Ammoina carbonate when heated to `200^(@)C` gives a mixture of `NH_(3)` and `CO_(2)` vapour with a density of `13.0` What is the degree of dissociation of ammonium carbonate ?

Options

A. `3//2`
B. `1//2`
C. 2
D. 1

Explanation

To find the degree of dissociation of ammonium carbonate when heated, we can follow these steps:

### Step 1: Determine the Molecular Weight of Ammonium Carbonate
Ammonium carbonate has the formula \( NH_4CO_3 \). To calculate its molecular weight:
- Nitrogen (N): 14 g/mol (1 atom)
- Hydrogen (H): 1 g/mol (4 atoms, so \( 4 \times 1 = 4 \))
- Carbon (C): 12 g/mol (1 atom)
- Oxygen (O): 16 g/mol (3 atoms, so \( 3 \times 16 = 48 \))

Now, summing these up:
\[ \text{Molecular weight of } NH_4CO_3 = 14 + 4 + 12 + 48 = 78 \text{ g/mol} \]

### Step 2: Calculate the Initial Vapor Density
The vapor density (D) is defined as half the molecular weight:
\[ D = \frac{\text{Molecular Weight}}{2} = \frac{78}{2} = 39 \]

### Step 3: Use the Given Final Vapor Density
The problem states that the final vapor density after heating is 13.

### Step 4: Set Up the Degree of Dissociation Formula
The degree of dissociation (\( \alpha \)) can be calculated using the formula:
\[ \alpha = \frac{D - d}{(N - 1) \cdot d} \]
Where:
- \( D \) = initial vapor density (39)
- \( d \) = final vapor density (13)
- \( N \) = number of moles of products formed

### Step 5: Determine the Number of Moles (N)
When ammonium carbonate decomposes, it produces ammonia (\( NH_3 \)) and carbon dioxide (\( CO_2 \)). The balanced reaction is:
\[ NH_4CO_3 \rightarrow 2NH_3 + CO_2 \]
From this reaction, we see that 1 mole of ammonium carbonate produces 2 moles of ammonia and 1 mole of carbon dioxide, totaling 3 moles of products. Therefore, \( N = 3 \).

### Step 6: Substitute Values into the Degree of Dissociation Formula
Now we can substitute the values into the formula:
\[ \alpha = \frac{39 - 13}{(3 - 1) \cdot 13} = \frac{26}{2 \cdot 13} = \frac{26}{26} = 1 \]

### Conclusion
The degree of dissociation of ammonium carbonate is \( \alpha = 1 \).

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