What are the hybrid orbitals of the carbon atoms in the following molecules? (a) \(\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}_{3}\) (b) \(\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}=\mathrm{CH}_{2}\) (c) \(\mathrm{CH}_{3}-\mathrm{C} \equiv \mathrm{C}-\mathrm{CH}_{2} \mathrm{OH}\) (d) \(\mathrm{CH}_{3} \mathrm{CH}=\mathrm{O}\) (e) \(\mathrm{CH}_{3} \mathrm{COOH}\)

Let's start by recognizing the types and numbers of bonds in each molecule: (a) Each carbon in \(\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}_{3}\) forms four single bonds. (b) The first carbon in \(\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}=\mathrm{CH}_{2}\) forms four single bonds, while the second forms one single, one double, and one single bond. (c) In \(\mathrm{CH}_{3}-\mathrm{C} \equiv \mathrm{C}-\mathrm{CH}_{2} \mathrm{OH}\), the first and third carbon atoms form four single bonds, the second forms one single and one triple bond. (d) The first carbon in \(\mathrm{CH}_{3} \mathrm{CH}=\mathrm{O}\) forms four single bonds, while the second forms two single bonds and one double bond. (e) There are two carbons in \(\mathrm{CH}_{3} \mathrm{COOH}\): the first forms four single bonds, while the second forms two singles and one double bond.

The type of hybridization can be determined on the basis of the types and numbers of bonds formed by the carbon atoms. For four single bonds (a sigma bond and three sigma bonds due to hybridization), it's sp3 hybridization. For a double bond (a sigma bond and a pi bond) and two single bonds, it's sp2 hybridization. For a triple bond (a sigma bond and two pi bonds) and a single bond, it's sp hybridization.

Applying step 2 to each molecule leads to: (a) For \(\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}_{3}\), both carbons have sp3 hybridization. (b) In \(\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}=\mathrm{CH}_{2}\), the first carbon has sp3 and the second has sp2 hybridization. (c) In \(\mathrm{CH}_{3}-\mathrm{C} \equiv \mathrm{C}-\mathrm{CH}_{2} \mathrm{OH}\), the first and third carbons have sp3, whereas the middle carbon has sp hybridization. (d) For \(\mathrm{CH}_{3} \mathrm{CH}=\mathrm{O}\), the first carbon has sp3 and the second has sp2 hybridization. (e) For \(\mathrm{CH}_{3} \mathrm{COOH}\), the first carbon has sp3 and the second has sp2 hybridization.