A nucleus ` ""_(Z)^(A) X` emits 2 `alpha` particles and 1 `beta` particle to form a nucleus ` ""_(85)^(222) R`. Find the atomic number and mass number of X.

Answer

To solve the problem, we need to determine the atomic number (Z) and mass number (A) of the nucleus \( _{Z}^{A}X \) that emits 2 alpha particles and 1 beta particle to form the nucleus \( _{85}^{222}R \).

### Step-by-Step Solution:

1. **Understand the Emission Process**:
- An alpha particle (α) consists of 2 protons and 2 neutrons, which means it decreases the mass number (A) by 4 and the atomic number (Z) by 2.
- A beta particle (β) is an electron emitted when a neutron is converted into a proton. This process does not change the mass number (A) but increases the atomic number (Z) by 1.

2. **Calculate Changes Due to Alpha Emission**:
- Since 2 alpha particles are emitted:
- Change in mass number due to 2 alpha particles = \( 2 \times 4 = 8 \)
- Change in atomic number due to 2 alpha particles = \( 2 \times 2 = 4 \)

3. **Calculate Changes Due to Beta Emission**:
- Since 1 beta particle is emitted:
- Change in mass number due to beta particle = 0 (no change)
- Change in atomic number due to beta particle = 1

4. **Combine the Changes**:
- Total change in mass number (A):
\[ A \text{ (of } X\text{)} - 8 = 222 \]
- Total change in atomic number (Z):
\[ Z \text{ (of } X\text{)} - 4 + 1 = 85 \]

5. **Set Up the Equations**:
- From the mass number equation:
\[ A - 8 = 222 \implies A = 222 + 8 = 230 \]
- From the atomic number equation:
\[ Z - 4 + 1 = 85 \implies Z - 3 = 85 \implies Z = 85 + 3 = 88 \]

6. **Final Answer**:
- The atomic number \( Z \) of nucleus \( X \) is 88.
- The mass number \( A \) of nucleus \( X \) is 230.

### Summary:
- Atomic number \( Z \) of \( X \) = 88
- Mass number \( A \) of \( X \) = 230