Select the correct relationship for the given equilibrium

Options

A. `N_(2(g)) + O_(2(g)) hArr 2 NO_((g)), K_(eq) = K_(1) , NO_((g)) hArr (1)/(2) N_(2(g)) + (1)/(2) O_(2(g)) , K_(eq) = K_(2) `thus , `K_(1) K_(2)^(2)` = 1
B. Free energy change is `Delta G_(1)^(@)` for ` 2NO_(2(g)) hArr N_(2)O_(4(g)) and Delta G_(2)^(@) "for" (1)/(2) N_(2) O_(4(g)) hArr NO_(2(g))` then `2 Delta G_(2)^(@) + Delta G_(1)^(@) ` = 0
C. Vont't Hoff equation is `Delta_(r) H^(@) [ (1)/(T_(2)) - (1)/(T_(1)) ] - (Delta G^(@))/(T_(2)) - (Delta_(r).Delta G^(@))/(T_(1))`
D. Fo the equilibrium `A_((g)) hArr 2B_((g)) , x = sqrt((K_(p))/(K_(p) + 4p))` (x = degree of dissociation)

Answer

To solve the problem of selecting the correct relationship for the given equilibrium, we will analyze the provided reactions and their associated equilibrium constants step by step.

### Step 1: Analyze the First Reaction
The first reaction is given as:
\[ \text{N}_2 + \text{O}_2 \rightleftharpoons 2 \text{NO} \]
Let the equilibrium constant for this reaction be \( K_1 \).

### Step 2: Analyze the Second Reaction
The second reaction is:
\[ 2 \text{NO} \rightleftharpoons \text{N}_2 + \text{O}_2 \]
The equilibrium constant for this reaction can be expressed as:
\[ K_2 = \frac{[\text{N}_2][\text{O}_2]}{[\text{NO}]^2} \]

### Step 3: Relationship Between \( K_1 \) and \( K_2 \)
From the first reaction, we can express \( K_1 \) as:
\[ K_1 = \frac{[\text{NO}]^2}{[\text{N}_2][\text{O}_2]} \]
For the second reaction, since it is the reverse of the first, we have:
\[ K_2 = \frac{1}{K_1} \]

### Step 4: Establishing the Relationship
We can relate \( K_1 \) and \( K_2 \) as follows:
\[ K_1 \cdot K_2 = 1 \]
This implies:
\[ K_1 \cdot K_2^2 = 1 \]

### Step 5: Verify the Given Options
- **Option A**: \( K_1 \cdot K_2^2 = 1 \) is correct based on our analysis.
- **Option B**: The relationship involving Gibbs free energy changes can be verified using the equations provided in the transcript. The relationship \( 2 \Delta G_2^0 + \Delta G_1^0 = 0 \) is also correct.
- **Option C**: The rearrangement of Gibbs free energy and the van't Hoff equation also checks out, confirming that this relationship is correct.
- **Option D**: The relationship involving the degree of dissociation \( \alpha \) is also valid as derived from the equilibrium expression.

### Conclusion
All options (A, B, C, D) are correct relationships for the given equilibrium.

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