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 ${K_{sp}} = 27{x^4}$
The value of rate constant is already given as \[2.8 \times {10^{ - 23}}\] , we can now equate both the ${K_{sp}}$ values.
$27{x^4} = 2.8 \times {10^{ - 23}}$
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.