To solve the problem regarding the decay sequence of Uranium-238, we need to identify the particles emitted (x1, x2, x3, and x4) during each step of the decay process.
### Step-by-Step Solution:
1. **Identify the first decay (Uranium-238 to Thorium-234)**:
- The decay is from \( ^{238}_{92}U \) to \( ^{234}_{90}Th \).
- The mass number decreases from 238 to 234 (a change of 4), and the atomic number decreases from 92 to 90 (a change of 2).
- This indicates the emission of an **alpha particle** (\( \alpha \)), which consists of 2 protons and 2 neutrons.
- Therefore, \( x_1 = \alpha \).
**Hint**: Remember that an alpha decay involves a decrease in both mass number by 4 and atomic number by 2.
2. **Identify the second decay (Thorium-234 to Protactinium-234)**:
- The decay is from \( ^{234}_{90}Th \) to \( ^{234}_{91}Pa \).
- The mass number remains the same (234), but the atomic number increases from 90 to 91 (a change of +1).
- This indicates the emission of a **beta particle** (\( \beta^- \)), which is an electron.
- Therefore, \( x_2 = \beta^- \).
**Hint**: Beta decay occurs when a neutron is converted into a proton, resulting in an increase in atomic number by 1 while the mass number remains unchanged.
3. **Identify the third decay (Protactinium-234 to Z)**:
- The decay is from \( ^{234}_{91}Pa \) to \( ^{234}_{Z} \).
- Again, the mass number remains the same (234), but the atomic number increases from 91 to Z (which is also +1).
- This again indicates the emission of a **beta particle** (\( \beta^- \)).
- Therefore, \( x_3 = \beta^- \).
**Hint**: Similar to the previous step, the emission of a beta particle indicates an increase in atomic number without a change in mass number.
4. **Identify the fourth decay (Z to Thorium-230)**:
- The decay is from \( ^{234}_{Z} \) to \( ^{230}_{90}Th \).
- The mass number decreases from 234 to 230 (a change of 4), and the atomic number decreases from Z to 90 (a change of 2).
- This indicates the emission of an **alpha particle** (\( \alpha \)).
- Therefore, \( x_4 = \alpha \).
**Hint**: Again, an alpha decay involves a decrease in both mass number by 4 and atomic number by 2.
### Summary of Results:
- \( x_1 = \alpha \)
- \( x_2 = \beta^- \)
- \( x_3 = \beta^- \)
- \( x_4 = \alpha \)
### Final Answer:
The correct options are those that identify:
- \( x_1 \) as an alpha particle,
- \( x_2 \) as a beta particle,
- \( x_3 \) as a beta particle,
- \( x_4 \) as an alpha particle.
