In the industrial production of which of the following, molecular hydrogen is obtained as a bye product.

We need to identify the industrial process that produces molecular hydrogen ($$H_2$$) as a by-product.

Option A: $$Na_2CO_3$$ - Sodium carbonate is manufactured by the Solvay process, which involves $$NaCl$$, $$NH_3$$, and $$CO_2$$. No $$H_2$$ is produced as a by-product.

Option B: $$NaOH$$ - Sodium hydroxide is manufactured by the Chlor-alkali process, which involves the electrolysis of concentrated brine (NaCl solution): $$2NaCl(aq) + 2H_2O(l) \xrightarrow{\text{electrolysis}} 2NaOH(aq) + Cl_2(g) + H_2(g)$$ In this process, $$H_2$$ gas is liberated at the cathode as a by-product, along with $$Cl_2$$ at the anode. The main product is $$NaOH$$.

Option C: Na metal - Sodium metal is produced by the Downs process (electrolysis of molten NaCl). No water is involved, so no $$H_2$$ is produced.

Option D: $$NaHCO_3$$ - Sodium bicarbonate is produced in the Solvay process. No $$H_2$$ is produced.

The correct answer is Option B: $$NaOH$$.