4 moles each of `SO_(2) "and" O_(2)` gases are allowed to react to form `SO_(3)` in a closed vessel. At equilibrium 25% of `O_(2)` is used up. The total number of moles of all the gases at equilibrium is

Options

A. 6.5
B. 7
C. 8
D. 2

Explanation

To solve the problem, we will follow these steps:

### Step 1: Write the balanced chemical equation.
The reaction between sulfur dioxide (SO₂) and oxygen (O₂) to form sulfur trioxide (SO₃) can be represented as:
\[ \text{SO}_2 + \frac{1}{2} \text{O}_2 \rightarrow \text{SO}_3 \]

### Step 2: Determine the initial moles of each gas.
According to the problem, initially, we have:
- Moles of SO₂ = 4 moles
- Moles of O₂ = 4 moles
- Moles of SO₃ = 0 moles (since it hasn't formed yet)

### Step 3: Calculate the amount of O₂ used up.
We are told that 25% of O₂ is used up at equilibrium. Therefore, the amount of O₂ used up can be calculated as:
\[ \text{Used O}_2 = 0.25 \times 4 = 1 \text{ mole} \]

### Step 4: Relate the moles of O₂ used to the moles of SO₂ consumed.
From the balanced equation, we see that 1 mole of SO₂ reacts with 0.5 moles of O₂. Therefore, if 1 mole of O₂ is consumed, the moles of SO₂ consumed (let's denote it as X) can be calculated as follows:
\[ \frac{X}{2} = 1 \quad \Rightarrow \quad X = 2 \text{ moles of SO}_2 \text{ consumed} \]

### Step 5: Calculate the moles of each gas at equilibrium.
Now we can find the moles of each gas at equilibrium:
- Moles of SO₂ at equilibrium = Initial moles - Moles consumed
\[ \text{Moles of SO}_2 = 4 - 2 = 2 \text{ moles} \]
- Moles of O₂ at equilibrium = Initial moles - Moles consumed
\[ \text{Moles of O}_2 = 4 - 1 = 3 \text{ moles} \]
- Moles of SO₃ at equilibrium = Moles produced (which is equal to moles of SO₂ consumed)
\[ \text{Moles of SO}_3 = 0 + 2 = 2 \text{ moles} \]

### Step 6: Calculate the total number of moles at equilibrium.
Now, we can find the total number of moles of all gases at equilibrium:
\[ \text{Total moles} = \text{Moles of SO}_2 + \text{Moles of O}_2 + \text{Moles of SO}_3 \]
\[ \text{Total moles} = 2 + 3 + 2 = 7 \text{ moles} \]

### Final Answer:
The total number of moles of all the gases at equilibrium is **7 moles**.

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