Alright, let's work through the nuclear equation provided to find the particle that completes it.
The incomplete nuclear equation given is:
[tex]\[
^{239}_{94}Pu + ^{1}_{0}n \rightarrow ^{100}_{40}Zr + 7 + 2^{1}_{0}n + X
\][/tex]
### Step 1: Determine the Initial Side
First, we look at the initial side of the reaction:
- Plutonium-239: [tex]\(_{94}^{239}Pu\)[/tex]
- Neutron: [tex]\(_{0}^{1}n\)[/tex]
### Step 2: Calculate the Total Mass and Atomic Numbers on the Initial Side
- Total mass number: [tex]\(239 + 1 = 240\)[/tex]
- Total atomic number: [tex]\(94 + 0 = 94\)[/tex]
### Step 3: Determine the Products Given
On the right side of the equation, we have:
- Zirconium-100: [tex]\(_{40}^{100}Zr\)[/tex]
- Unknown particle (7 i.e. a placeholder we'll have to figure out)
- 2 Neutrons: [tex]\(2 \times ^{1}_{0}n\)[/tex]
### Step 4: Calculate the Total Mass and Atomic Numbers on the Right Side (Excluding the Unknown Particle)
- Mass number of Zirconium (Zr): 100
- Atomic number of Zirconium (Zr): 40
- Mass number of 2 neutrons: [tex]\(2 \times 1 = 2\)[/tex]
- Atomic number of 2 neutrons: [tex]\(2 \times 0 = 0\)[/tex]
Adding these together:
- Total mass number on the right (excluding the unknown particle): [tex]\(100 + 2 + 7 = 109\)[/tex] (assuming the placeholder has a mass of 7)
- Total atomic number on the right (excluding the unknown particle): [tex]\(40 + 0 + 7 = 47\)[/tex] (the placeholder should account for atomic number 7)
### Step 5: Determine the Mass and Atomic Numbers for the Missing Particle
To balance the equation:
- We need the mass number to total 240 on both sides.
- We need the atomic number to total 94 on both sides.
### Calculation:
To balance the mass numbers:
[tex]\[ 240 - 109 = 131 \][/tex]
To balance the atomic numbers:
[tex]\[ 94 - 47 = 47 \][/tex]
So the missing particle must have:
- Mass number: 131
- Atomic number: 47
### Step 6: Verify the Given Options
Now, let's look at the given options:
1. [tex]\({ }_{54}^{138} \times e\)[/tex]
2. [tex]\({ }_{54}^{139} \times e\)[/tex]
3. [tex]\({ }_{54}^{119} Xe\)[/tex]
4. [tex]\({ }_{52}^{138} Te\)[/tex]
None of the given particles match the calculated mass number and atomic number (47, 131), which indicates there is no suitable particle provided. Therefore, I conclude that the correct particle is not in the provided options.
