Equal number of moles of A and B are allowed to react with each other till it reaches equilibrium.
`2A+BhArrC+D`
The value of `K_(c)` for this equilibrium can never be

Options

A. `gt1`
B. `lt1`
C. `=1`
D. `oo`

Answer

To solve the problem, we need to analyze the reaction and the equilibrium constant \( K_c \) for the given reaction:

\[ 2A + B \rightleftharpoons C + D \]

### Step 1: Understanding the Reaction
The reaction involves two moles of substance A reacting with one mole of substance B to produce one mole of substance C and one mole of substance D.

### Step 2: Setting Up the Initial Conditions
Let’s assume we start with \( n \) moles of A and \( n \) moles of B. Since the question states that equal numbers of moles of A and B are allowed to react, we can denote the initial concentrations as follows:

- Initial concentration of A = \( n \)
- Initial concentration of B = \( n \)
- Initial concentration of C = 0
- Initial concentration of D = 0

### Step 3: Change in Concentration at Equilibrium
Let \( x \) be the amount of A that reacts at equilibrium. According to the stoichiometry of the reaction:

- Change in A = \( -2x \)
- Change in B = \( -x \)
- Change in C = \( +x \)
- Change in D = \( +x \)

At equilibrium, the concentrations will be:

- Concentration of A = \( n - 2x \)
- Concentration of B = \( n - x \)
- Concentration of C = \( x \)
- Concentration of D = \( x \)

### Step 4: Expression for \( K_c \)
The equilibrium constant \( K_c \) for the reaction can be expressed as:

\[ K_c = \frac{[C][D]}{[A]^2[B]} \]

Substituting the equilibrium concentrations into the expression:

\[ K_c = \frac{x \cdot x}{(n - 2x)^2 \cdot (n - x)} = \frac{x^2}{(n - 2x)^2 \cdot (n - x)} \]

### Step 5: Analyzing the Possible Values of \( K_c \)
1. **Positive Values**: Since concentrations cannot be negative, \( K_c \) must be positive.
2. **Zero**: If the reaction does not proceed at all (i.e., no products are formed), then \( K_c \) approaches zero. However, this situation is not possible since we are starting with equal moles of A and B, and they will react to some extent.
3. **Infinity**: If the reaction goes to completion, meaning all A and B are converted to C and D, then \( K_c \) would approach infinity. This is theoretically possible but depends on the specific conditions of the reaction.

### Conclusion
The value of \( K_c \) can never be zero because there will always be some products formed as long as the reaction is allowed to proceed with equal moles of A and B. Thus, the answer is:

**The value of \( K_c \) can never be zero.**