Problem 33 - Chemistry: The Central Science in SI Units, Global Edition

Question

By what factor does \(\left[\mathrm{H}^{+}\right]\) change for a pH change of (a) 3.0 units, (b) 0.3 units?

Step-by-Step Solution

Answer

The concentration of hydrogen ions (\([\mathrm{H}^{+}]\)) changes by a factor of (a) \(10^{-3}\) or \(0.001\) for a pH change of 3.0 units, and (b) \(10^{-0.3}\) or approximately \(0.501\) for a pH change of 0.3 units.

Step 1: Calculate the initial and final pH values

We are given that the pH change is 3.0 units. Let the initial pH be \(pH_{i}\) and the final pH be \(pH_{f}\). So, we can write \(pH_{f} = pH_{i} + 3\).

Step 2: Use the pH formula to find the change in H+ concentration

We know that \(pH = -\log_{10}([\mathrm{H}^{+}])\). Thus, we have: Initial H+ concentration: \([\mathrm{H}^{+}]_{i}=10^{-pH_{i}}\) Final H+ concentration: \([\mathrm{H}^{+}]_{f}=10^{-pH_{f}}=10^{-(pH_{i}+3)}\)

Step 3: Calculate the factor by which the H+ concentration changes

We have to find the factor \(\frac{[\mathrm{H}^{+}]_{f}}{[\mathrm{H}^{+}]_{i}}\): \(\frac{[\mathrm{H}^{+}]_{f}}{[\mathrm{H}^{+}]_{i}} = \frac{10^{-(pH_{i}+3)}}{10^{-pH_{i}}}\) By dividing exponents: \(\frac{[\mathrm{H}^{+}]_{f}}{[\mathrm{H}^{+}]_{i}} = 10^{-3}\) So the H+ concentration changes by a factor of \(10^{-3}\) or \(0.001\). (b) For a pH change of 0.3 units:

Step 1: Calculate the initial and final pH values

We are given that the pH change is 0.3 units. We can write \(pH_{f} = pH_{i} + 0.3\).

Step 2: Use the pH formula to find the change in H+ concentration

We have: Initial H+ concentration: \([\mathrm{H}^{+}]_{i}=10^{-pH_{i}}\) Final H+ concentration: \([\mathrm{H}^{+}]_{f}=10^{-pH_{f}}=10^{-(pH_{i}+0.3)}\)

Step 3: Calculate the factor by which the H+ concentration changes

We have to find the factor \(\frac{[\mathrm{H}^{+}]_{f}}{[\mathrm{H}^{+}]_{i}}\): \(\frac{[\mathrm{H}^{+}]_{f}}{[\mathrm{H}^{+}]_{i}} = \frac{10^{-(pH_{i}+0.3)}}{10^{-pH_{i}}}\) By dividing exponents: \(\frac{[\mathrm{H}^{+}]_{f}}{[\mathrm{H}^{+}]_{i}} = 10^{-0.3}\) So the H+ concentration changes by a factor of \(10^{-0.3}\) or approximately \(0.501\).

Key Concepts

Hydrogen Ion Concentration

Hydrogen ion concentration, represented as \([\mathrm{H}^{+}]\), is a crucial measure in chemistry. It helps us understand the acidity or alkalinity of a solution. When we talk about the concentration of hydrogen ions, we're essentially discussing the number of hydrogen ions present in a given volume of solution.

Here's why it's important:

• High hydrogen ion concentration indicates an acidic solution.
• Low hydrogen ion concentration suggests a basic or alkaline solution.

Consider how hydrogen ions change when a pH shifts. Using formulas, we can calculate how these concentrations vary. For instance, if the pH changes by 3, the concentration shifts by a factor of 1000! Understanding these changes can help you predict how solutions will react.

Logarithmic Scale

The logarithmic scale is a unique way to express quantities that vary over large ranges. In terms of pH, this scale is used to measure hydrogen ion concentrations in solutions.

An essential feature:

• Every whole number change in pH corresponds to a tenfold change in hydrogen ion concentration.

For example, a shift from pH 3 to pH 4 results in a \([\mathrm{H}^{+}]\) change by a factor of 10. This is because the mathematical relationship is determined using base 10 logarithms:

\[ pH = -\log_{10}([\mathrm{H}^{+}]) \]

Therefore, small changes in pH imply substantial shifts in hydrogen ion concentrations, making the logarithmic scale effective in illustrating these differences.

Acid-Base Chemistry

Acid-base chemistry deals with the study of acids and bases, including their behaviors and reactions in solutions. It's fundamental to understand how pH affects solutions on a molecular level.

Here are some key points:

• Acids release hydrogen ions (\([\mathrm{H}^{+}]\)) in solution, thereby increasing acidity.
• Bases remove hydrogen ions, reducing acidity and increasing pH.
• pH is a scale used to specify acidity or basicity, ranging from 0-14.

The equilibrium between acids and bases in water determines the hydrogen ion concentration and thus the pH. Changes in pH, as seen in specific exercises, reflect the underlying shifts between these ion concentrations. Practically, this can influence various chemical reactions and the properties of substances in everyday life.