In the nuclear reaction `U_92^238 to Pb_82^206 + xHe_2^4 + y beta_(-1)^0` the value of x and y are respectively ………

Options

A. 8, 6
B. 6, 4
C. 6, 8
D. 8, 10

Answer

To solve the nuclear reaction \( U_{92}^{238} \rightarrow Pb_{82}^{206} + x He_{2}^{4} + y \beta_{-1}^{0} \), we need to determine the values of \( x \) (the number of helium nuclei or alpha particles emitted) and \( y \) (the number of beta particles emitted).

### Step-by-Step Solution:

1. **Identify the Mass and Atomic Numbers**:
- Uranium (U) has a mass number of 238 and an atomic number of 92.
- Lead (Pb) has a mass number of 206 and an atomic number of 82.
- Helium (He) has a mass number of 4 and an atomic number of 2.
- Beta particles (\( \beta \)) have a mass number of 0 and an atomic number of -1.

2. **Calculate the Change in Mass Number**:
- The initial mass number is 238 (from U).
- The final mass number is 206 (from Pb).
- The difference in mass number is:
\[ 238 - 206 = 32 \]

3. **Determine the Contribution of Alpha Particles**:
- Each alpha particle (He) has a mass number of 4.
- Let \( x \) be the number of alpha particles emitted. The total mass number contributed by alpha particles is \( 4x \).
- Thus, we can set up the equation:
\[ 4x + 0y = 32 \]
- This simplifies to:
\[ 4x = 32 \implies x = \frac{32}{4} = 8 \]

4. **Calculate the Change in Atomic Number**:
- The initial atomic number is 92 (from U).
- The final atomic number is 82 (from Pb).
- The difference in atomic number is:
\[ 92 - 82 = 10 \]

5. **Determine the Contribution of Alpha and Beta Particles**:
- Each alpha particle decreases the atomic number by 2. Therefore, for \( x \) alpha particles, the contribution to the atomic number is \( -2x \).
- Each beta particle increases the atomic number by 1. Therefore, for \( y \) beta particles, the contribution to the atomic number is \( +1y \).
- We can set up the equation:
\[ -2x + y = -10 \]
- Substituting \( x = 8 \):
\[ -2(8) + y = -10 \implies -16 + y = -10 \implies y = 6 \]

6. **Final Values**:
- The values of \( x \) and \( y \) are:
\[ x = 8 \quad \text{and} \quad y = 6 \]

### Conclusion:
The values of \( x \) and \( y \) are respectively \( 8 \) and \( 6 \).

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