Ammonium carbamate when heated tto `200^(@)C` gives a mixture of `NH_(3) and CO_(2)` vapour with a density of 13. what is the degree of dissociation of ammonium carbamate?

Options

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

Answer

To solve the problem of determining the degree of dissociation of ammonium carbamate when heated, we can follow these steps:

### Step 1: Write the decomposition reaction
Ammonium carbamate (NH2COONH4) decomposes into ammonia (NH3) and carbon dioxide (CO2) upon heating. The reaction can be represented as:
\[ \text{NH}_2\text{COONH}_4 \rightarrow \text{NH}_3 + \text{CO}_2 \]

### Step 2: Calculate the molecular weight of ammonium carbamate
To find the molecular weight (M) of ammonium carbamate, we sum the atomic weights of its constituent atoms:
- Nitrogen (N): 14 g/mol (2 Nitrogens = 28 g/mol)
- Hydrogen (H): 1 g/mol (4 Hydrogens = 4 g/mol)
- Carbon (C): 12 g/mol (1 Carbon = 12 g/mol)
- Oxygen (O): 16 g/mol (2 Oxygens = 32 g/mol)

Thus, the total molecular weight is:
\[ M = 28 + 4 + 12 + 32 = 76 \, \text{g/mol} \]

### Step 3: Calculate the initial vapor density
The vapor density (D) of the gas can be calculated using the formula:
\[ D = \frac{M}{2} \]
Substituting the molecular weight:
\[ D = \frac{76}{2} = 38 \]

### Step 4: Identify the final density and number of moles
The problem states that the density of the mixture of ammonia and carbon dioxide is given as 13 g/L.

The total number of moles (n) produced from the decomposition of one mole of ammonium carbamate is 2 (1 mole of NH3 and 1 mole of CO2).

### Step 5: Use the degree of dissociation formula
The degree of dissociation (α) can be calculated using the formula:
\[ \alpha = \frac{D - d}{n - 1} \cdot d \]
Where:
- \( D \) = initial vapor density = 38
- \( d \) = density of the mixture = 13
- \( n \) = total number of moles = 2

Substituting the values:
\[ \alpha = \frac{38 - 13}{2 - 1} \cdot 13 \]
\[ \alpha = \frac{25}{1} \cdot 13 \]
\[ \alpha = 25 \cdot 13 \]
\[ \alpha = 25 \]

### Step 6: Conclusion
The degree of dissociation of ammonium carbamate is 25.