Relationship between Solubility and Solubility product

Relationship between solubility (S) and solubility product (K_sp)

Consider M_qA_r a sparingly soluble salt.
Where
q = Number of cations (M^r+) and
r = Number of anions (A^q-)

That is we have in dissolved state


M_qA_r ↔ qM^r+ + r A^q-


Then

K_sp = [M^r+]^q [A^q-]^r

If solubility is S, according to the definition of solubility product

We have
[M^r+]^q = q.S mol/dm³
[A^q-]^r = r.S mol/dm³

Hence K_sp = [q.S] ^q [r.S] ^r

= S^q+r. q^q.r^r


For example for the salt, calcium Phophate, Ca₃(PO₄)₂


Ca₃(PO₄)₂ ↔ 3Ca_aq²⁺ + 2PO_4(aq)^3-


K_sp = [Ca²⁺] ³ [PO₄^3-]²

= S³⁺².3³.2²
= 108S⁵

Past JEE Question

For a sparingly soluble salt A_pB_q, the relationship of its solubility product (K_sp) with its solubility (s) is

a. K_sp = s^p+q.p^p.q^q
b. K_sp = s^p+q.p^q.q^p
c. K_sp = s^pq.p^p.q^q
d. K_sp = s^pq.(pq)^p+q)

(2001)

Answer: a