To predict the missing component in the nuclear equation:
[tex]\[
{ }_{19}^{40} K \rightarrow X + { }_{-1}^{0} e
\][/tex]
we need to understand the process involved, which is beta decay. Beta decay occurs when a neutron in the nucleus of an atom converts into a proton, emitting an electron (beta particle) and an antineutrino in the process.
Given the initial element is Potassium-40 ([tex]\( { }_{19}^{40} K \)[/tex]), it will undergo beta decay:
1. Identify the initial atomic number (proton number) and mass number of Potassium:
[tex]\[
\text{Atomic number} = 19, \quad \text{Mass number} = 40
\][/tex]
2. Understand the contribution of the beta particle ([tex]\( { }_{-1}^{0} e \)[/tex]):
- The beta particle has an atomic number of -1 (its charge is -1 equivalent to a proton).
- The mass number of the beta particle is 0 (electrons have negligible mass).
3. Conservation of mass and charge:
- To balance the atomic number, the initial atomic number of the parent nucleus is 19.
- The emitted beta particle has an atomic number of -1.
Therefore, if [tex]\( X \)[/tex] represents the newly formed element after decay,
[tex]\[
\text{Atomic number of } X = 19 - (-1) = 19 + 1 = 20
\][/tex]
- The mass number remains conserved in beta decay. Since the initial mass number is 40,
[tex]\[
\text{Mass number of } X = 40
\][/tex]
Thus, the newly formed element (X) after beta decay must have:
- An atomic number of 20
- A mass number of 40
This corresponds to the element Calcium-40 ([tex]\( { }_{20}^{40} Ca \)[/tex]).
Therefore, the missing component in the nuclear equation is:
[tex]\[
{ }_{20}^{40} Ca
\][/tex]
