Nitrogen Gas Evolved from Nylon Monomer via Dumas Method

To find the amount of nitrogen gas evolved, we follow these steps:

1. Identification of Monomer ($\mathbf{X}$): Nylon 6,6 is a polymer formed by the condensation polymerization of two monomers:

   • Adipic acid: $\text{HOOC}-(\text{CH}_2)_4-\text{COOH}$
   • Hexamethylenediamine: $\text{H}_2\text{N}-(\text{CH}_2)_6-\text{NH}_2$

   Since the monomer $\mathbf{X}$ gives a positive carbylamine test, it must be a primary amine. Thus, monomer $\mathbf{X}$ is hexamethylenediamine ($\text{C}_6\text{H}_{16}\text{N}_2$).

2. Stoichiometry of Nitrogen Gas Evolution in Dumas Method: In the Dumas method, all the nitrogen present in the compound is converted into molecular nitrogen gas ($\text{N}_2$).

   From the chemical formula of hexamethylenediamine ($\text{C}_6\text{H}_{16}\text{N}_2$), each molecule contains $2$ nitrogen atoms: $\text{C}_6\text{H}_{16}\text{N}_2 \xrightarrow{\text{Dumas Method}} \text{N}_2\text{(g)} + \dots$

   Therefore, $1\text{ mole of }\mathbf{X}$ produces $1\text{ mole of }\text{N}_2$ gas.

3. Calculation of the Mass of $\text{N}_2$ Gas:

   • Moles of $\mathbf{X} = 10\text{ moles}$
   • Moles of $\text{N}_2\text{ gas evolved} = 10\text{ moles}$
   • Molar mass of $\text{N}_2 = 2 \times 14\text{ g/mol} = 28\text{ g/mol}$

   $\text{Mass of }\text{N}_2\text{ gas} = 10\text{ mol} \times 28\text{ g/mol} = 280\text{ g}$

Final Answer: $280$