A nucleus of an element `._84 X^202` emits an `alpha-`particle first a `beta-`particle next and then a gamma photon. The final nucleus formed has an atomic number

Options

A. 200
B. 199
C. 83
D. 198

Explanation

To solve the problem step by step, we will analyze the decay process of the nucleus \( _{84}X^{202} \) as it emits an alpha particle, followed by a beta particle, and finally a gamma photon.

### Step 1: Identify the initial values
The initial nucleus is given as \( _{84}X^{202} \), where:
- Atomic number (Z) = 84
- Mass number (A) = 202

### Step 2: Alpha particle emission
When an alpha particle is emitted, the following changes occur:
- The mass number decreases by 4 (since an alpha particle has a mass number of 4).
- The atomic number decreases by 2 (since an alpha particle consists of 2 protons and 2 neutrons).

Calculating the new values after alpha emission:
- New mass number \( A' = 202 - 4 = 198 \)
- New atomic number \( Z' = 84 - 2 = 82 \)

So after the emission of the alpha particle, we have:
\[ _{82}Y^{198} \]

### Step 3: Beta particle emission
Next, a beta particle is emitted. The emission of a beta particle results in:
- The mass number remains unchanged.
- The atomic number increases by 1 (since a beta particle is an electron emitted from a neutron, which converts into a proton).

Calculating the new values after beta emission:
- New mass number remains \( A' = 198 \)
- New atomic number \( Z'' = 82 + 1 = 83 \)

So after the emission of the beta particle, we have:
\[ _{83}Z^{198} \]

### Step 4: Gamma photon emission
Finally, a gamma photon is emitted. The emission of a gamma photon does not change the mass number or the atomic number:
- Mass number remains \( A'' = 198 \)
- Atomic number remains \( Z'' = 83 \)

### Final Result
After all emissions, the final nucleus formed is:
\[ _{83}Z^{198} \]

Thus, the atomic number of the final nucleus is **83**.

### Summary of Steps:
1. Start with the initial nucleus \( _{84}X^{202} \).
2. After alpha emission: \( _{82}Y^{198} \).
3. After beta emission: \( _{83}Z^{198} \).
4. Gamma emission does not change the values.
5. Final atomic number is **83**.