This problem involves determining the number of amino groups in an organic compound based on the molecular mass change after acylation with acetyl chloride (CH₃COCl).

Concept Explanation:

Acylation is a reaction where an amino group (-NH₂) reacts with acetyl chloride to form an amide. The reaction for each amino group is:

$R-{\mathrm{NH}}_2+{\mathrm{CH}}_3\mathrm{COCl}\to R-\mathrm{NH}-\mathrm{CO}-{\mathrm{CH}}_3+\mathrm{HCl}$

For each amino group that undergoes acylation, the molecular mass increases because the -H from -NH₂ is replaced by the -COCH₃ group (acetyl group).

Step-by-Step Solution:

Step 1: Calculate the mass change per amino group.

Molecular mass of acetyl chloride (CH₃COCl) = 12 + 3*1 + 12 + 16 + 35.5 = 78.5 g/mol.

After reaction, the acetyl group (-COCH₃) is added, and one H atom is removed.

Mass of acetyl group (-COCH₃) = 12 + 16 + 12 + 3*1 = 43 g/mol.

Mass of H atom removed = 1 g/mol.

So, net mass increase per amino group = 43 - 1 = 42 g/mol.

Step 2: Find total mass increase.

Initial molecular mass = 180 g/mol.

Final molecular mass after acylation = 390 g/mol.

Total mass increase = 390 - 180 = 210 g/mol.

Step 3: Determine number of amino groups.

Let $n$ be the number of amino groups.

Total mass increase = $n$ × (mass increase per amino group)

$210=n\times 42$

$n=\frac{210}{42}=5$

So, the number of amino groups per molecule is 5.

Related Topics and Formulae:

Related Topics: Acylation reactions, molecular mass calculations in organic chemistry, stoichiometry in functional group transformations.

Key Formula: For each amino group acylated, net mass increase = (Mass of -COCH₃ group) - (Mass of H atom) = 43 - 1 = 42 g/mol.

Number of amino groups = (Total molecular mass increase) / 42