Let's break down the question step by step to understand which nucleus completes the given equation:
The initial nucleus is denoted as [tex]\({ }_{48}^{99}Cd\)[/tex], which tells us:
- The atomic number (number of protons) is 48.
- The atomic mass (number of protons + neutrons) is 99.
The reaction involves a beta-plus decay, emitting a positron [tex]\({ }_{+1}^0 e\)[/tex]. During beta-plus decay, a proton in the nucleus is converted into a neutron, which means:
- The atomic number increases by 1 due to the conversion of a proton to a neutron (effectively adding one to the number of protons).
- The atomic mass remains unchanged because the total number of nucleons (protons + neutrons) stays the same.
So, the changes can be summarized as follows:
- The number of protons after the decay increases from 48 to 49.
- The atomic mass remains 99.
Therefore, we have a nucleus with:
- 49 protons,
- an atomic mass of 99.
Checking the options provided:
A. [tex]\({ }_4^{99} Es\)[/tex] -- Incorrect, does not match atomic number or element.
B. [tex]\({ }_{4:}^{99} Ag\)[/tex] -- Incorrect, does not match atomic number or element.
C. [tex]\({ }_{49}^{99} Cd\)[/tex] -- Incorrect, has the wrong element.
D. [tex]\({ }_{49}^{99} In\)[/tex] -- Correct, has the right atomic number and mass, matching indium's values.
Thus, the correct nucleus that completes the equation is indeed
[tex]\[
\boxed{{}_{49}^{99} \text{In}}
\][/tex]
