The radioisotope carbon-14 emits beta particles and has a half-life of 5730 years - Leaving Cert Chemistry - Question d - 2018

In beta decay, a neutron in the nucleus transforms into a proton. During this process, a beta particle (which is an electron) is emitted from the nucleus.

The balanced equation for the beta decay of a carbon-14 nucleus can be written as:

Given that the half-life of carbon-14 is 5730 years, the number of carbon-14 atoms halves every 5730 years. If the analysis shows that the fragment contains 1.5 × 10²¹ carbon-14 atoms, then it would have contained:

$2 \times (1.5 \times 10^{21}) = 3.0 \times 10^{21} \text{ atoms 5730 years ago.}$

To determine the count 5730 years prior to that, we double it again:

$2 \times (3.0 \times 10^{21}) = 6.0 \times 10^{21} \text{ atoms 11460 years ago.}$

This suggests that the fragment originally contained 3.0 × 10²² carbon-14 atoms before the last measurement.

To calculate the mass of carbon-14 contained in 3.0 × 10²² atoms, we first need to determine moles of carbon-14 using Avogadro's number:

$moles = \frac{3.0 \times 10^{22} \text{ atoms}}{6.022 \times 10^{23} \text{ atoms/mole}} \approx 0.0498 \text{ moles}$

Then, using the molar mass of carbon-14 (approximately 14 g/mol), we find the mass:

$mass = moles \times molar \: mass \approx 0.0498 \times 14 \approx 0.698 g$

Thus, the mass of carbon-14 would be approximately 0.698 grams.