To determine which dimethylcyclobutane is optically active, we need to analyze the four given structures: trans 1,2-dimethylcyclobutane, cis 1,2-dimethylcyclobutane, trans 1,3-dimethylcyclobutane, and cis 1,3-dimethylcyclobutane.
### Step-by-Step Solution:
1. **Draw the Structures**:
- **Trans 1,2-Dimethylcyclobutane**: In this structure, the two methyl groups (CH₃) are on opposite sides of the cyclobutane ring.
- **Cis 1,2-Dimethylcyclobutane**: Here, the two methyl groups are on the same side of the cyclobutane ring.
- **Trans 1,3-Dimethylcyclobutane**: In this case, the methyl groups are on opposite sides but at different positions (1 and 3) on the ring.
- **Cis 1,3-Dimethylcyclobutane**: The methyl groups are on the same side but at different positions (1 and 3) on the ring.
2. **Analyze Optical Activity**:
- **Trans 1,2-Dimethylcyclobutane**: This structure does not have a plane of symmetry, which means it can exist in two non-superimposable mirror images (enantiomers). Therefore, it is optically active.
- **Cis 1,2-Dimethylcyclobutane**: This structure has a plane of symmetry, which means it is superimposable on its mirror image and is therefore optically inactive.
- **Trans 1,3-Dimethylcyclobutane**: This structure also has a plane of symmetry, making it optically inactive.
- **Cis 1,3-Dimethylcyclobutane**: Similar to trans 1,3, this structure has a plane of symmetry and is also optically inactive.
3. **Conclusion**:
- The only optically active compound among the options is **Trans 1,2-Dimethylcyclobutane**.
### Final Answer:
**Trans 1,2-Dimethylcyclobutane is optically active.**
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