Question

The two solids $CuBr\left(s\right)$) and $AgBr\left(s\right)$ are only very slightly soluble in water: $K_{sp}\left(CuBr\right)=4.2\times {10}^{-8}$ and$K_{sp}\left(AgBr\right)=7.7\times {10}^{-12}$ . Some$CuBr\left(s\right)$ and $AgBr\left(s\right)$are both mixed into a quantity of water that is then stirred until it is saturated with respect to both solutes. Next, a small amount of $KBr$is added and dissolves completely. Compute the ratio of$\left[C{u}^{+}\right]$ to $\left[A{g}^{+}\right]$after the system re-establishes equilibrium.

Short answer

The ratio of $\left[A{g}^{-}\right]$and$\left[C{u}^{+}\right]$ is$2.3\times {10}^2$

Step-by-step solution

Reaction.

It’s given that:

$\text{AgBr(s)}\rightleftharpoons {\text{Ag}}^{\text{+}}{\text{(aq) + Br}}^{\text{-}}\text{(aq)}$

${\text{K}}_{\text{sp}}{\text{= [Ag}}^{\text{+}}{\text{][Br}}^{\text{-}}\text{]}$

$K_{sp}\left(AgBr\right)=7.7\times {10}^{-13}$

As we know that this two are solid so,

$$\begin{gathered} \left[A{g}^{-}\right]=s \\ [B{r}^{}-]=s \end{gathered}$$

Let’s find the value of :

$$\begin{gathered} \left[A{g}^{-}\right]\left[B{r}^{-}\right]=S\times S=7.7\times {10}^{-13} \\ {S}^2=7.7\times {10}^{-13} \\ S=8.8\times {10}^{-7} \\ \left[A{g}^{-}\right]=8.8\times {10}^{-7}M \end{gathered}$$

Now by mixing this two solution we get:

$\text{CuBr(s)}\rightleftharpoons {\text{Cu}}^{\text{+}}{\text{(aq) + Br}}^{\text{-}}\text{(aq)}$

${\text{K}}_{\text{sp}}{\text{= [Cu}}^{\text{+}}{\text{][Br}}^{\text{-}}\text{]}$

${\text{K}}_{\text{sp}}\text{(CuBr)}$$=4.2\times {10}^{-8}$

Finding value for Cu and Br.

$$\begin{gathered} {\text{[Cu}}^{\text{+}}\text{]=S} \\ {\text{[Br}}^{\text{-}}\text{]=s} \end{gathered}$$

$$\begin{gathered} \left[C{u}^{+}\right]\left[B{r}^{-}\right]=s\times s=4.2\times {10}^{-8} \\ {S}^2=4.2\times {10}^{-8} \\ S=2.0\times {10}^{-4} \\ \left[C{u}^{+}\right]=4.2\times {10}^{-4}M \end{gathered}$$

Now computing the ratio of $\left[A{g}^{-}\right]and\left[C{u}^{+}\right]$we will get

$\frac{\left[C{u}^{+}\right]}{\left[A{g}^{-}\right]}=\frac{2.0\times {10}^{-4}M}{8.8\times {10}^{-7}M}=2.3\times {10}^2$

$Therefore,Theratioof\left[A{g}^{-}\right]and\left[C{u}^{+}\right]is2.3\times {10}^2.$