Group 17: describe question

A student adds solid sodium chloride to an excess of cold concentrated sulfuric acid. The mixture effervesces and a misty white fume is seen. The student writes the equation:

$$\ce{NaCl(s) + H2SO4(l) -> NaHSO4(s) + HCl(g)}$$

The student then repeats the experiment with solid sodium iodide in place of sodium chloride. The misty fume is again seen, but the mixture also turns brown and a gas with a sharp, choking smell is produced.

Describe and explain why, unlike iodide, chloride is not oxidised by concentrated sulfuric acid.