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If the cyclobutane ring was to adopt a planar conformation, then in
the cis-isomer, the two substituents would be on the same face of
the ring and there would be a steric repulsion between them as shown in
the 3D structure below. In the trans-isomer however, the two substituents
will be on opposite faces of the four-membered ring and there will be no
interaction between them as shown below. Thus, in this case the trans-isomer
would be thermodynamically more stable.
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The reaction occurs because the hydrogen atom next to the carbonyl of the ester (in both the starting material and product) is acidic and can be removed by a base such as sodium methoxide. This generates an enolate which can be reprotonated from either face to either regenerate the starting material or give the product. The product is the thermodynamically more stable isomer, so eventually the trans-isomer (starting material) is converted into the cis-isomer (product).
