The K_sp values of Ag₂CrO₄ and AgBr are 32x and 4y respectively. The ratio of solubilities S₁/S₂ is

[Here, S1 is solubility of Ag2CrO4 and S2 is solubility of AgBr]

The correct option is:
$\frac{x^{\frac{1}{3}}}{y^{\frac{1}{2}}}$

To find the ratio of the solubilities, let us calculate the solubility of each salt in terms of its solubility product ($K_{\mathrm{sp}}$).

1. Solubility of Ag₂CrO₄ (S₁):
Ag₂CrO₄ dissociates in aqueous solution as follows:
${\mathrm{Ag}}_2{\mathrm{CrO}}_4\left(s\right)\rightleftharpoons 2{\mathrm{Ag}}^{+}\left(\mathrm{aq}\right)+{\mathrm{CrO}}_4^{2-}\left(\mathrm{aq}\right)$

If the solubility of Ag₂CrO₄ is S₁ mol/L, then at equilibrium:
$\left[{\mathrm{Ag}}^{+}\right]=2S_1$
$\left[{\mathrm{CrO}}_4^{2-}\right]=S_1$

The solubility product expression is:
$K_{\mathrm{sp}}\left({\mathrm{Ag}}_2{\mathrm{CrO}}_4\right)={\left[{\mathrm{Ag}}^{+}\right]}^2\left[{\mathrm{CrO}}_4^{2-}\right]$
Substituting the equilibrium concentrations:
$32x={\left(2S_1\right)}^2\left(S_1\right)$
$32x=4S_1^3$
Dividing by 4:
$8x=S_1^3$
Taking the cube root on both sides:
$S_1={\left(8x\right)}^{\frac{1}{3}}=2x^{\frac{1}{3}}$

2. Solubility of AgBr (S₂):
AgBr dissociates in aqueous solution as follows:
$\mathrm{AgBr}\left(s\right)\rightleftharpoons {\mathrm{Ag}}^{+}\left(\mathrm{aq}\right)+{\mathrm{Br}}^{-}\left(\mathrm{aq}\right)$

If the solubility of AgBr is S₂ mol/L, then at equilibrium:
$\left[{\mathrm{Ag}}^{+}\right]=S_2$
$\left[{\mathrm{Br}}^{-}\right]=S_2$

The solubility product expression is:
$K_{\mathrm{sp}}\left(\mathrm{AgBr}\right)=\left[{\mathrm{Ag}}^{+}\right]\left[{\mathrm{Br}}^{-}\right]$
Substituting the equilibrium concentrations:
$4y=\left(S_2\right)\left(S_2\right)$
$4y=S_2^2$
Taking the square root on both sides:
$S_2={\left(4y\right)}^{\frac{1}{2}}=2y^{\frac{1}{2}}$

3. Ratio of solubilities (S₁/S₂):
Now, let us calculate the ratio:
$\frac{S_1}{S_2}=\frac{2x^{\frac{1}{3}}}{2y^{\frac{1}{2}}}$
Cancelling out the common factor of 2 from the numerator and denominator, we get:
$\frac{S_1}{S_2}=\frac{x^{\frac{1}{3}}}{y^{\frac{1}{2}}}$