Question

The iceman known as "Ötzi" was discovered in a high mountain pass on the Austrian-Italian border. Samples of his hair and bones had carbon- 14 activity that was about \(50 \%\) of that present in new hair or bone. Carbon- 14 is a beta emitter. a. How long ago did "Ötzi" live if the half-life for \(\mathrm{C}-14\) is \(5730 \mathrm{y} ?\) b. Write a balanced nuclear equation for the decay of carbon- 14 .

Short answer

Ötzi lived approximately 5730 years ago. The balanced nuclear equation is\[ ^{14}_6C \rightarrow ^{14}_7N + \beta^{-} \].

Step-by-step solution

- Understanding the Half-Life Concept

The half-life of a radioactive isotope is the time it takes for half of the radioactive nuclei in a sample to decay. For Carbon-14, this half-life is 5730 years.

- Determine the Fraction of Carbon-14 Remaining

Ötzi's carbon-14 activity is 50% of the original activity. This means that one half-life has passed because the original amount has reduced to half.

- Calculate the Time Since Ötzi's Death

Since the fraction of remaining Carbon-14 is 50%, and one half-life corresponds to 5730 years, the time since Ötzi’s death is \(\text{Time} = 1 \times 5730 \text{ years} = 5730 \text{ years} \)

- Write the Nuclear Decay Equation

For the decay of Carbon-14 (^14C), which is a beta emitter, the nuclear equation is: \[ ^{14}_6C \rightarrow ^{14}_7N + \beta^{-} \]This shows Carbon-14 decaying into Nitrogen-14 and an electron (beta particle).

Key concepts

Half-Life

In nuclear physics, the concept of 'half-life' is crucial to understanding radioactive decay. The half-life of a radioactive isotope is the time required for half of the radioactive atoms in a sample to decay. This measure helps determine the age of ancient objects and fossilized remains.
For instance, Carbon-14 has a half-life of 5730 years. This means that every 5730 years, half of the Carbon-14 in a sample decays. Over multiple half-lives, less and less Carbon-14 remains in the sample.
When looking at Ötzi, the iceman, his carbon-14 activity is at 50% of that found in new hair or bone. By knowing the half-life, we can determine that one half-life period has elapsed since his death. Thus, Ötzi lived approximately 5730 years ago.

Radioactive Decay

Radioactive decay is a process where unstable atomic nuclei lose energy by emitting radiation. This can include alpha particles, beta particles, or gamma rays. Over time, this decay changes the original element (parent) into a different element (daughter).
Carbon-14 is particularly important in archaeology and geology for dating fossils and artifacts. It decays over time, allowing scientists to estimate the age of objects. The discovery of Ötzi offers a fascinating glimpse into the application of radioactive decay for uncovering historical timelines.
By measuring the decrease in Carbon-14 activity in Ötzi's remains, scientists could calculate the approximate time since his death. This forms the basis for radiocarbon dating, a pivotal tool in historical sciences.

Nuclear Equation

When dealing with radioactive decay, nuclear equations are used to represent the transformation of nuclei. They show the initial isotope and the particles emitted during decay.
The nuclear equation for the decay of Carbon-14, a beta emitter, is as follows:
\[ ^{14}_6C \rightarrow ^{14}_7N + \beta^{-} \]
This equation illustrates that a carbon-14 atom (\(^{14}_6C\)) changes into a nitrogen-14 atom (\(^{14}_7N\)) and emits a beta particle, which is an electron. This balanced equation helps to visualize the transmutation from carbon to nitrogen during radioactive decay.

Beta Emission

Beta emission is a type of radioactive decay where a beta particle is emitted from an atomic nucleus. A beta particle can be an electron (\(\beta^-\)) or a positron (\(\beta^+\)). In the case of Carbon-14, it emits a beta-minus particle.
Beta emission happens when a neutron in the nucleus converts into a proton and an electron. The electron is then ejected from the nucleus.
This process increases the number of protons in the nucleus by one, effectively changing the element. For Carbon-14, the emission of a beta-minus particle transforms it into Nitrogen-14. This form of decay helps with dating organic materials and unravels the history of ancient objects like Ötzi the iceman.
The nuclear equation representing beta emission for Carbon-14 is:
\[ ^{14}_6C \rightarrow ^{14}_7N + \beta^{-} \]