`1` mol of `N_(2)` is mixed with `3` mol of `H_(2)` in a litre container. If `50%` of `N_(2)` is converted into ammonia by the reaction `N_(2)(g)+3H_(2)(g) hArr 2NH_(3)(g)`, then the total number of moles of gas at the equilibrium are

Options

A. `1.5`
B. `4.5`
C. `3.0`
D. `6.0`

Explanation

To solve the problem step by step, we will analyze the reaction and the changes in the number of moles of the gases involved.

### Step 1: Write down the initial amounts of reactants
We start with:
- 1 mole of \(N_2\)
- 3 moles of \(H_2\)

### Step 2: Determine the degree of dissociation
It is given that 50% of \(N_2\) is converted into ammonia (\(NH_3\)). Therefore, the degree of dissociation (\(\alpha\)) is:
\[ \alpha = 50\% = \frac{50}{100} = 0.5 \]

### Step 3: Write the balanced chemical equation
The balanced chemical equation for the reaction is:
\[ N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \]

### Step 4: Calculate the change in moles at equilibrium
If 50% of \(N_2\) is converted, then:
- Moles of \(N_2\) that reacted = \(0.5 \times 1 = 0.5\) moles
- Moles of \(H_2\) that reacted = \(3 \times 0.5 = 1.5\) moles (since it takes 3 moles of \(H_2\) for every mole of \(N_2\))
- Moles of \(NH_3\) produced = \(2 \times 0.5 = 1\) mole (since 2 moles of \(NH_3\) are produced for every mole of \(N_2\))

### Step 5: Calculate the moles at equilibrium
Now we can calculate the moles of each substance at equilibrium:
- Moles of \(N_2\) at equilibrium = Initial moles - Moles reacted
\[ N_2 = 1 - 0.5 = 0.5 \text{ moles} \]
- Moles of \(H_2\) at equilibrium = Initial moles - Moles reacted
\[ H_2 = 3 - 1.5 = 1.5 \text{ moles} \]
- Moles of \(NH_3\) at equilibrium = Moles produced
\[ NH_3 = 1 \text{ mole} \]

### Step 6: Calculate the total moles at equilibrium
Now, we sum the moles of all gases at equilibrium:
\[ \text{Total moles} = \text{Moles of } N_2 + \text{Moles of } H_2 + \text{Moles of } NH_3 \]
\[ \text{Total moles} = 0.5 + 1.5 + 1 = 3.0 \text{ moles} \]

### Final Answer
The total number of moles of gas at equilibrium is:
\[ \boxed{3.0} \]