The mechanism involves a cyclic manganate ester, so the addition is stereospecific.
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(a) Propene
"CH"_3"CH=CH"_2 stackrelcolor (blue)("alk. MnO"_4^-)(→) "CH"_3stackrelcolor (blue)(✳)("C")"HOHCH"_2"OH"
Note that the starred carbon atom is chiral, so the product is a racemic mixture of (R)- and (S)-propane-1,2-diol.
(b) cis-But-2-ene
"CH"_3"CH=CHCH"_3 stackrelcolor (blue)("alk. MnO"_4^-)(→) "CH"_3stackrelcolor (red)(✳)("C")"HOH"stackrelcolor (red)(✳)("C")"HOHCH"_3
Here we have two chiral carbons.
If you replace the wedged "H" atom in the diagram above with a "CH"_3 group, you have the mechanism for the reaction with cis-but-2-ene.
Note that the two "CH"_3 groups are both "in front", so the molecule has an internal plane of symmetry.
The product is meso-butane-2,3-diol.
(c) trans-But-2-ene
"CH"_3"CH=CHCH"_3 stackrelcolor (blue)("alk. MnO"_4^-)(→) "CH"_3stackrelcolor (red)(✳)("C")"HOH"stackrelcolor (red)(✳)("C")"HOHCH"_3
(the same equation)
But here the second "CH"_3 group replaces the "H" atom on the dashed bond.
Both "C" atoms are chiral, but there is no longer a mirror plane.
The product is a racemic mixture of (2R,3R)- and (2S,3S)-butane-1,2-diol.
