An element P disintegrates by `alpha` emission and the new element suffers two further disintegrations, both by `beta` emission, to form an element Q. Explain the fact that P and Q are isotopes.

Answer

To solve the problem, we need to analyze the disintegration process of element P and how it leads to the formation of element Q. We will use the concepts of alpha and beta decay to understand why P and Q are isotopes.

### Step-by-Step Solution:

1. **Identify Element P**:
- Let the atomic number of element P be \( Z \) and its mass number be \( A \).
- Therefore, we can represent element P as \( P(A, Z) \).

2. **Alpha Emission**:
- When element P undergoes alpha emission, it emits an alpha particle, which consists of 2 protons and 2 neutrons.
- This process decreases the atomic number by 2 and the mass number by 4.
- The new element formed after this emission can be represented as \( X(A-4, Z-2) \).

3. **Beta Emission**:
- The new element \( X \) then undergoes two beta emissions. In beta decay, a neutron is converted into a proton, which increases the atomic number by 1 while the mass number remains unchanged.
- After the first beta emission, the atomic number increases by 1:
- New element after first beta emission: \( Y(A-4, Z-1) \).
- After the second beta emission, the atomic number increases by another 1:
- Final element Q after second beta emission: \( Q(A-4, Z) \).

4. **Comparison of Elements P and Q**:
- Now we have:
- Element P: \( P(A, Z) \)
- Element Q: \( Q(A-4, Z) \)
- Here, both elements P and Q have the same atomic number \( Z \) but different mass numbers \( A \) and \( A-4 \).

5. **Conclusion**:
- Since isotopes are defined as elements that have the same atomic number but different mass numbers, we conclude that elements P and Q are isotopes of each other.

### Final Representation:
- Element P: \( P(A, Z) \)
- Element Q: \( Q(A-4, Z) \)
- Therefore, P and Q are isotopes because they have the same atomic number \( Z \) but different mass numbers \( A \) and \( A-4 \).