Ionic Equilibrium: Edge Cases and Subtle Traps (1)

easy • 2 min read
Tags Ionic Equilibrium

Question

Calculate the pH of 10−8 M10^{-8}\,\text{M} HCl solution. Many students answer pH = 8. Why is that wrong, and what’s the correct answer?

Solution — Step by Step

If the answer were pH = 8, the solution would be basic. But HCl is a strong acid — even at low concentration it cannot make a solution basic. So pH must be less than 7.

At very low acid concentrations, the H+\text{H}^+ from water (10−7 M10^{-7}\,\text{M}) becomes comparable to the acid contribution. We can’t ignore water.

Let total [H+]=x[\text{H}^+] = x. From HCl: 10−810^{-8}. From water: equal H+\text{H}^+ and OH−\text{OH}^- produced, say yy each.

Charge balance: [H+]=[OH−]+[Cl−][\text{H}^+] = [\text{OH}^-] + [\text{Cl}^-]. So x=y′+10−8x = y' + 10^{-8} where y′=[OH−]y' = [\text{OH}^-].

Equilibrium: x⋅y′=10−14x \cdot y' = 10^{-14}.

Substituting: x(x−10−8)=10−14x(x - 10^{-8}) = 10^{-14}, giving x2−10−8x−10−14=0x^2 - 10^{-8}x - 10^{-14} = 0.

Solving: x=10−8+10−16+4×10−142≈1.05×10−7 Mx = \dfrac{10^{-8} + \sqrt{10^{-16} + 4\times 10^{-14}}}{2} \approx 1.05 \times 10^{-7}\,\text{M}.

pH=−log⁡(1.05×10−7)≈6.98\text{pH} = -\log(1.05 \times 10^{-7}) \approx 6.98

The correct pH is approximately 6.986.98, slightly acidic — never basic.

Why This Works

Whenever the acid concentration approaches or drops below 10−7 M10^{-7}\,\text{M}, water’s autoionisation matters. The standard formula pH =−log⁡[HCl]= -\log[\text{HCl}] assumes the acid dominates, which fails here. The right approach combines mass balance, charge balance, and KwK_w.

Alternative Method

Approximation: at very dilute acid, total [H+]≈[HCl]⋅contribution+10−14[\text{H}^+] \approx \sqrt{[\text{HCl}] \cdot \text{contribution} + 10^{-14}}. Numerically, 10−8⋅10−7+10−14≈1.05×10−7\sqrt{10^{-8}\cdot 10^{-7} + 10^{-14}} \approx 1.05 \times 10^{-7}. Same answer.

The trap: applying pH =−log⁡C= -\log C blindly. Always sanity check: a strong acid solution can never be basic. If you’re getting pH > 7 for HCl, redo the calculation accounting for water.

Common Mistake

Reporting pH = 8. The fix: whenever C<10−6 MC < 10^{-6}\,\text{M}, account for water. JEE Main 2023 had a 44-mark question on 10−9 M10^{-9}\,\text{M} HCl with pH near 77.

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