The ordinary rate is
[tex]R = k [A] [ B ]^2[/tex]
or, equivalently,
[tex]R = k \frac{n_A}{V} \left( \frac{n_B}{V} \right)^2 \propto \frac{1}{V^3}[/tex]
Therefore for reactions in different volumes
[tex]\frac{R´}{R} = \left( \frac{V}{V´} \right)^3 [/tex]
if [itex]V' = 2V[/itex] we obtain the 1/8 mentioned by curiouscat