1. Key idea: secondary hydrogen

A secondary hydrogen is a hydrogen attached to a secondary carbon, i.e. a carbon bonded to two other carbons.

So we must find the alkane which has exactly two hydrogens on secondary carbons.

2. Check each option

Option A: $2,2,4,5$-Tetramethylheptane

Write the main chain of heptane:

$C_1-C_2-C_3-C_4-C_5-C_6-C_7$

Substituents:

• at $C_2$: two methyl groups
• at $C_4$: one methyl group
• at $C_5$: one methyl group

Now determine the type of each main-chain carbon:

• $C_1, C_7$: primary
• $C_2$: bonded to $C_1, C_3$, and two methyl groups $\Rightarrow$ quaternary
• $C_3$: bonded to $C_2$ and $C_4$ $\Rightarrow$ secondary, so it has 2 secondary H
• $C_4$: bonded to $C_3, C_5$, and one methyl group $\Rightarrow$ tertiary, so 1 tertiary H
• $C_5$: bonded to $C_4, C_6$, and one methyl group $\Rightarrow$ tertiary, so 1 tertiary H
• $C_6$: bonded to $C_5$ and $C_7$ $\Rightarrow$ secondary, so it has 2 secondary H

Total secondary hydrogens $=2+2=4$.

So, Option A is not correct.

Option B: $2,2,4,4$-Tetramethylhexane

Main chain:

$C_1-C_2-C_3-C_4-C_5-C_6$

Substituents:

• at $C_2$: two methyl groups
• at $C_4$: two methyl groups

Carbon types:

• $C_2$: quaternary
• $C_4$: quaternary
• $C_3$: bonded to $C_2$ and $C_4$ only $\Rightarrow$ secondary, so it has 2 secondary H
• $C_1, C_5, C_6$ and all methyl substituent carbons are primary except $C_5$ is bonded to $C_4$ and $C_6$, so actually $C_5$ is secondary

Let us correct that carefully:

• $C_1$: primary
• $C_2$: quaternary
• $C_3$: secondary $\Rightarrow 2$ secondary H
• $C_4$: quaternary
• $C_5$: bonded to $C_4$ and $C_6$ $\Rightarrow$ secondary $\Rightarrow 2$ secondary H
• $C_6$: primary

Total secondary hydrogens $=2+2=4$.

So, Option B is not correct.

Option C: $4$-Ethyl-$3,4$-dimethyloctane

Main chain:

$C_1-C_2-C_3-C_4-C_5-C_6-C_7-C_8$

Substituents:

• at $C_3$: one methyl
• at $C_4$: one methyl and one ethyl

Carbon types:

• $C_3$: bonded to $C_2, C_4$, and methyl $\Rightarrow$ tertiary
• $C_4$: bonded to $C_3, C_5$, methyl, ethyl $\Rightarrow$ quaternary
• $C_2$: secondary $\Rightarrow 2$ secondary H
• $C_5$: secondary $\Rightarrow 2$ secondary H
• $C_6$: secondary $\Rightarrow 2$ secondary H
• $C_7$: secondary $\Rightarrow 2$ secondary H
• in the ethyl substituent, the carbon attached to $C_4$ is also secondary? No — it is bonded to $C_4$ and terminal $CH_3$, so yes, that carbon is secondary $\Rightarrow 2$ secondary H

Thus total secondary hydrogens are much more than 2.

So, Option C is not correct.

Option D: $2,2,3,3$-Tetramethylpentane

Main chain:

$C_1-C_2-C_3-C_4-C_5$

Substituents:

• at $C_2$: two methyl groups
• at $C_3$: two methyl groups

Carbon types:

• $C_2$: bonded to $C_1, C_3$, and two methyl groups $\Rightarrow$ quaternary
• $C_3$: bonded to $C_2, C_4$, and two methyl groups $\Rightarrow$ quaternary
• $C_4$: bonded to $C_3$ and $C_5$ only $\Rightarrow$ secondary, so it has 2 secondary H
• $C_1, C_5$, and all methyl substituent carbons are primary

Therefore total secondary hydrogens $=2$.

So, Option D is correct.

3. Final answer

The alkane having exactly two secondary hydrogens is:

$\boxed{\text{D: }2,2,3,3\text{-Tetramethylpentane}}$

4. Comparison with stored correct answer

Stored correct answer: D

My derived answer: D

Hence, they agree.