Problem 33 - Chemistry: The Central Science

Question

By what factor does \(\left[\mathrm{H}^{+}\right]\)change for a pH change of (a) \(2.00\) units, (b) \(0.50\) units?

Step-by-Step Solution

Answer

For a pH change of (a) 2.00 units, the [H+] concentration changes by a factor of 0.01. For a pH change of (b) 0.50 units, the [H+] concentration changes by a factor of 0.3162.

Step 1: Review the formula for pH

The formula for pH is given by: \( \textrm{pH} = -\log_{10} [\textrm{H}^+] \) Where pH is the measure of acidity or basicity, and [H+] represents the concentration of hydronium ions in the solution.

Step 2: Write the formula for the change in pH

The change in pH is can be written as: \( \Delta \textrm{pH} = \textrm{pH}_{1} - \textrm{pH}_{2} \) Where \( \textrm{pH}_{1} \) is the initial pH, \( \textrm{pH}_{2} \) is the final pH, and \( \Delta \textrm{pH} \) is the change in pH.

Step 3: Write the formula for the change in H+ concentration

We can rewrite the pH formula given in step 1 by taking the antilog to find the concentration of the hydronium ions: \( [\textrm{H}^+] = 10^{-\textrm{pH}} \) So, the change in the concentration of H+ ions can be written as: \( \frac{[\textrm{H}^+]_{2}}{[\textrm{H}^+]_{1}} = \frac{10^{-\textrm{pH}_{2}}}{10^{-\textrm{pH}_{1}}} \)

Step 4: Calculate pH change and corresponding H+ concentration change

(a) For a 2.00 unit increase in pH: \( \Delta \textrm{pH} = 2.00 \) Substitute into the formula in step 3: \( \frac{[\textrm{H}^+]_{2}}{[\textrm{H}^+]_{1}} = \frac{10^{-\textrm{pH}_{2}}}{10^{-\textrm{pH}_{1}}} = 10^{\frac{-\textrm{pH}_{2}+\textrm{pH}_{1}}{\underline{\phantom{xx}}}} = 10^{-2.00} = 0.01 \) In this case, the [H+] concentration changes by a factor of 0.01 (reduces). (b) For a 0.50 unit increase in pH: \( \Delta \textrm{pH} = 0.50 \) Substitute into the formula in step 3: \( \frac{[\textrm{H}^+]_{2}}{[\textrm{H}^+]_{1}} = \frac{10^{-\textrm{pH}_{2}}}{10^{-\textrm{pH}_{1}}} = 10^{\frac{-\textrm{pH}_{2}+\textrm{pH}_{1}}{\underline{\phantom{xx}}}} = 10^{-0.50} = 0.3162 \) In this case, the [H+] concentration changes by a factor of 0.3162 (reduces).

Step 5: Conclusion

For a pH change of (a) 2.00 units, the [H+] concentration changes by a factor of 0.01. For a pH change of (b) 0.50 units, the [H+] concentration changes by a factor of 0.3162.

Key Concepts

Hydronium Ion Concentration

The hydronium ion, represented by \([H^+]\), is a key player in acid-base chemistry. Hydronium ions form when an acid dissolves in water and its hydrogen ions (H^+) associate with water molecules. The concentration of these ions is crucial in determining the acidity of a solution.

When it comes to measuring \([H^+]\) in a solution, we use the formula: \([H^+]= 10^{-\text{pH}}\), which connects directly to the pH value of the solution. A change in the pH directly impacts the concentration of the hydronium ions.

• A decrease in pH represents an increase in \([H^+]\) concentration, indicating more acidic conditions.
• An increase in pH points to a decrease in \([H^+]\) concentration, reflecting more basic or alkaline conditions.

Understanding this relationship is fundamental in predicting how a solution will behave chemically due to changes in hydronium ion levels.

Acid-Base Equilibrium

Acid-base equilibrium is a balancing act of sorts, where acids and bases maintain a particular concentration of ions in a solution. Acids provide hydrogen ions ( H^+ ), while bases supply hydroxide ions ( OH^− ). The equilibrium is governed by how these ions interact.

The autoionization of water is a classic example, where water molecules dissociate into H^+ and OH^− . The product of these concentrations is constant at a given temperature, symbolized by K_w, and is significant for maintaining equilibrium in pure water and solutions.

• The formula K_w = [H⁺][OH⁻] is fundamental, where K_w is approximately 1.0 × 10^{-14} at 25°C.
• This equilibrium constant explains how, in a neutral solution, the concentrations of H^+ and OH^− will be equal, each being 1.0 × 10^{-7} M.

In practical terms, understanding these equilibriums helps chemists and students alike predict reaction outcomes and balance solution compositions.

pH Calculation

Calculating pH is a vital skill in chemistry, providing insight into the acidity or basicity of a solution. The formula for pH is \(\text{pH} = -\log_{10} [\text{H}^+]\), which allows us to determine the pH from the hydronium ion concentration.

To calculate pH effectively, follow these steps:

• Determine the hydronium ion concentration, \([H^+]\), of the solution.
• Apply the pH formula to find the pH value.
• If needed, use the inverse operation, the antilog, to solve for \([H^+]\), when given the pH.

Let's say you have a solution with a hydronium ion concentration of 1.0 × 10^{-3} M. Using the formula, you would find \(\text{pH} = -\log_{10} (1.0 × 10^{-3})= 3\). Hence, the solution is acidic. Understanding this process aids in the analysis and classification of different substances, enhancing your comprehension of chemical properties and reactions.