For the given reaction at constant pressure,
`{:(,nA(g)hArr,A_n(g)),("Initial moles",1,0),("Final moles",1-alpha,alpha/n):}`
Then the correct relation between initial density `(d_i)` and final density `(d_f)` of the system is :

Options

A. `[(n-1)/(n)][(d_(f)-d_(i))/(d_(f))]=alpha`
B. `(n)/(n-1)([d_(f)-d_(i)])/(d_(f))=alpha`
C. `[(n-1)/(n)][(d_(i)-d_(f))/(d_(i))]=alpha`
D. `(1)/((n-1))[(d_(i)-d_(f))/(d_(i))]=alpha`

Explanation

To solve the problem, we need to establish the relationship between the initial density \( d_i \) and the final density \( d_f \) of the system given the reaction:

\[ N_A(g) \rightleftharpoons A_n(g) \]

### Step-by-Step Solution:

1. **Identify Initial and Final Moles:**
- Initially, we have 1 mole of \( N_A \) and 0 moles of \( A_n \).
- After the reaction, the moles are \( 1 - \alpha \) for \( N_A \) and \( \frac{\alpha}{n} \) for \( A_n \).

2. **Use the Ideal Gas Law:**
- At constant temperature and pressure, the volume of an ideal gas is directly proportional to the number of moles. Therefore, we can say:
\[ V \propto n \]
- Thus, we can express the mass of the gas in terms of density and volume:
\[ \text{Mass} = \text{Density} \times \text{Volume} \]

3. **Express Initial Mass:**
- The initial mass of the system can be expressed as:
\[ m_i = d_i \times V_i \]
- Since \( V_i \) is proportional to the initial moles (1 mole), we can write:
\[ m_i = d_i \times 1 = d_i \]

4. **Express Final Mass:**
- The final mass of the system can be expressed as:
\[ m_f = d_f \times V_f \]
- The final volume \( V_f \) is proportional to the total final moles, which is:
\[ V_f \propto (1 - \alpha) + \frac{\alpha}{n} \]
- Therefore, we can express the final mass as:
\[ m_f = d_f \times \left( (1 - \alpha) + \frac{\alpha}{n} \right) \]

5. **Set Up the Conservation of Mass Equation:**
- Since mass is conserved, we have:
\[ m_i = m_f \]
- Substituting the expressions for \( m_i \) and \( m_f \):
\[ d_i = d_f \times \left( (1 - \alpha) + \frac{\alpha}{n} \right) \]

6. **Rearranging the Equation:**
- Rearranging gives:
\[ \frac{d_i}{d_f} = (1 - \alpha) + \frac{\alpha}{n} \]
- This can be simplified further, but it already gives us a relationship between \( d_i \) and \( d_f \).

7. **Final Relation:**
- The final relationship can be expressed as:
\[ d_i = d_f \left( (1 - \alpha) + \frac{\alpha}{n} \right) \]

### Final Answer:
The correct relation between the initial density \( d_i \) and the final density \( d_f \) of the system is:
\[ d_i = d_f \left( (1 - \alpha) + \frac{\alpha}{n} \right) \]