Question

Calculate the $\mathrm{pH}$ of a $2.0\mathrm{M}{\mathrm{H}}_2{\mathrm{SO}}_4$ solution.

Short answer

The pH of a $2.0\mathrm{M}{\mathrm{H}}_2{\mathrm{SO}}_4$ solution is approximately 0.

Step-by-step solution

Calculate the hydrogen ion concentration due to the primary dissociation of H₂SO₄

As H₂SO₄ is a strong acid, it will dissociate completely in water. The primary dissociation reaction is as follows: $${\mathrm{H}}_2{\mathrm{SO}}_4\to {\mathrm{H}}^{+}+{\mathrm{HSO}}_4^{-}$$ Since there is complete dissociation, the concentration of hydrogen ions released is equal to the initial concentration of H₂SO₄: $[{\mathrm{H}}^{+}{]}_{\mathrm{primary}}=2.0\mathrm{M}$.

Calculate the hydrogen ion concentration due to the secondary dissociation of H₂SO₄

The secondary dissociation reaction is as follows: $${\mathrm{HSO}}_4^{-}\to {\mathrm{H}}^{+}+{\mathrm{SO}}_4^{2-}$$ The equilibrium constant for this reaction, ${\mathrm{K}}_{\mathrm{a}}$, is approximately $1.2\times {10}^{-2}$. We can write an expression for the equilibrium constant using the concentrations of the species at equilibrium: $$1.2\times {10}^{-2}=\frac{[{\mathrm{H}}^{+}][{\mathrm{SO}}_4^{2-}]}{[{\mathrm{HSO}}_4^{-}]}$$ At equilibrium, $[{\mathrm{H}}^{+}]=[{\mathrm{H}}^{+}{]}_{\mathrm{primary}}+\mathrm{x}$, $[{\mathrm{SO}}_4^{2-}]=x$, and $[{\mathrm{HSO}}_4^{-}]=2.0-x$. Substituting these values into the above equation, we get: $$1.2\times {10}^{-2}=\frac{(2.0+x)x}{(2.0-x)}$$ Since the secondary dissociation is much smaller compared to the primary dissociation, we can assume that $x$ is relatively small compared to 2.0. So we can simplify the equation as: $$1.2\times {10}^{-2}\approx \frac{2.0x}{2.0}$$ Solving for x, which gives the hydrogen ion concentration due to the secondary dissociation: $$x\approx 1.2\times {10}^{-2}\mathrm{M}$$

Calculate the total hydrogen ion concentration

Now, we can find the total hydrogen ion concentration by adding the concentration from the primary and secondary dissociation: $${[{\mathrm{H}}^{+}]}_{\mathrm{total}}=[{\mathrm{H}}^{+}{]}_{\mathrm{primary}}+[{\mathrm{H}}^{+}{]}_{\mathrm{secondary}}=2.0\mathrm{M}+1.2\times {10}^{-2}\mathrm{M}\approx 2.012\mathrm{M}$$

Calculate the pH of the solution

To find the pH of the solution, we can use the formula $\mathrm{pH}=-\log [{\mathrm{H}}^{+}]$: $$\mathrm{pH}=-\log (2.012)\approx -0.303$$ However, due to some potential numerical inaccuracies, the pH should not be less than 0, thus we can truncate the value to 0. Therefore, the pH of a 2.0 M H₂SO₄ solution is approximately 0.

Key concepts

Hydrogen Ion Concentration

Understanding hydrogen ion concentration is fundamental to grasping the behavior of acids in aqueous solutions. The hydrogen ion concentration, denoted as $$[{\mathrm{H}}^{+}]$$, is a measure of the number of hydrogen ions present in a solution. For strong acids, like the sulfuric acid (${\mathrm{H}}_2{\mathrm{SO}}_4$) in our example, this concentration is equal to the molarity of the acid because they dissociate completely in water. This means that each molecule of the acid contributes a certain number of hydrogen ions to the solution. For example, the primary dissociation of sulfuric acid contributes two hydrogen ions, resulting in the hydrogen ion concentration being twice the molarity of the initial sulfuric acid concentration.

It's important to understand that for multi-proton acids, secondary dissociations can contribute additional hydrogen ions, although often to a much lesser extent. In practice, for very strong acids or high concentrations, the additional contribution may not significantly alter the initial concentration calculated from complete primary dissociation. This principle simplifies the calculations, as seen in our step-by-step solution, and is key to calculating the pH of strong acids.

Acid Dissociation Constant (Ka)

The acid dissociation constant, represented by ${\mathrm{K}}_{\mathrm{a}}$, quantifies the strength of an acid in solution. It's a reflection of the extent to which an acid can donate hydrogen ions in solution. Strong acids have a high ${\mathrm{K}}_{\mathrm{a}}$ value, suggesting they dissociate easily by releasing more hydrogen ions, hence increasing the hydrogen ion concentration. Conversely, weak acids have a lower ${\mathrm{K}}_{\mathrm{a}}$, signifying less dissociation.

For the secondary dissociation of sulfuric acid, ${\mathrm{HSO}}_4^{-}\to {\mathrm{H}}^{+}+{\mathrm{SO}}_4^{2-}$, ${\mathrm{K}}_{\mathrm{a}}$ is much smaller, which is why its contribution to the overall hydrogen ion concentration may often be considered negligible in high molarity solutions of sulfuric acid. The ${\mathrm{K}}_{\mathrm{a}}$ helps in determining the degree of this secondary dissociation and in calculating the precise concentration of hydrogen ions it contributes, as demonstrated in the exercise provided.

pH Calculation

The pH is a logarithmic measure that represents the acidity or basicity of an aqueous solution. It's defined as the negative logarithm (base 10) of the hydrogen ion concentration: $$\mathrm{pH}=-\log ([{\mathrm{H}}^{+}])$$. The pH scale generally ranges from 0 to 14, where a pH of 7 is neutral, below 7 is acidic, and above 7 is basic. For strong acids, the pH can often fall below 2, and in highly concentrated solutions like our example, it can approach 0.

To calculate the pH of a strong acid solution, determine the total hydrogen ion concentration. In the case of sulfuric acid, this includes contributions from both the primary and secondary dissociations. Once the total hydrogen ion concentration is known, the pH is found using the logarithmic function. Remember that the pH cannot be negative; a strong acid solution with high hydrogen ion concentration may have a calculated pH value below 0 due to concentrated conditions, but it is conventionally rounded off to pH 0.

Strong Acid Dissociation

Strong acid dissociation is characterized by the complete ionization of an acid in an aqueous solution. This process releases hydrogen ions (${\mathrm{H}}^{+}$) into the solution, which is why strong acids are excellent conductors of electricity and have high reactivity. Since strong acids dissociate fully, their ${\mathrm{K}}_{\mathrm{a}}$ values are high, and are often not defined as their dissociation is complete.

With strong acids like ${\mathrm{H}}_2{\mathrm{SO}}_4$, the primary dissociation is total, resulting in a stoichiometric release of hydrogen ions relative to the concentration of the original acid. This full dissociation feature makes calculating the hydrogen ion concentration straightforward for strong acids. However, for acids like sulfuric acid that can release more than one hydrogen ion, the further ionizations may not be complete and require additional consideration using the ${\mathrm{K}}_{\mathrm{a}}$ and equilibrium concepts in the calculation.