The given compound is \[ \text{CH}_3\text{-CH}_2\text{-CH(CH}_3\text{)-CH}_2\text{-CH(CH}_3\text{)-CH}_2\text{-CH}_3 \]. To determine the correct IUPAC name, we need to follow these steps:
- Identify the longest carbon chain. The longest continuous chain of carbon atoms in the given compound has 7 carbon atoms, making it a heptane.
- Number the carbon atoms in the principal chain. Start numbering from the end that gives the substituents the lowest possible numbers. In this case, numbering from the left gives the methyl groups the positions 3 and 5.
- Determine and name the substituents:
- At carbon 3, there is a methyl group (\( \text{CH}_3 \)).
- At carbon 5, there is another methyl group (\( \text{CH}_3 \)).
- Combine the substituents and parent chain. The compound has two methyl groups, hence it is named 3,5-dimethylheptane.
Thus, the correct IUPAC name for the compound is 3,5-dimethylheptane.
Let's justify the rejection of incorrect options:
- 3-methyl-5-ethylhexane: This name suggests an incorrect structure as our given structure does not have an ethyl group.
- 3-ethyl-5-methylheptane: This is incorrect because there is no ethyl group in the given compound.
- 2,4-diethylhexane: This is also incorrect due to the absence of ethyl groups in the structure.
Therefore, the only correct answer is 3,5-dimethylheptane.