The toal number of isomers including stereoisomers for 1,3-
dimethyl cyclohexane is:

Options

A. (a) 2, cis and trans both optically inactive
B. (b) 3, one cis and two optically active trans froms
C. (c) 3, one trans and two optically active cis froms
D. (d) 4, two cis (optically active) and two trans (optically active)

Answer

To determine the total number of isomers, including stereoisomers, for 1,3-dimethylcyclohexane, we can follow these steps:

### Step 1: Understand the structure of 1,3-dimethylcyclohexane
1,3-dimethylcyclohexane consists of a cyclohexane ring with two methyl (CH3) groups attached to the 1st and 3rd carbon atoms.

### Step 2: Draw the cyclohexane ring
Draw a hexagon to represent the cyclohexane ring. Label the carbon atoms from 1 to 6.

### Step 3: Place the methyl groups
Attach the methyl groups to the 1st and 3rd carbon atoms. This gives us two possible arrangements based on the position of the methyl groups:
- **Cis** (both methyl groups on the same side of the ring)
- **Trans** (methyl groups on opposite sides of the ring)

### Step 4: Identify the isomers
1. **Cis isomer**: Both methyl groups are on the same side of the cyclohexane ring. This is optically inactive due to the presence of a plane of symmetry.
2. **Trans isomer**: Methyl groups are on opposite sides of the cyclohexane ring. This is optically active because it does not have a plane of symmetry.

### Step 5: Consider stereoisomers
For the trans isomer, there are two enantiomers (mirror images) because the two methyl groups can be arranged in two different ways (R,S) and (S,R).

### Step 6: Count the total isomers
- 1 **Cis isomer** (optically inactive)
- 2 **Trans isomers** (optically active enantiomers)

Thus, the total number of isomers including stereoisomers for 1,3-dimethylcyclohexane is:
- 1 (Cis) + 2 (Trans) = **3 isomers**

### Final Answer:
The total number of isomers including stereoisomers for 1,3-dimethylcyclohexane is **3**.

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