The equilibrium `A(g) +4B(g) hArr AB_4(g)` is attained by mixing equal moles of A and B in a one litre vessel Then at equilibrium

Options

A. [A]=[B]
B. `[A] gt [B]`
C. `[A] lt [B]`
D. `[AB_4] gt [A]`

Answer

To solve the problem, we will analyze the equilibrium reaction given:

**Reaction:**
\[ A(g) + 4B(g) \rightleftharpoons AB_4(g) \]

**Step 1: Initial Moles**
We start with equal moles of A and B in a 1-liter vessel. Let’s assume we have 1 mole of A and 1 mole of B initially.

**Step 2: Change in Moles at Equilibrium**
Let \( \alpha \) be the extent of the reaction that proceeds to the right (the amount of A that reacts). According to the stoichiometry of the reaction:
- For every 1 mole of A that reacts, 4 moles of B react.
- Therefore, at equilibrium:
- Moles of A left = \( 1 - \alpha \)
- Moles of B left = \( 1 - 4\alpha \)
- Moles of \( AB_4 \) formed = \( \alpha \)

**Step 3: Concentration at Equilibrium**
Since the volume of the vessel is 1 liter, the concentration (C) can be calculated as:
- Concentration of A at equilibrium:
\[ [A] = \frac{1 - \alpha}{1} = 1 - \alpha \]

- Concentration of B at equilibrium:
\[ [B] = \frac{1 - 4\alpha}{1} = 1 - 4\alpha \]

- Concentration of \( AB_4 \) at equilibrium:
\[ [AB_4] = \frac{\alpha}{1} = \alpha \]

**Step 4: Analyzing the Concentrations**
Now we need to compare the concentrations:
- Since \( \alpha \) is a positive value (the reaction proceeds to the right), \( 1 - 4\alpha \) will decrease faster than \( 1 - \alpha \) as \( \alpha \) increases.
- If \( \alpha \) is small (less than 0.25), \( [B] \) will still be positive, but as \( \alpha \) approaches 0.25, \( [B] \) approaches zero.

**Step 5: Conclusions**
1. **Comparison of Concentrations**:
- For small values of \( \alpha \), \( 1 - \alpha > 1 - 4\alpha \) implies \( [A] > [B] \).
- As \( \alpha \) increases, \( [B] \) decreases more rapidly than \( [A] \).
- Therefore, we conclude that \( [A] > [B] \) at equilibrium.

2. **Options Analysis**:
- Option 1: \( [A] = [B] \) - Incorrect.
- Option 2: \( [A] > [B] \) - Correct.
- Option 3: \( [A] < [B] \) - Incorrect.
- Option 4: \( [AB_4] > [A] \) - Uncertain without specific values for \( \alpha \).

Thus, the correct answer is **Option 2: \( [A] > [B] \)**.

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