The equation `alpha =(D-d)/((n-1)d)` is correctly matched for

Options

A. `A hArr nB//2 =nC//3`
B. `A hArr nB//3 +(2n//3)C`
C. `A to (n//2)B+(n//4)C`
D. `A hArr (n//2)B+C`

Answer

To solve the equation \( \alpha = \frac{D - d}{(n - 1)d} \) and determine the correct context in which it applies, we need to analyze the terms involved in the equation and the chemical reactions that fit this description.

### Step-by-Step Solution:

1. **Understanding the Terms:**
- \( \alpha \): Degree of dissociation, which indicates the fraction of the reactant that has dissociated into products.
- \( D \): Vapor density before dissociation.
- \( d \): Vapor density after dissociation.
- \( n \): Number of moles of gaseous products formed from 1 mole of reactant.

2. **Identifying the Reaction Type:**
- The equation is applicable to a reaction where one mole of a reactant dissociates into \( n \) moles of products. This is essential for applying the formula correctly.

3. **Analyzing the Given Options:**
- We need to find a reaction where the reactant is 1 mole and the total number of moles of products is \( n \).
- Let's evaluate the provided equations one by one.

4. **Evaluating the First Option:**
- Reaction: \( A \rightleftharpoons \frac{n}{3}B + \frac{2n}{3}C \)
- Reactant moles: 1 (for \( A \))
- Product moles: \( \frac{n}{3} + \frac{2n}{3} = n \)
- This fits the criteria, so this option is valid.

5. **Evaluating the Second Option:**
- Reaction: \( A \rightleftharpoons \frac{n}{2}B + \frac{n}{4}C \)
- Reactant moles: 1 (for \( A \))
- Product moles: \( \frac{n}{2} + \frac{n}{4} = \frac{3n}{4} \)
- This does not fit since \( \frac{3n}{4} \neq n \).

6. **Evaluating the Third Option:**
- Reaction: \( A \rightleftharpoons \frac{n}{3}B + \frac{2n}{3}C \) (same as the first option)
- This has already been validated as correct.

7. **Evaluating the Fourth Option:**
- Reaction: \( A \rightleftharpoons \frac{n}{2}B + C \)
- Reactant moles: 1 (for \( A \))
- Product moles: \( \frac{n}{2} + 1 \)
- This does not fit since \( \frac{n}{2} + 1 \neq n \) unless \( n = 2 \), which is not a general case.

8. **Conclusion:**
- The only valid reaction that fits the equation \( \alpha = \frac{D - d}{(n - 1)d} \) is the first option, where the total number of moles of products equals \( n \) when starting from 1 mole of reactant.

### Final Answer:
The equation \( \alpha = \frac{D - d}{(n - 1)d} \) is correctly matched for the reaction \( A \rightleftharpoons \frac{n}{3}B + \frac{2n}{3}C \).