Step 1: Why the C-X bond is polar.
Picture the electron cloud between carbon and the halogen. Because the halogen holds electrons more tightly, the cloud is lop-sided towards X. The result is a permanent dipole: a fractional negative charge on X and a matching fractional positive charge on C. This is a covalent bond with ionic character, that is a polar covalent bond, and the electron-poor carbon is where reactions begin.
Step 2: How the bond changes down the halogens.
Two opposing trends matter. Polarity tracks electronegativity, largest for C-F. Bond strength tracks orbital overlap, which worsens as the halogen grows, so C-I is the longest and weakest link while C-F is the shortest and strongest.
Step 3: A concrete sp3 case.
Take chloromethane, CH3Cl. Its carbon uses four equivalent sp3 orbitals arranged tetrahedrally, one of which overlaps with chlorine. That carbon-chlorine link is therefore an sp3 C-X bond.
Step 4: Carbylamine equation.
Aniline illustrates the test just as well as a simple alkyl amine. Warmed with trichloromethane and alcoholic potash it yields phenyl isocyanide.
C6H5NH2 + CHCl3 + 3KOH → C6H5NC + 3KCl + 3H2O
Step 5: Hinsberg equation.
The Hinsberg reagent is benzenesulphonyl chloride. Reacting it with methylamine (a primary amine) swaps the chlorine for the amine and releases HCl.
C6H5SO2Cl + CH3NH2 → C6H5SO2NHCH3 + HCl
The sulphonamide formed keeps one N-H, so it is soluble in KOH.