If PQ, T2U2, RS2 and VW4 have the same Ksp, which has the lowest solubility in moles of solute per litre? If the solubility products of the following ionic compounds all have the same numerical value, and all solubilities have values between zero and one, which one would have the lowest solubility in moles of solute per litre?

$\ce{PQ}$ 

$\ce{RS2}$

$\ce{T2U2}$

$\ce{VW_4}$   



Their chemical equations and $\ce{Ksp}$ values would be:


$\ce{PQ &lt;-&gt; P +Q}$  and $\ce{Ksp=[P][Q]=[P][P]=[P]^2}$
$\ce{RS2  &lt;-&gt; R +2S}$ and  $\ce{Ksp=[R][2S]^2=[R][2R]^2=4[R]^3}$
$\ce{T2U2  &lt;-&gt; 2T +2U}$ and $\ce{Ksp=[2T]^2[2U]^2=[2T]^2[2T]^2=16[T]^4}$ (although the textbook says $\ce{[T]^4}$ which I don't understand)
$\ce{VW4  &lt;-&gt; V + 4W}$ and $\ce{Ksp=[V]^2[4W]^4=[W]^2[4W]^4=256[W]^5}$ (although the textbook says $\ce{256[V]^5}$ but I am pretty sure its equivalent whether it is written in terms of V or W)


FYI my thought process here is that:


Based on the chemical equation RS2⟷R+2S 
The concentration of one product (S) is twice that of (R). 
So if there is X moles of RS2, and there is X moles of R and 2X moles of S
Hence in the Ksp=[A]^n⋅[B]^m , A=X and B=2X 




Now the textbook says 


  As $\ce{[P][P]=4[R]^3=[T]^4=256[V]^5}$
  [P] must be the lowest because $\ce{[P]^2}$  has the same value as  $\ce{4[R]^3 and [T]^4 and 256[V]^5}$   


I can't get my head around this. I would have thought this proves
$\ce{[P]}$ has the highest value not the lowest. Why is [P] the LOWEST?

E.g. noting that all values are between $0$ and $1$ (as defined in the question)

$\ce{[P]^2=256[V]^5}$  (the other two options are in the middle so don't matter)

If both sides equaled e.g. $0.25$ (which is between $0$ and $1$)

$\ce{[P]^2=0.25 and 256[V]^5=0.25}$

${[P]=0.5}$ and ${[V]=(0.25/256)}^{1/5} = 0.25$

Showing [P]>[V], so [P] (moles/L) is the greatest solubility, not the lowest.

Is anyone able to spot my error?



Question and answer as shown in book below (I have typed up the relevant parts above but sometimes I interpret parts of the textbook question incorrectly (making my stack exchange question invalid or misleading):
Easier just to migrate my comment into an answer. Your assumption
$$\ce{[PQ]} &gt; \ce{[VW4]}$$ is not valid based on
$$\ce{[PQ]^{2}} = 256\ce{[VW4]}^{5}$$

For $y = a^{x}$, if $a&gt;1$, larger $x$ leads to larger $y$. But for $a&lt;1$, larger $x$ leads to smaller $y$, and the question indicates that solubilities are between zero and one, i.e., small.

If you plug in $K_{\mathrm{sp}}= 10^{-10}$, you get $\ce{[PQ]}=10^{-5}$ and $\ce{[VW4]} = 3.3\cdot 10^{-3}$.

As I said in my comment, I believe that this is a very poorly designed question: the constraints are too broad and there is a typo in the key. Someone was very sloppy...