Question

The $K_{sp}$ value for $Gd(OH{)}_3$ is $2.8\times {10}^{-23}$ , what is the $pH$ value of $Gd(OH{)}_3$ in a saturated solution?
a) $6.08$
b) $$5.08$$
c) $$8.47$$
d) $$4.08$$

Answer 1

Hint: The given compound Gadolinium trihydroxide dissociates to Gadolinium and hydroxyl ions. Write the equation for this reaction and calculate the value of $K_{sp}$ from it. This way we can find the concentration of $OH$ and hence its $pH$ value.

Complete step-by-step answer:First we write the equation for dissociation reaction of Gadolinium trihydroxide.
$Gd(OH{)}_3\rightleftarrows G{d}^{+3}+3O{H}^{-}$
From this equation, we can write the formula for $K_{sp}$ using the concentrations of the dissociated ions as follows.
$K_{sp}=[G{d}^{+3}][O{H}^{-}{]}^3$
Now, let us assume that the concentrations of the dissociated ions is $x$
The value of rate constant now becomes, $K_{sp}=x\times (3x{)}^3$
We can solve this and write as $K_{sp}=27x^4$
The value of rate constant is already given as $$2.8\times {10}^{-23}$$ , we can now equate both the $K_{sp}$ values.
$27x^4=2.8\times {10}^{-23}$
By solving this we get, $x^4\simeq {10}^{-24}$
$x={10}^{-6}$
Now that we have the value of $x$ i.e. concentration of $O{H}^{-}$ we can calculate $pOH$
$$[OH-]=3x=3\times {10}^{-6}$$
Taking $\log$ on both sides, we get $\log OH=\log 3\times {10}^{-6}$
$\log OH=-5.52$
We know that $pOH$ is equal to $-\log OH$ and hence the value of $pOH$ is $5.52$
Now in order to calculate the $pH$ value, we have to substitute the value of $pOH$ in the formula below.
$pH+pOH=14\Rightarrow pH=14-pOH$
$pH=14-5.52=8.48$
Therefore the $pH$ value for the given compound is $8.48$ which means that the compound is basic in nature.

Note: In order to find the $pH$ value of the given substance, we calculate the concentration of $O{H}^{-}$ from the given rate constant value. This concentration of $O{H}^{-}$ is then used to calculate the $pOH$ value. The resultant $pH$ value is $8.48$ which is above seven meaning that the compound Gadolinium trihydroxide is basic in nature.