One reaction for the production of adipic acid, \(\mathrm{HOOC}\left(\mathrm{CH}_{2}\right)_{4} \mathrm{COOH},\) used in the manufacture of nylon, involves the oxidation of cyclohexanone, \(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{O},\) in a nitric acid solution. Assume that dinitrogen monoxide is also formed, and write a balanced equation for this reaction.

First, write down the unbalanced chemical reaction: \(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{O} + \mathrm{HNO}_{3} \rightarrow \mathrm{HOOC(CH}_{2}\mathrm{)_{4}COOH} + \mathrm{N}_{2} \mathrm{O}\).

Start by balancing the carbon atoms. There are 6 carbon atoms on the left side, therefore there should be 6 on the right side. However, there's only 1 atom in the adipic acid, which means multiply adipic acid by 6. Now balance the hydrogen atoms. On the left, there are 10, and on the right, there are 14 (from 6 adipic acid molecules and 2 from dinitrogen monoxide). Add 9 water molecules on the left to balance the hydrogen atoms. The equation now is \(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{O} + 9\mathrm{H}_{2}\mathrm{O} + \mathrm{HNO}_{3} \rightarrow 6\mathrm{HOOC(CH}_{2}\mathrm{)_{4}COOH} + 2\mathrm{N}_{2} \mathrm{O}\). Now, balance the oxygen atom: we have 1 (from cyclohexanone) + 18 (from water molecules) + 3 (from nitric acid) on the left, and 72 (from adipic acid) + 2 (from dinitrogen monoxide) on the right. Thus, we need 51 Nitric acid molecule on the left to balance the Oxygen atoms. The last step is balancing nitrogen atoms. We have 51 on the left (from nitric acid) and 2 on the right (from dinitrogen monoxide), add 49 dinitrogen monoxide on the right to balance. This gives the final balanced equation: \(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{O} + 9\mathrm{H}_{2}\mathrm{O} + 51\mathrm{HNO}_{3} \rightarrow 6\mathrm{HOOC(CH}_{2}\mathrm{)_{4}COOH} + 51\mathrm{N}_{2} \mathrm{O}\).

Always make sure to verify that all the atoms balance in the final equation. In this case, on both sides, there are 6 carbon atoms, 56 hydrogen atoms, and 102 oxygen atoms, and 51 nitrogen atoms. This confirms that the equation is balanced.