Carbon- 14 dating is a method used to determine the age of historical artifacts by examining the ratio of two isotopes of carbon (carbon- 14 and carbon-12). A living plant consumes carbon dioxide in the photosynthesis process and incorporates the carbon, including \({ }^{14} \mathrm{C}\), into its molecules. As long as a plant lives, the \({ }^{14} \mathrm{C} /{ }^{12} \mathrm{C}\) ratio in its molecules remains the same as in the atmosphere because of its continuous uptake of carbon. However, as soon as a tree is cut to make a wooden bowl or a flax plant is harvested to make linen, the \({ }^{14} \mathrm{C}^{12} \mathrm{C}\) ratio begins to decrease because of the radioactive decay of \({ }^{14} \mathrm{C}\left({ }^{12} \mathrm{C}\right.\) is stable). By comparing the current \({ }^{14} \mathrm{C} /{ }^{12} \mathrm{C}\) ratio to the presumed ratio when the artifact was made, one can estimate the age of the artifact. For carbon-14 and carbon- 12 , how many protons and neutrons are in each nucleus? Assuming neutral atoms, how many electrons are present in an atom of carbon- 14 and in an atom of carbon-12?

Step by step solution

01

Understanding atomic structure and isotopes

: An atom is composed of a nucleus (which contains protons and neutrons) and electrons that orbit around the nucleus. The number of protons in an atom is called its atomic number (Z), while the sum of protons and neutrons is called its mass number (A). Isotopes are atoms of the same element (having the same atomic number) but with different mass numbers due to a different number of neutrons.

02

Find the atomic number (Z) for carbon

: The atomic number of carbon is denoted as Z, which is the number of protons in its nucleus. By looking at the periodic table, we find that the atomic number of carbon (C) is 6, meaning that all carbon atoms have 6 protons.

03

Determine the mass number (A) for carbon-14 and carbon-12

: For carbon-14, the mass number (A) is given as 14, while for carbon-12, the mass number (A) is given as 12. This indicates that carbon-14 has 14 nucleons (protons and neutrons) in its nucleus, while carbon-12 has 12 nucleons in its nucleus.

04

Find the number of neutrons for each isotope

: To find the number of neutrons in each isotope, we need to subtract the atomic number (number of protons) from the mass number (number of protons and neutrons). For carbon-14: \(n_{14} = A_{14} - Z = 14 - 6 = 8\) For carbon-12: \(n_{12} = A_{12} - Z = 12 - 6 = 6\) Hence, carbon-14 has 8 neutrons and carbon-12 has 6 neutrons.

05

Find the number of electrons for each isotope

: Since the atoms of carbon-14 and carbon-12 are neutral, the number of electrons is equal to the number of protons in each isotope. Recall that carbon has an atomic number of 6, which means that all carbon atoms, regardless of their isotopic forms, have 6 electrons. Thus, both carbon-14 and carbon-12 have 6 electrons each. By following these steps, we have determined that carbon-14 has 6 protons, 8 neutrons, and 6 electrons, while carbon-12 has 6 protons, 6 neutrons, and 6 electrons.

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