Match each sparingly soluble salt with its correct solubility product (K_{sp}) expression,… — JAMB UTME

Match each sparingly soluble salt with its correct solubility product ( $K_{sp}$ ) expression, where $s$ represents the molar solubility of the salt in $\text{mol dm}^{-3}$ .

Calcium sulfate ( $\text{CaSO}_4$ ) $K_{sp} = s^2$
Silver chromate ( $\text{Ag}_2\text{CrO}_4$ ) $K_{sp} = 4s^3$
Aluminium hydroxide ( $\text{Al(OH)}_3$ ) $K_{sp} = 27s^4$
Calcium phosphate ( $\text{Ca}_3(\text{PO}_4)_2$ ) $K_{sp} = 108s^5$

Answer

Calcium sulfate matches

K_{sp} = s^2

; Silver chromate matches

K_{sp} = 4s^3

; Aluminium hydroxide matches

K_{sp} = 27s^4

; Calcium phosphate matches

K_{sp} = 108s^5

Each salt dissociates according to its stoichiometry

A_x B_y \rightleftharpoons x A^{y+} + y B^{x-}

, giving

K_{sp} = x^x y^y s^{x+y}

. Calcium sulfate (a 1:1 salt) gives

s^2

; silver chromate (a 2:1 salt) gives

4s^3

; aluminium hydroxide (a 1:3 salt) gives

27s^4

; and calcium phosphate (a 3:2 salt) gives

108s^5

Step-by-Step Solution

Write the balanced dissolution equilibrium for a generic sparingly soluble salt

A_x B_y

A_x B_y(s) \rightleftharpoons x A^{y+}(aq) + y B^{x-}(aq)

This establishes the stoichiometry of the dissolved ions in solution.

Express the concentration of each constituent ion in terms of molar solubility

s

[A^{y+}] = xs

and

[B^{x-}] = ys

Molar solubility

s

represents the moles of salt dissolved per

\text{dm}^3

of saturated solution.

Substitute the ionic concentrations into the solubility product expression

K_{sp} = [A^{y+}]^x [B^{x-}]^y

K_{sp} = (xs)^x (ys)^y = x^x y^y s^{(x+y)}

This yields the general mathematical relationship between

K_{sp}

and

s

for any ionic solid.

Apply the general formula to each specific salt based on its stoichiometric coefficients

x

and

y

For

\text{CaSO}_4

(

1:1

K_{sp} = 1^1 \cdot 1^1 \cdot s^{1+1} = s^2

. For

\text{Ag}_2\text{CrO}_4

(

2:1

K_{sp} = 2^2 \cdot 1^1 \cdot s^{2+1} = 4s^3

. For

\text{Al(OH)}_3

(

1:3

K_{sp} = 1^1 \cdot 3^3 \cdot s^{1+3} = 27s^4

. For

\text{Ca}_3(\text{PO}_4)_2

(

3:2

K_{sp} = 3^3 \cdot 2^2 \cdot s^{3+2} = 108s^5

Matching each salt with its stoichiometric coefficients produces the correct mathematical relationship.

Key Concept

Derivation of Solubility Product (

K_{sp}

) expressions from stoichiometric dissolution equilibria.

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