Question

Calculate the \(\mathrm{pH}\) of each of the following solutions containing a strong acid in water. a. \(2.0 \times 10^{-2} M \mathrm{HNO}_{3}\) b. \(4.0 \mathrm{M} \mathrm{HNO}_{3}\) c. \(6.2 \times 10^{-12} \mathrm{M} \mathrm{HNO}_{3}\)

Short answer

In summary, the pH values of the given HNO\(_3\) solutions are as follows: a. \(2.0 \times 10^{-2} M \mathrm{HNO}_{3} \Rightarrow pH \approx 1.70\) b. \(4.0 \mathrm{M} \mathrm{HNO}_{3} \Rightarrow pH \approx 0.398\) c. \(6.2 \times 10^{-12} \mathrm{M} \mathrm{HNO}_{3} \Rightarrow pH \approx 11.21\)

Step-by-step solution

Identify the concentration of hydronium ions

Since HNO\(_{3}\) is a strong acid and we can assume 100% dissociation in water, the concentration of H\(_{3}\)O\(^{+}\) ions will be equal to the given concentration of HNO\(_3\). So, \(\mathrm{[H_{3}O^{+}]} = 2.0 \times 10^{-2} \mathrm{M}\).

Calculate pH

Now, substitute the concentration of H\(_{3}\)O\(^{+}\) in the pH formula: \(pH = -\log (2.0 \times 10^{-2})\) After calculating the value, we get: \(pH \approx 1.70\) #b. Calculating pH for 4.0 M HNO₃ solution#

Identify the concentration of hydronium ions

For 4.0 M HNO\(_3\), the concentration of H\(_{3}\)O\(^{+}\) ions will be equal to the given concentration of HNO\(_3\). So, \(\mathrm{[H_{3}O^{+}]} = 4.0 \mathrm{M}\).

Calculate pH

Substitute the concentration of H\(_{3}\)O\(^{+}\) in the pH formula: \(pH = -\log (4.0)\) After calculating the value, we get: \(pH \approx 0.398\) #c. Calculating pH for 6.2 x 10⁻¹² M HNO₃ solution#

Identify the concentration of hydronium ions

For 6.2 x 10⁻¹² M HNO\(_3\), the concentration of H\(_{3}\)O\(^{+}\) ions will be equal to the given concentration of HNO\(_3\). So, \(\mathrm{[H_{3}O^{+}]} = 6.2 \times 10^{-12} \mathrm{M}\).

Calculate pH

Substitute the concentration of H\(_{3}\)O\(^{+}\) in the pH formula: \(pH = -\log (6.2 \times 10^{-12})\) After calculating the value, we get: \(pH \approx 11.21\) In summary, the pH values of the given HNO\(_3\) solutions are as follows: a. \(2.0 \times 10^{-2} M \mathrm{HNO}_{3} \Rightarrow pH \approx 1.70\) b. \(4.0 \mathrm{M} \mathrm{HNO}_{3} \Rightarrow pH \approx 0.398\) c. \(6.2 \times 10^{-12} \mathrm{M} \mathrm{HNO}_{3} \Rightarrow pH \approx 11.21\)

Key concepts

Strong Acids

When we talk about strong acids, we refer to their complete dissociation in water. Strong acids like nitric acid (HNO₃) dissociate 100%, providing a straightforward calculation for hydronium ions concentration. This means every molecule of HNO₃ breaks apart into its ions, specifically hydrogen ions (H⁺), which immediately become hydronium ions (H₃O⁺) when they interact with water. This characteristic makes strong acids easy to work with for pH calculations, since the concentration of the acid is equal to its hydronium ion concentration.

Key strong acids include:

• Hydrochloric acid (HCl)
• Nitric acid (HNO₃)
• Sulfuric acid (H₂SO₄)

These acids are used in various industries and laboratory settings due to their reactivity and predictability in aqueous solutions.

Hydronium Ion Concentration

The concept of hydronium ion concentration is fundamental in understanding acidity in solutions. When a strong acid like nitric acid dissolves in water, it releases hydrogen ions that form hydronium ions (H₃O⁺). This concentration of hydronium ions directly affects the pH of the solution.

Mathematically, if the concentration of the strong acid in solution is known, the hydronium ion concentration can be given as:

• \[[H₃O⁺] = [ ext{concentration of strong acid}]\]

It’s important to note that for very dilute solutions, such as \(6.2 \times 10^{-12} M HNO₃\), the contribution of hydronium ions from the water itself can affect the pH, leading to interesting results where the pH might appear neutral or even basic as in the present case.

Logarithmic Functions in pH Calculations

Logarithmic functions play a crucial role in calculating the pH of a solution. The pH scale is logarithmic, not linear, which means each whole number change on the pH scale represents a tenfold change in hydronium ion concentration. The formula to calculate pH is derived from the base-10 logarithm of the hydronium ion concentration:

• \[pH = -\log [H₃O⁺]\]

The negative sign in the formula is because we want a scale where higher concentrations of hydronium ions (and thus more acidic solutions) correspond to lower pH values.

When calculating, for example, the pH of a solution with hydronium ions concentration \(2.0 \times 10^{-2} M\),

Using the formula:

• \[pH = -\log(2.0 \times 10^{-2})\]Resulting in \(pH \approx 1.70\).

This illustrates how logarithmic functions simplify expressions involving large ranges of ion concentrations into a more practical and manageable scale.