Identify Identical Newman Projection Conformers

To determine which option represents a pair of identical molecules, we determine the IUPAC structural formulas for each of the given Newman projections ($\mathbf{P}$, $\mathbf{Q}$, $\mathbf{R}$, and $\mathbf{S}$).

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1. Analysis of Molecule $\mathbf{P}$

• Front Carbon ($\text{C}_{\text{front}}$): Attached to two methyl groups ($-\text{CH}_3$) and one ethyl group ($-\text{C}_2\text{H}_5$). $\text{C}_{\text{front}} = -\text{C}(\text{CH}_3)_2(\text{C}_2\text{H}_5)$
• Back Carbon ($\text{C}_{\text{back}}$): Attached to two methyl groups ($-\text{CH}_3$) and one hydroxyl group ($-\text{OH}$). $\text{C}_{\text{back}} = -\text{C}(\text{OH})(\text{CH}_3)_2$
• IUPAC Name: 2,3,3-trimethylpentan-2-ol.

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2. Analysis of Molecule $\mathbf{Q}$

• Front Carbon ($\text{C}_{\text{front}}$): Attached to two hydrogen atoms ($-\text{H}$) and one methyl group ($-\text{CH}_3$), which together form an ethyl group: $\text{C}_{\text{front}} = -\text{CH}_2\text{CH}_3 = -\text{C}_2\text{H}_5$
• Back Carbon ($\text{C}_{\text{back}}$): Attached to:
  1. One hydrogen atom ($-\text{H}$)
  2. One ethyl group ($-\text{C}_2\text{H}_5$)
  3. One $2$-hydroxypropan-$2$-yl group ($-\text{C}(\text{OH})(\text{CH}_3)_2$)
  4. The front carbon group ($-\text{C}_2\text{H}_5$)

Thus, the central carbon ($\text{C}_{\text{back}}$) is bonded to two identical ethyl groups ($-\text{C}_2\text{H}_5$), one hydrogen atom ($-\text{H}$), and one $-\text{C}(\text{OH})(\text{CH}_3)_2$ group.

• IUPAC Name: 3-ethyl-2-methylpentan-2-ol.

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3. Analysis of Molecule $\mathbf{R}$

• Front Carbon ($\text{C}_{\text{front}}$): Attached to two methyl groups ($-\text{CH}_3$) and one hydroxyl group ($-\text{OH}$): $\text{C}_{\text{front}} = -\text{C}(\text{OH})(\text{CH}_3)_2$
• Back Carbon ($\text{C}_{\text{back}}$): Attached to:
  1. One hydrogen atom ($-\text{H}$)
  2. Two ethyl groups ($-\text{C}_2\text{H}_5$)
  3. The front carbon group ($-\text{C}(\text{OH})(\text{CH}_3)_2$)

Thus, the central carbon ($\text{C}_{\text{back}}$) is bonded to two identical ethyl groups ($-\text{C}_2\text{H}_5$), one hydrogen atom ($-\text{H}$), and one $-\text{C}(\text{OH})(\text{CH}_3)_2$ group.

• IUPAC Name: 3-ethyl-2-methylpentan-2-ol.

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4. Analysis of Molecule $\mathbf{S}$

• Front Carbon ($\text{C}_{\text{front}}$): Attached to two hydrogen atoms ($-\text{H}$) and one methyl group ($-\text{CH}_3$), forming an ethyl group ($-\text{C}_2\text{H}_5$).
• Back Carbon ($\text{C}_{\text{back}}$): Attached to one ethyl group ($-\text{C}_2\text{H}_5$), one isopropyl group ($-\text{CH}(\text{CH}_3)_2$), and one hydroxyl group ($-\text{OH}$).
• IUPAC Name: 3-ethyl-2,4-dimethylpentan-3-ol.

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Conclusion

Both $\mathbf{Q}$ and $\mathbf{R}$ represent 3-ethyl-2-methylpentan-2-ol: $\text{HO}-\text{C}(\text{CH}_3)_2-\text{CH}(\text{CH}_2\text{CH}_3)_2$

Since the molecule has no stereocenters (C2 is attached to two identical methyl groups and C3 is attached to two identical ethyl groups), it is achiral. Therefore, $\mathbf{Q}$ and $\mathbf{R}$ represent the identical molecule.

Correct Option: (C) $\mathbf{Q}$ and $\mathbf{R}$