Let's break down the process of what happens when element X undergoes the emissions mentioned:
### Step-by-Step Solution:
1. Information Given:
- Atomic number of element X: \( Z_X = 92 \)
- Mass number of element X: \( A_X = 238 \)
2. Emission of an [tex]$\alpha$[/tex]-particle:
- An [tex]$\alpha$[/tex]-particle consists of 2 protons and 2 neutrons.
- When element X emits an [tex]$\alpha$[/tex]-particle, its atomic number decreases by 2 and its mass number decreases by 4.
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Atomic number after [tex]$\alpha$[/tex]-decay: \( Z_{Y\alpha} = Z_X - 2 = 92 - 2 = 90 \)
Mass number after [tex]$\alpha$[/tex]-decay: \( A_{Y\alpha} = A_X - 4 = 238 - 4 = 234 \)
```
Therefore, after the emission of the [tex]$\alpha$[/tex]-particle, the new element formed (let's call it Yα) has:
- Atomic number \( Z_{Y\alpha} = 90 \)
- Mass number \( A_{Y\alpha} = 234 \)
3. Emission of a [tex]$\beta$[/tex]-particle:
- A [tex]$\beta$[/tex]-particle is a high-energy electron ejected from the nucleus when a neutron decays into a proton.
- When an element undergoes [tex]$\beta$[/tex]-decay:
- The number of protons (atomic number) increases by 1.
- The mass number remains unchanged.
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Atomic number after [tex]$\beta$[/tex]-decay: \( Z_{Y\beta} = Z_{Y\alpha} + 1 = 90 + 1 = 91 \)
Mass number after [tex]$\beta$[/tex]-decay: \( A_{Y\beta} = A_{Y\alpha} = 234 \)
```
Thus, after the emission of the [tex]$\beta$[/tex]-particle, the new element formed (let's call it Yβ) has:
- Atomic number \( Z_{Y\beta} = 91 \)
- Mass number \( A_{Y\beta} = 234 \)
4. Emission of [tex]$\gamma$[/tex]-rays:
- [tex]$\gamma$[/tex]-rays do not affect the atomic number or the mass number of an element because they are simply high-energy photons.
```
Atomic number after [tex]$\gamma$[/tex]-emission: \( Z_{Y\gamma} = Z_{Y\beta} = 91 \)
Mass number after [tex]$\gamma$[/tex]-emission: \( A_{Y\gamma} = A_{Y\beta} = 234 \)
```
Therefore, after the emission of the [tex]$\gamma$[/tex]-rays, the final element formed maintains:
- Atomic number \( Z_{Y\gamma} = 91 \)
- Mass number \( A_{Y\gamma} = 234 \)
### Conclusion:
After the emission of an [tex]$\alpha$[/tex]-particle, a [tex]$\beta$[/tex]-particle, and [tex]$\gamma$[/tex]-rays from the element X, the resulting element has:
- Atomic number: 91
- Mass number: 234
Thus, the respective proton (atomic) and nucleon (mass) numbers of the new element formed are 91 and 234 respectively.
