull if the half-life of carbon-14 is about 5600 years.

Given,

$$\begin{gathered} \frac{dN}{dt}鈭�N \\ \frac{dN}{dt}=-位N \end{gathered}$$

$$\begin{gathered} \frac{dN}{N}=-位 \\ 鈭�\frac{dN}{N}=-位鈭�dt \\ lnN=-位t+ln{N}_0 \end{gathered}$$

where, N₀is an arbitrary constant.

$$\begin{gathered} lnN-ln{N}_0=-位t \\ ln\left(\frac{N}{N_0}\right)=-位t \\ \frac{N}{N_0}=e^{-位t} \\ N=N_0{e}^{-位t}路路路路路路\left(1\right) \end{gathered}$$

One will use this formula to solve the question.

The half-life of carbon-14 is given as 5600 years. The formula for finding the half-life is,

$t_{\frac{1}{2}}=\frac{\ln 2}{位}$

Here,$t_{\frac{1}{2}}=5600years$

Thus,$5600=\frac{\ln 2}{位}$

$位=\frac{\ln 2}{5600}路路路路路路\left(2\right)$ .

One will use this value of $位$in step4 to find the estimated age of the skull.

Let theoriginal amount of carbon-14be N₀ and let the amount of remaining carbon-14 in the burnt wood of the campfire be N, which is given as 2% of the original amount,

i.e., N = 0.02 N₀

Using the equation (1),

$$\begin{gathered} \left(0.02\right)N_0=N_0{e}^{-位t} \\ 0.02=e^{-位t} \\ {e}^{位t}=\frac{100}{2} \\ {e}^{位t}=50 \\ 位t=ln50 \\ t=\frac{ln50}{位} \end{gathered}$$

Now, using the value of $位$from equation (2),

$$\begin{gathered} t=\frac{ln50}{ln2}路\left(5600\right) \\ t=31606years \end{gathered}$$

Hence, the estimated age of the skull is 31,606 years.