Question

When determining the pH of ${\mathrm{H}}_2{\mathrm{SO}}_4$ solutions, sometimes the ${\mathrm{H}}^{+}$ contribution from ${\mathrm{HSO}}_4^{-}$ can be ignored by the 5$\mathrm{\%}$ rule. At what concentrations of an ${\mathrm{H}}_2{\mathrm{SO}}_4$ solution can the ${\mathrm{H}}^{+}$ contribution from ${\mathrm{HSO}}_4^{-}$ be ignored when determining the pH of the solution?

Short answer

The H⁺ contribution from the second dissociation of HSO₄⁻ can be ignored while determining the pH of the solution when the concentration of H₂SO₄ is more than 0.228 M.

Step-by-step solution

Dissociation Equations of H₂SO₄

H₂SO₄ is a strong acid with two hydrogen ions that dissociate in water stepwise. 1. First dissociation: H₂SO₄ → H⁺ + HSO₄⁻ 2. Second dissociation: HSO₄⁻ → H⁺ + SO₄²⁻

Determine the Concentration of Each Ion

Let the concentration of H₂SO₄ be C. In the first dissociation, one mole of H⁺ and HSO₄⁻ will be formed per mole of H₂SO₄. For the first dissociation: [H₂SO₄] = C [H⁺] = C [HSO₄⁻] = C For the second dissociation, the reaction is between HSO₄⁻ and water, and one mole of H⁺ and SO₄²⁻ formed per mole of HSO₄⁻. K₂ = $\frac{[H^{+}][S{O}_4^{2-}]}{[HS{O}_4^{-}]}$ C₂ = [SO₄²⁻] = [H⁺ from second dissociation]

Use the 5% Rule to Find When the H⁺ Contribution from the Second Dissociation Can Be Ignored

We want the H⁺ concentration from the second dissociation to be less than 5% of the total H⁺ concentration. Fraction, X = $\frac{[H^{+}fromseconddissociation]}{Total[H^{+}]}$ = $\frac{C_2}{C+C_2}$ According to the 5% rule, we need to find when X < 0.05. $0.05>$($\frac{C_2}{C+C_2}$) $0.05(C+C_2)>C_2$ $0.05C+0.05C_2>C_2$ $0.05C>0.95C_2$ $\frac{C_2}{C}<\frac{0.05}{0.95}$ Now, using the expression for K₂: K₂ = $\frac{C_2(C+C_2)}{C}$ Substitute the inequality relation, K₂ > $\frac{0.05}{0.95}$ Find the concentration of H₂SO₄ for which the H⁺ contribution from the second dissociation can be ignored. K₂ = 0.012, find C. C > $\frac{0.012(0.95)}{0.05}$ C > 0.228 M

Conclusion

The H⁺ contribution from the second dissociation of HSO₄⁻ can be ignored while determining the pH of the solution when the concentration of H₂SO₄ is more than 0.228 M.

Key concepts

Sulfuric Acid

Sulfuric acid, with the chemical formula ${\mathrm{H}}_2{\mathrm{SO}}_4$, is one of the most important industrial chemicals. It's used in vast quantities to manufacture fertilizers, explosives, and in the petroleum refining process.
Beyond its industrial importance, sulfuric acid is notable in chemistry due to its strong acidic nature and ability to donate protons readily.
In aqueous solutions, sulfuric acid is capable of dissociating completely in two steps, producing hydrogen ions $({\mathrm{H}}^{+})$ in each step.

• The first dissociation: ${\mathrm{H}}_2{\mathrm{SO}}_4\to {\mathrm{H}}^{+}+{\mathrm{HSO}}_4^{-}$
• The second dissociation: ${\mathrm{HSO}}_4^{-}\to {\mathrm{H}}^{+}+{\mathrm{SO}}_4^{2-}$

The strong nature of sulfuric acid allows it to dissociate almost completely in water, especially in the first step. This is why it’s often simply referred to as a strong acid.

pH Calculation

Calculating the pH of a sulfuric acid solution involves understanding its dissociation in water. pH is a measure of the hydrogen ion concentration in a solution and is calculated using the formula:$$\mathrm{pH}=-\log [{\mathrm{H}}^{+}]$$For sulfuric acid, the immediate dissociation of the first hydrogen ion means that in most cases, the initial concentration of ${\mathrm{H}}_2{\mathrm{SO}}_4$ is equal to the ${\mathrm{H}}^{+}$ concentration after the first dissociation. Thus, for low concentrations of sulfuric acid, the pH is straightforward to compute. However, at higher concentrations, both dissociations contribute to the ${\mathrm{H}}^{+}$ concentration in varying degrees.
This dual contribution needs to be considered carefully, particularly at concentrations where ignoring the second dissociation might lead to inaccuracies in pH calculation.

5% Rule

The 5% rule is a useful guideline in acid-base chemistry for determining when a particular contribution to an overall concentration can be ignored. Specifically, for polyprotic acids like sulfuric acid, it helps identify when the ${\mathrm{H}}^{+}$ contribution from a secondary dissociation is negligible.When your calculations show that the hydrogen ions from the second dissociation contribute less than 5% of the total hydrogen ion concentration, you can ignore them for pH calculation purposes.
Mathematically, you express this as:

• $\text{Fraction}\text{, }\mathrm{X}=\frac{[{\mathrm{H}}^{+}\text{ from 2nd dissociation}]}{\text{Total }[{\mathrm{H}}^{+}]}$
• To ignore the contribution, $\mathrm{X}<0.05$

This simplification keeps calculations manageable without a significant loss of accuracy.

First and Second Dissociation

Sulfuric acid is a diprotic acid, having two protons that can dissociate. These dissociations happen in two distinct steps, each with its own characteristics:

• First dissociation: This is the immediate and complete dissociation of the first hydrogen ion, resulting in the formation of ${\mathrm{HSO}}_4^{-}$. This step is characterized by a very high equilibrium constant, implying full dissociation.
• Second dissociation: ${\mathrm{HSO}}_4^{-}$ dissociates into ${\mathrm{H}}^{+}$ and ${\mathrm{SO}}_4^{2-}$, with a much lower equilibrium constant $(K_2)$, meaning it is not as complete or immediate as the first dissociation.

Understanding the nature of these dissociations helps in making accurate calculations especially in varying concentration levels of sulfuric acid solutions.

Strong Acid

Sulfuric acid is a classic example of a strong acid, meaning it completely dissociates in solutions. This complete dissociation is what primarily characterizes strong acids and differentiates them from weak acids.

• For the first dissociation, sulfuric acid acts much like any other strong acid, with nearly full dissociation at typical concentrations.
• The second dissociation, while not as robust, still adds to the hydrogen ion concentration, but requires consideration under specific conditions as guided by rules like the 5% rule.

In practice, these properties mean that sulfuric acid solutions typically have low pH levels and require careful handling and precise calculations to account for all contributions to the hydrogen ion concentration.