Provide the structures for:
$1.$ $MnO_4^{-}$
$2.$ $MnO_4^{2-}$

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(N/A) $1.$ The permanganate ion $(MnO_4^-)$ has a tetrahedral geometry where the manganese atom is at the center,bonded to four oxygen atoms. It involves $d\pi-p\pi$ bonding.
$2.$ The manganate ion $(MnO_4^{2-})$ also has a tetrahedral geometry with the manganese atom at the center,bonded to four oxygen atoms. It is paramagnetic due to the presence of one unpaired electron.

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Describe the preparation of potassium permanganate. State its physical properties,structure,and uses. Discuss its chemical properties and provide chemical reactions demonstrating the oxidizing character of $KMnO_{4}$.

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$Cr_2O_7^{2-} / H^{\oplus}$ $\xrightarrow[200^{\circ}C]{H_2O_2} 'X'$ $\xrightarrow{\text{On Standing}} 'Y'_{(aq. sol)}$
Select the correct option for $'X'$ and $'Y'$

Fill in the blanks:
$(1)$ ......... acid is added to perform titrations with $KMnO_4$ and $K_2Cr_2O_7$.
$(2)$ Crystals of potassium dichromate are of ......... color,while crystals of potassium chromate are of ......... color.
$(3)$ The color of $MnO_4^{2-}$ is ......... and the color of $MnO_4^-$ is ......... .

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