### Solution:
1. Identify the given elements and particles in the nuclear equation:
- Starting with the left side:
- Plutonium-239 ([tex]\( _{94}^{239}Pu \)[/tex])
- One neutron ([tex]\( _{0}^{1}n \)[/tex])
- On the right side:
- Xenon-134 ([tex]\( _{54}^{134}Xe \)[/tex])
- Three neutrons ([tex]\( 3 \cdot _{0}^{1}n \)[/tex])
2. Conservation of mass number:
- The total mass number on the left side:
- Mass number of Plutonium-239: 239
- Mass number of one neutron: 1
- Total mass number on the left: [tex]\( 239 + 1 = 240 \)[/tex]
- The total mass number on the right side:
- Mass number of Xenon-134: 134
- Mass number of three neutrons: [tex]\( 3 \times 1 = 3 \)[/tex]
- Total mass number contributed by known entities on the right: [tex]\( 134 + 3 = 137 \)[/tex]
To balance the mass number:
- Total mass number must be 240, and currently, we have 137.
- So, the missing element must have a mass number: [tex]\( 240 - 137 = 103 \)[/tex]
3. Conservation of atomic number:
- The total atomic number on the left side:
- Atomic number of Plutonium-239: 94
- Atomic number of one neutron: 0 (neutrons have no charge)
- Total atomic number: [tex]\( 94 + 0 = 94 \)[/tex]
- The total atomic number on the right side:
- Atomic number of Xenon-134: 54
- Atomic number of three neutrons: [tex]\( 3 \times 0 = 0 \)[/tex]
- Total atomic number contributed by known entities on the right: 54
To balance the atomic number:
- Total atomic number must be 94, and currently, we have 54.
- So, the atomic number of the missing element: [tex]\( 94 - 54 = 40 \)[/tex]
4. Identify the missing element:
- The missing element has an atomic number of 40 and a mass number of 103.
- The element with atomic number 40 is Zirconium ([tex]\( Zr \)[/tex]).
5. Selecting the correct isotope:
- Given isotopes of Zirconium are:
- [tex]\( _{40}^{101}Zr \)[/tex]
- [tex]\( _{40}^{105}Zr \)[/tex]
- [tex]\( _{40}^{103}Zr \)[/tex]
- The correct isotope must have a mass number of 103.
Therefore, the missing element in the nuclear equation is:
[tex]\[
_{40}^{103} Zr
\][/tex]
So, the balanced nuclear equation will be:
[tex]\[
_{94}^{239}Pu + _{0}^{1}n \rightarrow _{54}^{134}Xe + _{40}^{103}Zr + 3 _{0}^{1}n
\][/tex]
