Understanding effect of ph on solubility of salt

In the following picture why Ksp is not simply S2- ? Why is f included with one species only? Please explain the last part.

What does f represent and why it is used in molar solubility?

• I am also very confused on what is happening here. I have no idea what f is suppose to be and I don;t know how they got that expression for Ksp. I will will write up an answer on how I would do this which is hopefully much clearer – Nanoputian Apr 26 '16 at 7:27
• $f$ is defined in the question as the fraction of $[\ce{X}]_{tot}$ that is ionized. It's just a shorthand abbreviation for $\frac{[\ce{X-}]}{[\ce{X-}] + [\ce{HX}]}$. By charge conservation, the concentration of metal $\ce{M+}$ is equal to $f S$, so $K_{sp}$ can be written as $[S] * f[S]$. – Curt F. Apr 27 '16 at 9:52

Let assume we have the following equilibria: $$\ce{MX \rightleftharpoons M+ + X-}$$ $$\ce{HX \rightleftharpoons H+ + X-}$$ The expression for their respective equilibrium constants are the following: $$\mathrm{K_{sp} = [M^+][X-]}$$ $$\mathrm{K_a = \frac{[H^+][X^-]}{[HX]}}$$ We also know the following (let S be solubility): $$\mathrm{S = [M^+] = [X^-]_{tot}}$$ Now we just need one more equation before we can start solving for S which is a mass balance. Since all the $\mathrm{X^-}$ that is produced either reacts with water to form $\mathrm{HX}$ or remains as $\mathrm{X^-}$, we get the following equation : $$\mathrm{[X^-]_{tot} = [X^-] + [HX]}$$ Now lets put $\mathrm{[X^-]}$ and $\mathrm{[HX]}$ in terms of $\mathrm{[M^+]}$: $$\mathrm{[X^-] = \frac{K_{sp}}{[M^+]}}$$ $$\mathrm{[HX] = \frac{[H^+][X^-]}{K_a} = \frac{K_{sp}[H^+]}{K_a[M+]}}$$ Now let plugs those values into our mass balance and replace $\mathrm{[X^-]_{tot}}$ and $\mathrm{[M^+]}$ with S: $$\mathrm{[X^-]_{tot} = \frac{K_{sp}}{[M^+]} + \frac{K_{sp}[H^+]}{K_a[M+]}}$$ $$\mathrm{S = \frac{K_{sp}}{S} + \frac{K_{sp}[H^+]}{K_a\times S}}$$ $$\mathrm{S^2 = K_{sp} + \frac{K_{sp}[H^+]}{K_a}}$$ $$\mathrm{S = \sqrt{K_{sp} + \frac{K_{sp}[H^+]}{K_a}}}$$ This gives you the exact same expression as the one on the sheet but hopefully this is much clearer for you.