aqueous neutral
aqueous acidic
both aqueous acidic and neutral
both aqueous acidic and faintly alkaline
The reaction of permanganate \( (\text{MnO}_4^-) \) with thiosulphate \( (\text{S}_2\text{O}_3^{2-}) \) depends on the pH of the medium. Permanganate can be reduced to different manganese species depending on the conditions.
In a neutral or weakly alkaline medium, permanganate is reduced to manganese(IV) oxide \( (\text{MnO}_2) \):
\( \text{MnO}_4^-\text{(aq)} + 2\text{H}_2\text{O(l)} + 3e^- \rightarrow \text{MnO}_2\text{(s)} + 4\text{OH}^-\text{(aq)} \)
In this reaction, the oxidation state of manganese changes from +7 in \( \text{MnO}_4^- \) to +4 in \( \text{MnO}_2 \), a change of 3 units.
In an acidic medium, permanganate is reduced to manganese(II) ions \( (\text{Mn}^{2+}) \):
\( \text{MnO}_4^-\text{(aq)} + 8\text{H}^+\text{(aq)} + 5e^- \rightarrow \text{Mn}^{2+}\text{(aq)} + 4\text{H}_2\text{O(l)} \)
Here, the oxidation state of manganese changes from +7 to +2, a change of 5 units.
Since the question specifies a change in oxidation state of 3, the reaction must be occurring in a neutral or weakly alkaline medium.
A bob of mass \(m\) is suspended at a point \(O\) by a light string of length \(l\) and left to perform vertical motion (circular) as shown in the figure. Initially, by applying horizontal velocity \(v_0\) at the point ‘A’, the string becomes slack when the bob reaches at the point ‘D’. The ratio of the kinetic energy of the bob at the points B and C is: 