Question

A living plant contains about the same fraction of carbon-14asatmospheric carbon dioxide. The observed rate of decay of carbon-14 from a living plant is 15.3 counts per minute per gram of carbon. How many counts per minute per gram of carbon will be measured from a 15,000-year-oldsample? Will radiocarbon dating work well for small samples of 10 mg or less? (For 14C, ${\mathit{t}}_{\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.}$=5730 years.)

Short answer

2.49 counts per minute per gram of carbon will be measured.

No, radiocarbon dating will not work well for samples of 10mg or less.

Step-by-step solution

Calculating the decay constant K.

The rate constant is evaluated as:

$\begin{array}{c}\mathbf{k}\mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{693}}{{\mathbf{t}}_{{\displaystyle \raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.}}}\\ =\frac{0.693}{5730years}\\ =1.21\times {10}^{-4}{\text{years}}^{-1}\end{array}$

Determining whether radiocarbon dating will work well for samples of 10mg or less or not.

We can calculate the decay rate occurring at time t using the first-order integrated rate law equation,

$\mathbf{ln}\frac{{\left[\mathbf{N}\right]}_{\mathbf{t}}}{{\left[\mathbf{N}\right]}_{\mathbf{0}}}\mathbf{=}\mathbf{-}\mathbf{kt}$

Where,

• k = Rate constant
• t = time required
• ${\left[\text{N}\right]}_t$= Concentration of N after time t
• ${\left[\text{N}\right]}_0$= Initial concentration of N.

The known values are substituted,

$\begin{array}{c}\ln \left(\frac{x}{15.3}\right)=-1.21\times {10}^{-4}\times 15000\\ \ln x-\ln 15.3=-1.815\\ \ln x=0.912\\ x=2.49\end{array}$

Thus, 2.49counts per minute per gram of carbon will be measured.

In case of 10mg carbon,

$10\text{mg×}\left(\frac{1g}{1000\text{mg}}\right)\times \left(\frac{2.49\text{counts/min}}{1g}\right)=2.49\times {10}^{-2}\text{counts/min}$

To observe one disintegration, the time required will be 40 minutes. This will be very hard to measure.Hence, radiocarbon dating will not work well for very small samples, that is, 10mg or less.