For a container contining `A(g),B(g),C(g) "&" D(g)` with rigid walls, an experiment is carried upon. This experiment involves increase in temperature of container in stepsof `1^(@)C` and system is allowed to attain equilibrium, followed by calculation of `K_(1) "&" K_(2)` at each step, where `K_(1) "&" K_(2)` are equilibrium constants for reaction (1) & (2) respectively.
`A(g)+2B(g)hArrC(g)+D(g)` ....(1)
`C(g)+D(g)hArrA(g)+2B(g)` .....(2)
Select the graph showing correct relationship-

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the two equilibrium reactions given and their corresponding equilibrium constants. Let's break it down step by step.

### Step 1: Write the Equilibrium Expressions
For the two reactions provided, we need to write the expressions for the equilibrium constants \( K_1 \) and \( K_2 \).

1. **For Reaction (1)**:
\[ A(g) + 2B(g) \rightleftharpoons C(g) + D(g) \]
The equilibrium constant \( K_1 \) is given by:
\[ K_1 = \frac{[C][D]}{[A][B]^2} \]

2. **For Reaction (2)**:
\[ C(g) + D(g) \rightleftharpoons A(g) + 2B(g) \]
The equilibrium constant \( K_2 \) is given by:
\[ K_2 = \frac{[A][B]^2}{[C][D]} \]

### Step 2: Relate \( K_1 \) and \( K_2 \)
From the expressions for \( K_1 \) and \( K_2 \), we can see that they are inversely related:
\[ K_2 = \frac{1}{K_1} \]
This means that if \( K_1 \) increases, \( K_2 \) must decrease, and vice versa.

### Step 3: Understand the Effect of Temperature
As the temperature of the system is increased, the equilibrium constants \( K_1 \) and \( K_2 \) will change according to the nature of the reactions (endothermic or exothermic). However, since we are not given specific information about the enthalpy changes of the reactions, we can only conclude that the relationship \( K_1 \cdot K_2 = 1 \) holds true at all temperatures.

### Step 4: Graphical Representation
Given that \( K_1 \) and \( K_2 \) are inversely related, we can conclude that the graph representing their relationship will show a hyperbolic curve where:
- When \( K_1 \) is high, \( K_2 \) is low.
- When \( K_2 \) is high, \( K_1 \) is low.

### Conclusion
Thus, the correct graph that represents the relationship between \( K_1 \) and \( K_2 \) is the one that shows an inverse relationship.

### Final Answer
The correct option is **Graph 3** (or option C).

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