This question asks us to identify a specific alkane from a list that contains primary ($1^\circ$), tertiary ($3^\circ$), and quaternary ($4^\circ$) carbon atoms, while excluding any secondary ($2^\circ$) carbon atoms. Let's break down how to determine the type of each carbon atom in an alkane and then analyze each given option.

Understanding Carbon Atom Types in Alkanes

In organic chemistry, carbon atoms in alkane structures are classified based on the number of other carbon atoms they are directly bonded to:

• Primary ($1^\circ$) Carbon: A carbon atom bonded to exactly one other carbon atom. These are typically found at the ends of alkane chains (methyl groups, $CH_3$).
• Secondary ($2^\circ$) Carbon: A carbon atom bonded to exactly two other carbon atoms. These are usually found within the carbon chain ($CH_2$ groups).
• Tertiary ($3^\circ$) Carbon: A carbon atom bonded to exactly three other carbon atoms. These are branching points in the carbon skeleton ($CH$ groups).
• Quaternary ($4^\circ$) Carbon: A carbon atom bonded to exactly four other carbon atoms. These are also branching points, but without any attached hydrogen atoms ($C$ atoms).

We need to find the alkane structure that has all three types ($1^\circ$, $3^\circ$, $4^\circ$) and none of the secondary ($2^\circ$) type.

Analyzing Alkane Structures and Carbon Types

Let's examine the structure and carbon types for each option.

Analysis of 2,3-Dimethylpentane

The structure of 2,3-Dimethylpentane is:

$CH_3 - CH(CH_3) - CH(CH_3) - CH_2 - CH_3$

Let's identify the types of carbons:

• The two terminal $CH_3$ groups (at positions 1 and 5) are Primary ($1^\circ$).
• The two methyl groups attached as branches (at positions 2 and 3) are Primary ($1^\circ$).
• The carbon at position 2 is bonded to three other carbons (C1, C3, and the methyl branch), making it Tertiary ($3^\circ$).
• The carbon at position 3 is bonded to three other carbons (C2, C4, and the methyl branch), making it Tertiary ($3^\circ$).
• The carbon at position 4 ($CH_2$) is bonded to two other carbons (C3 and C5), making it Secondary ($2^\circ$).

Carbon types present: $1^\circ$, $2^\circ$, $3^\circ$. This structure contains a secondary ($2^\circ$) carbon, so it does not meet the criteria.

Analysis of 2,2-Dimethylpentane

The structure of 2,2-Dimethylpentane is:

$CH_3 - C(CH_3)_2 - CH_2 - CH_2 - CH_3$

Let's identify the types of carbons:

• The terminal $CH_3$ group (at position 1) is Primary ($1^\circ$).
• The terminal $CH_3$ group (at position 5) is Primary ($1^\circ$).
• The two methyl groups attached at position 2 are Primary ($1^\circ$).
• The carbon at position 2 is bonded to four other carbons (C1, C3, and the two methyl branches), making it Quaternary ($4^\circ$).
• The carbon at position 3 ($CH_2$) is bonded to two other carbons (C2 and C4), making it Secondary ($2^\circ$).
• The carbon at position 4 ($CH_2$) is bonded to two other carbons (C3 and C5), making it Secondary ($2^\circ$).

Carbon types present: $1^\circ$, $2^\circ$, $4^\circ$. This structure contains secondary ($2^\circ$) carbons, so it does not meet the criteria.

Analysis of 3-Ethyl-2-methylpentane

The structure of 3-Ethyl-2-methylpentane is:

$CH_3 - CH(CH_3) - CH(CH_2CH_3) - CH_2 - CH_3$

Let's identify the types of carbons:

• The terminal $CH_3$ group (at position 1) is Primary ($1^\circ$).
• The terminal $CH_3$ group of the ethyl substituent is Primary ($1^\circ$).
• The $CH_3$ group attached as a branch at position 2 is Primary ($1^\circ$).
• The terminal $CH_3$ group (at position 5) is Primary ($1^\circ$).
• The carbon at position 2 is bonded to three other carbons (C1, C3, and the methyl branch), making it Tertiary ($3^\circ$).
• The carbon at position 3 is bonded to three other carbons (C2, C4, and the $CH_2$ of the ethyl group), making it Tertiary ($3^\circ$).
• The $CH_2$ group of the ethyl substituent is bonded to two other carbons (C3 and its attached $CH_3$), making it Secondary ($2^\circ$).
• The carbon at position 4 ($CH_2$) is bonded to two other carbons (C3 and C5), making it Secondary ($2^\circ$).

Carbon types present: $1^\circ$, $2^\circ$, $3^\circ$. This structure contains secondary ($2^\circ$) carbons, so it does not meet the criteria.

Analysis of 2,2,3-Trimethylbutane

The structure of 2,2,3-Trimethylbutane is:

$CH_3 - C(CH_3)_2 - CH(CH_3) - CH_3$

Let's identify the types of carbons:

• The terminal $CH_3$ group (at position 1) is Primary ($1^\circ$).
• The terminal $CH_3$ group (at position 4) is Primary ($1^\circ$).
• The two methyl groups attached at position 2 are Primary ($1^\circ$).
• The methyl group attached at position 3 is Primary ($1^\circ$).
• The carbon at position 2 is bonded to four other carbons (C1, C3, and the two methyl branches), making it Quaternary ($4^\circ$).
• The carbon at position 3 is bonded to three other carbons (C2, C4, and its methyl branch), making it Tertiary ($3^\circ$).

Carbon types present: $1^\circ$, $3^\circ$, $4^\circ$. This structure contains primary ($1^\circ$), tertiary ($3^\circ$), and quaternary ($4^\circ$) carbons, and crucially, it contains no secondary ($2^\circ$) carbons.

Final Conclusion on Alkane Carbon Types

By analyzing the structure and classifying the carbon atoms in each option, we found that 2,2,3-Trimethylbutane is the only alkane that satisfies the condition of containing primary ($1^\circ$), tertiary ($3^\circ$), and quaternary ($4^\circ$) carbons, while having no secondary ($2^\circ$) carbons.

Here is a summary table: