How Old is the Shroud of Turin?

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

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
Because 92 percent of the $^{14} C$ remains, $N / N_{O} = 0.92$ . Therefore, the equation $N = N_{O} e^{- λ t}$ can be used to find $λ t$ . We also know that the half-life of $^{14} C$ is 5,730 years, and so once $λ t$ is known, we can find $λ$ and then find t as requested. Here, we assume that the decrease in $^{14} C$ is solely due to nuclear decay.
Solution
Solving the equation $N = N_{O} e^{- λ t}$ for $N / N_{O}$ gives

$\frac{N}{N_{O}} = e^{- λ t} .$
22.58

Thus,

$0.92 = e^{- λ t} .$
22.59

Taking the natural logarithm of both sides of the equation yields

$\ln 0.92 = - λ t$
22.60

so that

$- 0.0834 = - λ t .$
22.61

Rearranging to isolate t gives

$t = \frac{0.0834}{λ} .$
22.62

Now, the equation $λ = \frac{0.693}{t_{1 / 2}}$ can be used to find $λ$ for $^{14} C$ . Solving for $λ$ and substituting the known half-life gives

$λ = \frac{0.693}{t_{1 / 2}} = \frac{0.693}{5,730 years} = 1.21 \times 10^{−4} y^{−1} .$
22.63

We enter that value into the previous equation to find t.

$t = \frac{0.0834}{1.21 \times 10^{−4}} = 690 years .$
22.64

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