
The reactivity of halides towards the S$_N$1 mechanism depends on the stability of the carbocation intermediate formed during the reaction:
Compound A forms a benzyl carbocation, which is highly stable due to resonance.
Compound B forms a primary carbocation, which is less stable but reacts due to iodine's leaving group strength.
Compound C forms a tertiary carbocation, which is very stable and reactive.
Compound D forms a primary carbocation, less stable but reacts due to bromine's moderate leaving group strength.
Thus, the halogens are ordered as per the leaving group stability in S$_N$1.
Calculate the potential for half-cell containing 0.01 M K\(_2\)Cr\(_2\)O\(_7\)(aq), 0.01 M Cr\(^{3+}\)(aq), and 1.0 x 10\(^{-4}\) M H\(^+\)(aq).

Let \( \alpha = \dfrac{-1 + i\sqrt{3}}{2} \) and \( \beta = \dfrac{-1 - i\sqrt{3}}{2} \), where \( i = \sqrt{-1} \). If
\[ (7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}, \] then the value of \( m \) is ___________.