Question

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

Short answer

The factor by which the concentration of H+ ions changes for a pH change of: (a) 2.00 units is 0.01. (b) 0.50 units is approximately 0.316.

Step-by-step solution

Understand and use the pH formula

The pH is the measure of acidity or basicity of a solution, and is given as the negative logarithm (base 10) of the hydrogen ion concentration [H+]. The formula is: \[pH = -\log_{10} [\mathrm{H}^{+}]\] To find the concentration of H+ ions, given the pH, we can rewrite the formula as follows: \[[\mathrm{H}^{+}] = 10^{-pH}\]

Find the factor for pH change of 2.00 units

For this case, we begin with an initial pH, let's call it \(pH_{1}\). When the pH changes by 2.00 units, the new pH is: \[pH_{2} = pH_{1} + 2.00\] Now, we need to find the initial and final H+ ion concentrations: \[[\mathrm{H}^{+}]_{1} = 10^{-pH_{1}}\] \[[\mathrm{H}^{+}]_{2} = 10^{-pH_{2}}\] We can now find the factor \(\mathrm{k}\) by which the concentration of H+ ions changes for a pH change of 2.00 units: \[\mathrm{k} = \frac{[\mathrm{H}^{+}]_{2}}{[\mathrm{H}^{+}]_{1}} = \frac{10^{-pH_{2}}}{10^{-pH_{1}}} = 10^{-(pH_{2}-pH_{1})} = 10^{-2}\] Therefore, the factor by which [H+] changes for a pH change of 2.00 units is 0.01.

Find the factor for pH change of 0.50 units

For this case, we have the same initial pH, \(pH_{1}\), and the new pH value for this situation is: \[pH_{3} = pH_{1} + 0.50\] Now, we need to find the initial and final H+ ion concentrations: \[[\mathrm{H}^{+}]_{1} = 10^{-pH_{1}}\] \[[\mathrm{H}^{+}]_{3} = 10^{-pH_{3}}\] We can now find the factor \(\mathrm{k}\) by which the concentration of H+ ions changes for a pH change of 0.50 units: \[\mathrm{k} = \frac{[\mathrm{H}^{+}]_{3}}{[\mathrm{H}^{+}]_{1}} = \frac{10^{-pH_{3}}}{10^{-pH_{1}}} = 10^{-(pH_{3}-pH_{1})} = 10^{-0.5}\] Therefore, the factor by which [H+] changes for a pH change of 0.50 units is approximately 0.316.

Final Answers

The factor by which the concentration of H+ ions changes for a pH change of: (a) 2.00 units is 0.01. (b) 0.50 units is approximately 0.316.

Key concepts

pH scale

Understanding the pH scale is critical when studying the properties of solutions, especially in acid-base chemistry. The pH scale is a numeric scale used to specify the acidity or basicity of an aqueous solution. It ranges typically from 0 to 14, where a pH of 7 is considered neutral, values less than 7 are acidic, and those greater than 7 are basic or alkaline.

For example, lemon juice has a pH around 2, which is acidic, while household ammonia has a pH around 11, indicating it's basic. Pure water, with a pH of 7, serves as the neutral midpoint. The importance of the pH scale lies in its ability to provide a quantitative measure of hydrogen ions present in a solution, which significantly influences the chemical reactions that can occur.

hydrogen ion concentration

Hydrogen ion concentration, expressed as \[\mathrm{H}^{+}\], is a measure of the number of moles of hydrogen ions per liter of solution. This concentration determines the acidity of a solution; the higher the concentration of hydrogen ions, the more acidic the solution is.

When the textbook asks, 'By what factor does \[\mathrm{H}^{+}\] change for a change in pH of 2.00 units?', it's inquiring about the change in the actual number of hydrogen ions in the solution. Since pH is negatively logarithmically related to \[\mathrm{H}^{+}\] concentration, a small change in pH can lead to a significant change in hydrogen ion concentration. Understanding this concept is fundamental for predicting how substances will react in a solution, particularly in acid-base reactions.

logarithmic relationships

The relationship between pH and hydrogen ion concentration is logarithmic, meaning that each unit change in pH corresponds to a tenfold change in hydrogen ion concentration. This can be seen in the pH formula \[pH = -\log_{10} \[\mathrm{H}^{+}\]\]. This logarithmic scale allows us to compress a wide range of ion concentrations into a small, manageable scale for practical use.

For instance, when the pH changes by 2 units, such as from 5 to 3, the \[\mathrm{H}^{+}\] concentration increases by a factor of 100, or \[10^2\]. If the pH changes by 0.50 units, the concentration increases by the square root of 10, or approximately 3.16. The logarithmic nature of the pH scale is essential for understanding how small changes in pH can imply significant changes in chemical behavior.

acid-base chemistry

Acid-base chemistry involves the study of acidic and basic substances and their chemical reactions with each other. Acids are substances that donate hydrogen ions \(\mathrm{H}^{+}\), while bases are substances that accept these ions. The pH level of a solution is a key attribute in this field, as it influences reaction rates, equilibrium, and the solubility of compounds.

The pH of a solution affects how acids and bases will react together. For example, a strong acid will significantly lower the pH of a solution, increasing the \[\mathrm{H}^{+}\] concentration, and can react with a strong base to neutralize the pH. Acid-base reactions are common in many biological and environmental processes, so understanding the principles of pH, \[\mathrm{H}^{+}\] concentration, and their interactions is vital for fields ranging from medicine to environmental science.