21.5 Half-life and activity  (Page 3/16)

How old is the shroud of turin?

Calculate the age of the Shroud of Turin given that the amount of ${}^{\text{14}}\text{C}$ found in it is 92% of that in living tissue.

Strategy

Knowing that 92% of the ${}^{\text{14}}\text{C}$ remains means that $N/N_0=0\text{.}\text{92}$ . Therefore, the equation $N=N_0{e}^{-\mathrm{\lambda t}}$ can be used to find $\mathrm{\lambda t}$ . We also know that the half-life of ${}^{\text{14}}\text{C}$ is 5730 y, and so once $\mathrm{\lambda t}$ is known, we can use the equation $\lambda =\frac{0\text{.}\text{693}}{t_{1/2}}$ to find $\lambda$ and then find $t$ as requested. Here, we postulate that the decrease in ${}^{\text{14}}\text{C}$ is solely due to nuclear decay.

Solution

Solving the equation $N=N_0{e}^{-\mathrm{\lambda t}}$ for $N/N_0$ gives

Thus,

Taking the natural logarithm of both sides of the equation yields

so that

Rearranging to isolate $t$ gives

Now, the equation $\lambda =\frac{0\text{.}\text{693}}{t_{1/2}}$ can be used to find $\lambda$ for ${}^{\text{14}}\text{C}$ . Solving for $\lambda$ and substituting the known half-life gives

We enter this value into the previous equation to find $t$ :

Discussion

This dates the material in the shroud to 1988–690 = a.d. 1300. Our calculation is only accurate to two digits, so that the year is rounded to 1300. The values obtained at the three independent laboratories gave a weighted average date of a.d. $\text{1320}\pm \text{60}$ . The uncertainty is typical of carbon-14 dating and is due to the small amount of ${}^{\text{14}}\text{C}$ in living tissues, the amount of material available, and experimental uncertainties (reduced by having three independent measurements). It is meaningful that the date of the shroud is consistent with the first record of its existence and inconsistent with the period in which Jesus lived.