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Why does using the Lindlar's catalyst form cis-alkenes whereas $\ce{Na/NH3}$ forms trans-alkenes?

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In Lindlar's catalyst, it is a matter of the physical way in which the molecules react. Firstly, $\ce{H2}$ dissociates into atomic hydrogen and binds to the catalyst, and the alkyne coordinates to the metal surface. Since the surface of the catalyst can be thought of as a vast plane, all carbon to metal bonds must be facing down onto the plane. The hydrogen atoms then get inserted in place of these carbon to metal bonds, and the result is syn-addition.

In the $\ce{Na/NH3}$ reduction, when the electrons are first added to the triple bond and the $\mathrm{sp}$ hybridization is broken, the groups bonds on either side the triple bond repel each other into the lowest energy conformation, the trans position. This is why the $\ce{H}$ atoms being donated by the $\ce{NH3}$ add through anti-addition.

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