Step-by-step Derivation:

1. Identify the Monomers of Nylon 6,6: Nylon 6,6 is a condensation polymer formed from two monomers, each containing 6 carbon atoms. These monomers are:

   • Hexamethylenediamine: H₂N-(CH₂)₆-NH₂
   • Adipic acid: HOOC-(CH₂)₄-COOH

2. Identify the Monomer (X): The problem states that monomer (X) gives a positive carbylamine test. The carbylamine test (also known as Hoffmann's isocyanide test) is a specific test for primary amines (-NH₂ group).

   • Hexamethylenediamine has two primary amine groups, so it will give a positive carbylamine test.
   • Adipic acid is a carboxylic acid and does not have an amine group, so it will give a negative test. Therefore, the monomer (X) is hexamethylenediamine.

3. Determine the Chemical Formula of Monomer (X): The chemical formula for hexamethylenediamine, H₂N-(CH₂)₆-NH₂, is C₆H₁₆N₂. From the formula, we can see that one molecule of hexamethylenediamine contains two nitrogen atoms.

4. Relate Monomer (X) to Nitrogen Gas in the Dumas Method: The Dumas method is used for the quantitative estimation of nitrogen. In this method, all the nitrogen present in the organic compound is converted into nitrogen gas (N₂). The monomer (X) has the formula C₆H₁₆N₂. It contains 2 nitrogen atoms per molecule. These 2 nitrogen atoms combine to form one molecule of nitrogen gas (N₂). So, the stoichiometric relationship is: $1 \text{ molecule of } C₆H₁₆N₂ \longrightarrow 1 \text{ molecule of } N₂$ On a molar basis: $1 \text{ mole of } C₆H₁₆N₂ \longrightarrow 1 \text{ mole of } N₂$

5. Calculate the Moles of Nitrogen Gas Evolved: We are given that 10 moles of monomer (X) are analyzed. Based on the stoichiometry from Step 4, 10 moles of X will produce 10 moles of N₂ gas. Moles of N₂ evolved = 10 mol.

6. Calculate the Mass of Nitrogen Gas Evolved: The mass of a substance is calculated by multiplying its moles by its molar mass.

   • Atomic mass of Nitrogen (N) = 14 amu. So, the molar mass of N is 14 g/mol.
   • Molar mass of nitrogen gas (N₂) = 2 × 14 g/mol = 28 g/mol.

   Mass of N₂ evolved = Moles of N₂ × Molar mass of N₂ Mass of N₂ evolved = $10 \text{ mol} × 28 \text{ g/mol}$ Mass of N₂ evolved = 280 g.

Final Answer:

The amount of nitrogen gas evolved is 280 grams.