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Question: The solubility product of \({\text{Cr}}{\left( {{\text{OH}}} \right)_3}\) at \(298{\text{ K}}\) is \...

The solubility product of Cr(OH)3{\text{Cr}}{\left( {{\text{OH}}} \right)_3} at 298 K298{\text{ K}} is 6.0×10−316.0 \times {10^{ - 31}}. The concentration of hydroxide ions in a saturated solution of Cr(OH)3{\text{Cr}}{\left( {{\text{OH}}} \right)_3} will be:
A) (2.22×10−31)1/4{\left( {2.22 \times {{10}^{ - 31}}} \right)^{{\text{1/4}}}}
B) (4.86×10−29)1/4{\left( {4.86 \times {{10}^{ - 29}}} \right)^{{\text{1/4}}}}
C) (18×10−31)1/2{\left( {18 \times {{10}^{ - 31}}} \right)^{{\text{1/2}}}}
D) (18×10−31)1/4{\left( {18 \times {{10}^{ - 31}}} \right)^{{\text{1/4}}}}

Explanation

Solution

We know that the solubility product of any salt at any temperature is the product of the molar concentration of its constituent ions. The concentration of ions is raised to the number of ions produced on dissociation of one molecule of the salt.

Complete solution:
We know that the solubility product of any salt at any temperature is the product of the molar concentration of its constituent ions. The concentration of ions is raised to the number of ions produced on dissociation of one molecule of the salt
We know that the solubility of a salt at any temperature is calculated from its solubility product.
Now, we are given a salt Cr(OH)3{\text{Cr}}{\left( {{\text{OH}}} \right)_3}. The salt dissociates as follows:
Cr(OH)3⇌Cr3++3OH−{\text{Cr}}{\left( {{\text{OH}}} \right)_3} \rightleftharpoons {\text{C}}{{\text{r}}^{3 + }} + 3{\text{O}}{{\text{H}}^ - }
The solubility product of the salt Cr(OH)3{\text{Cr}}{\left( {{\text{OH}}} \right)_3} is given as follows:
KSP=[Cr3+][OH−]3{{\text{K}}_{{\text{SP}}}} = [{\text{C}}{{\text{r}}^{3 + }}]{[{\text{O}}{{\text{H}}^ - }]^3}
Where KSP{{\text{K}}_{{\text{SP}}}} is the solubility product.
For the salt Cr(OH)3{\text{Cr}}{\left( {{\text{OH}}} \right)_3}.
KSP=[Cr3+][OH−]3=s×(3s)3=27s4{{\text{K}}_{{\text{SP}}}} = [{\text{C}}{{\text{r}}^{3 + }}]{[{\text{O}}{{\text{H}}^ - }]^3} = s \times {\left( {3s} \right)^3} = 27{s^4}
Where ss is the solubility of the ions.
We are given that the solubility product of salt Cr(OH)3{\text{Cr}}{\left( {{\text{OH}}} \right)_3} is 6.0×10−316.0 \times {10^{ - 31}}. Thus,
27s4=6.0×10−3127{s^4} = 6.0 \times {10^{ - 31}}
33s4=6.0×10−31{3^3}{s^4} = 6.0 \times {10^{ - 31}}
Multiply both sides by 3. Thus,
(3s)4=18.0×10−31{\left( {3s} \right)^4} = 18.0 \times {10^{ - 31}}
3s=(18.0×10−31)1/43s = {\left( {18.0 \times {{10}^{ - 31}}} \right)^{1/4}}
Now, the concentration of hydroxide ions is,
[OH−]=3s[{\text{O}}{{\text{H}}^ - }] = 3s
Thus,
[OH−]=(18.0×10−31)1/4[{\text{O}}{{\text{H}}^ - }] = {\left( {18.0 \times {{10}^{ - 31}}} \right)^{1/4}}
Thus, the concentration of hydroxide ions in a saturated solution of Cr(OH)3{\text{Cr}}{\left( {{\text{OH}}} \right)_3} is (18×10−31)1/4{\left( {18 \times {{10}^{ - 31}}} \right)^{{\text{1/4}}}}.

Thus, the correct option is (D) (18×10−31)1/4{\left( {18 \times {{10}^{ - 31}}} \right)^{{\text{1/4}}}}.

Note: The solubility product is calculated using the concentration of the ions in which the salt has dissociated. Solubility factor depends on various factors such as temperature, pressure and nature of the electrolyte. The concentration of ions is affected by these factors and thus, the solubility product gets affected.