Sure, let's go through the detailed solution step by step.
Potassium-40 (K-40) is a radioactive isotope with 19 protons (atomic number 19) and 21 neutrons (since the mass number 40 is the sum of protons and neutrons).
When K-40 decays, it can undergo two types of decay processes:
1. Beta decay ([tex]\(\beta^-\)[/tex] decay): During this process, a neutron in the nucleus is converted into a proton and an electron (beta particle) is emitted. This increases the atomic number by 1 while keeping the mass number the same.
[tex]\[
{}_{19}^{40}K \rightarrow {}_{20}^{40}Ca + \beta^-
\][/tex]
Here, potassium-40 turns into calcium-40.
2. Electron capture (or [tex]\(\beta^+\)[/tex] decay): In this process, a proton is converted into a neutron by capturing an electron from the inner shell of the atom. This decreases the atomic number by 1 while the mass number remains the same.
[tex]\[
{}_{19}^{40}K + e^- \rightarrow {}_{18}^{40}Ar
\][/tex]
Here, potassium-40 turns into argon-40.
Now let's evaluate the given options:
1. [tex]\({}_{18}^{40} Ar\)[/tex]: This represents Argon-40, which has 18 protons. This is a possible product of the electron capture decay of potassium-40.
2. [tex]\({}^{40} K\)[/tex]: This is just another representation of Potassium-40, which is the original isotope itself. It cannot be a product of its own decay.
3. [tex]\({}_{20}^{42} Ca\)[/tex]: This represents Calcium-42, which has 20 protons. This cannot be a product of potassium-40 decay since the mass number changes from 40 to 42, which does not happen in either of the decay processes discussed.
4. 42 20: This is shorthand for [tex]\( {}_{20}^{42} Ca \)[/tex], which again represents Calcium-42 and cannot be a product of potassium-40 decay.
Given these considerations, the correct atomic symbol representing an isotope that results from the decay of [tex]\( K-40 \)[/tex] is:
[tex]\[ {}_{18}^{40} Ar \][/tex]
So, the correct answer is the first option:
[tex]\[ \boxed{ {}_{18}^{40} Ar } \][/tex]
