Certainly! Let's solve the given problem step-by-step.
The nuclear equation given is:
[tex]\[ {}_{82}^{235} U \rightarrow X + {}_{4}^{2} He \][/tex]
To identify the nuclide [tex]\( X \)[/tex], we first need to apply the principles of conservation of mass and atomic number in nuclear reactions. This means that the total number of protons and the total number of nucleons (protons + neutrons) must be the same on both sides of the equation.
1. Atomic Number Conservation:
- The atomic number (protons) of Uranium ([tex]\( U \)[/tex]) is 82.
- The atomic number of Helium ([tex]\( He \)[/tex]) is 2.
Let the atomic number of [tex]\( X \)[/tex] be [tex]\( Z_X \)[/tex].
From the conservation of atomic number:
[tex]\[ 82 = Z_X + 2 \][/tex]
Solving for [tex]\( Z_X \)[/tex]:
[tex]\[ Z_X = 82 - 2 = 80 \][/tex]
So, the atomic number of [tex]\( X \)[/tex] is 80.
2. Mass Number Conservation:
- The mass number (nucleons) of Uranium ([tex]\( U \)[/tex]) is 235.
- The mass number of Helium ([tex]\( He \)[/tex]) is 4.
Let the mass number of [tex]\( X \)[/tex] be [tex]\( A_X \)[/tex].
From the conservation of mass number:
[tex]\[ 235 = A_X + 4 \][/tex]
Solving for [tex]\( A_X \)[/tex]:
[tex]\[ A_X = 235 - 4 = 231 \][/tex]
So, the mass number of [tex]\( X \)[/tex] is 231.
3. Identifying the Nuclide [tex]\( X \)[/tex]:
Now we know that [tex]\( X \)[/tex] has:
- Atomic number [tex]\( Z_X = 80 \)[/tex]
- Mass number [tex]\( A_X = 231 \)[/tex]
We need to match this with one of the given options:
1. [tex]\( {}_{94}^{294} Pu \)[/tex]
2. [tex]\( {}_{90}^{235} Th \)[/tex]
3. [tex]\( {}_{94}^{289} Pu \)[/tex]
4. [tex]\( {}_{90}^{281} Th \)[/tex]
None of the provided options match an atomic number of 80 (which corresponds to the element Mercury, [tex]\( Hg \)[/tex]) and a mass number of 231.
Thus, the correct nuclide [tex]\( X \)[/tex] is not listed among the given options. The solution leads us to conclude that the nuclide symbol of [tex]\( X \)[/tex] is:
[tex]\[ {}_{80}^{231} Hg \][/tex]
However, since none of the given options match this conclusion, it can be inferred that none of the provided symbols correctly describe the nuclide [tex]\( X \)[/tex].
