40% of a mixture of 0.2 mol of `N_2 ` and 0.6 mol of `H_2` react to give `NH_3` according to the eqution, `N_2(g)+3H_2(g) hArr 2NH_3(g)`, at constant temperature and pressure. Then the ratio of the final volume to the initial volume of gases are

Options

A. `4:5`
B. `5:4`
C. `7:10`
D. `8:5`

Explanation

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

### Step 1: Write the balanced chemical equation
The balanced equation for the reaction is:
\[ N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \]

### Step 2: Determine the initial moles of reactants
From the problem, we have:
- Moles of \( N_2 = 0.2 \, \text{mol} \)
- Moles of \( H_2 = 0.6 \, \text{mol} \)

### Step 3: Calculate the total initial moles
The total initial moles of gases can be calculated as:
\[ \text{Initial moles} = \text{Moles of } N_2 + \text{Moles of } H_2 = 0.2 + 0.6 = 0.8 \, \text{mol} \]

### Step 4: Determine the amount of reactants that react
According to the problem, 40% of the mixture reacts. Thus, we need to calculate how many moles react:
- Total moles reacting = \( 40\% \) of total initial moles = \( 0.4 \times 0.8 = 0.32 \, \text{mol} \)

### Step 5: Determine the stoichiometry of the reaction
From the balanced equation, we know:
- 1 mole of \( N_2 \) reacts with 3 moles of \( H_2 \) to produce 2 moles of \( NH_3 \).

Let \( x \) be the moles of \( N_2 \) that react. Then:
- Moles of \( H_2 \) that react = \( 3x \)
- Moles of \( NH_3 \) produced = \( 2x \)

### Step 6: Set up the equations based on the reaction
Since we know that the total moles reacting is \( 0.32 \, \text{mol} \):
\[ x + 3x = 0.32 \]
\[ 4x = 0.32 \]
\[ x = 0.08 \]

### Step 7: Calculate the moles of each substance after the reaction
Using \( x = 0.08 \):
- Moles of \( N_2 \) remaining = \( 0.2 - 0.08 = 0.12 \)
- Moles of \( H_2 \) remaining = \( 0.6 - 3(0.08) = 0.6 - 0.24 = 0.36 \)
- Moles of \( NH_3 \) produced = \( 2(0.08) = 0.16 \)

### Step 8: Calculate the total final moles
Now, we can find the total moles of gases at equilibrium:
\[ \text{Final moles} = \text{Moles of } N_2 + \text{Moles of } H_2 + \text{Moles of } NH_3 \]
\[ \text{Final moles} = 0.12 + 0.36 + 0.16 = 0.64 \, \text{mol} \]

### Step 9: Calculate the ratio of final volume to initial volume
Since the volume of a gas is directly proportional to the number of moles at constant temperature and pressure, we can find the ratio of final volume to initial volume:
\[ \text{Ratio} = \frac{\text{Final moles}}{\text{Initial moles}} = \frac{0.64}{0.8} = \frac{64}{80} = \frac{4}{5} \]

### Final Answer
The ratio of the final volume to the initial volume of gases is:
\[ \frac{4}{5} \]

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