The structure of the compound with molecular formula `C_(8)H_(14)` that has 12 secondary and 2 tertiary H atoms is:

Options

A. (a) Bicyclo `[2.2.2]` octane
B. (b) 1,2-Dimethyl cyclohexane
C. (c) Bicyclo `[3.2.1]` octane
D. (d) Tricylo `[2.2.2.0]` octane

Explanation

To determine the structure of the compound with the molecular formula \( C_8H_{14} \) that has 12 secondary and 2 tertiary hydrogen atoms, we will analyze the options provided.

### Step 1: Understand the Definitions
- **Tertiary Hydrogen**: Hydrogen atoms attached to a tertiary carbon (a carbon bonded to three other carbons).
- **Secondary Hydrogen**: Hydrogen atoms attached to a secondary carbon (a carbon bonded to two other carbons).

### Step 2: Analyze Each Option

#### Option A: Bicyclo[2.2.2]octane
1. **Structure**: Draw the structure of bicyclo[2.2.2]octane.
2. **Identify Tertiary Carbons**: There are 2 tertiary carbons.
3. **Count Tertiary Hydrogens**: Each tertiary carbon has 1 hydrogen, giving us 2 tertiary hydrogens.
4. **Identify Secondary Carbons**: Count the secondary carbons.
5. **Count Secondary Hydrogens**: Each secondary carbon has 2 hydrogens. There are 6 secondary carbons, resulting in \( 6 \times 2 = 12 \) secondary hydrogens.

#### Conclusion for Option A:
- **Tertiary Hydrogens**: 2
- **Secondary Hydrogens**: 12
- **This option is valid.**

#### Option B: 1,2-Dimethylcyclohexane
1. **Structure**: Draw the structure of 1,2-dimethylcyclohexane.
2. **Identify Tertiary Carbons**: There are 2 tertiary carbons.
3. **Count Tertiary Hydrogens**: Each tertiary carbon has 1 hydrogen, giving us 2 tertiary hydrogens.
4. **Identify Secondary Carbons**: Count the secondary carbons.
5. **Count Secondary Hydrogens**: There are 4 secondary carbons, resulting in \( 4 \times 2 = 8 \) secondary hydrogens.

#### Conclusion for Option B:
- **Tertiary Hydrogens**: 2
- **Secondary Hydrogens**: 8
- **This option is invalid.**

#### Option C: Bicyclo[3.2.1]octane
1. **Structure**: Draw the structure of bicyclo[3.2.1]octane.
2. **Identify Tertiary Carbons**: There are 2 tertiary carbons.
3. **Count Tertiary Hydrogens**: Each tertiary carbon has 1 hydrogen, giving us 2 tertiary hydrogens.
4. **Identify Secondary Carbons**: Count the secondary carbons.
5. **Count Secondary Hydrogens**: There are 6 secondary carbons, resulting in \( 6 \times 2 = 12 \) secondary hydrogens.

#### Conclusion for Option C:
- **Tertiary Hydrogens**: 2
- **Secondary Hydrogens**: 12
- **This option is valid.**

#### Option D: Tricyclo[2.2.2.0]octane
1. **Structure**: Draw the structure of tricyclo[2.2.2.0]octane.
2. **Identify Tertiary Carbons**: There are 4 tertiary carbons.
3. **Count Tertiary Hydrogens**: Each tertiary carbon has 1 hydrogen, giving us 4 tertiary hydrogens.
4. **Identify Secondary Carbons**: Count the secondary carbons.
5. **Count Secondary Hydrogens**: This structure does not yield 12 secondary hydrogens.

#### Conclusion for Option D:
- **Tertiary Hydrogens**: 4
- **Secondary Hydrogens**: Not enough to meet the requirement.
- **This option is invalid.**

### Final Answer:
The structures that satisfy the conditions of having 12 secondary and 2 tertiary hydrogen atoms are:
- **Option A: Bicyclo[2.2.2]octane**
- **Option C: Bicyclo[3.2.1]octane**