Problem 77 - Thomas Calculus

Question

The pH of human blood normally falls between 7.37 and \(7.44 .\) Find the corresponding bounds for \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] .\)

Step-by-Step Solution

Answer

The bounds for \([\mathrm{H}_3\mathrm{O}^+]\) are approximately \(3.63 \times 10^{-8}\) to \(4.27 \times 10^{-8}\) M.

Step 1: Understand the Relationship Between pH and Hydronium Ion Concentration

The pH is calculated using the formula: \( \text{pH} = -\log_{10} \left[ \mathrm{H}_3\mathrm{O}^+ \right] \), where \( \left[ \mathrm{H}_3\mathrm{O}^+ \right] \) is the concentration of hydronium ions in moles per liter.

Step 2: Calculate Minimum Hydronium Ion Concentration

To find the lower bound for \( \left[ \mathrm{H}_3\mathrm{O}^+ \right] \), use the pH value 7.44: \[ 7.44 = -\log_{10} \left[ \mathrm{H}_3\mathrm{O}^+ \right] \]Rearrange to solve for \( \left[ \mathrm{H}_3\mathrm{O}^+ \right] \):\[ \left[ \mathrm{H}_3\mathrm{O}^+ \right] = 10^{-7.44} \]Calculate: \[ \left[ \mathrm{H}_3\mathrm{O}^+ \right] \approx 3.63 \times 10^{-8} \text{ M} \]

Step 3: Calculate Maximum Hydronium Ion Concentration

To find the upper bound for \( \left[ \mathrm{H}_3\mathrm{O}^+ \right] \), use the pH value 7.37:\[ 7.37 = -\log_{10} \left[ \mathrm{H}_3\mathrm{O}^+ \right] \]Rearrange to solve for \( \left[ \mathrm{H}_3\mathrm{O}^+ \right] \):\[ \left[ \mathrm{H}_3\mathrm{O}^+ \right] = 10^{-7.37} \]Calculate:\[ \left[ \mathrm{H}_3\mathrm{O}^+ \right] \approx 4.27 \times 10^{-8} \text{ M} \]

Step 4: Present the Final Range for Hydronium Ion Concentration

Combining the results from Step 2 and 3, the corresponding bounds for \( \left[ \mathrm{H}_3\mathrm{O}^+ \right] \) in human blood are:\( 3.63 \times 10^{-8} \text{ M} \leq \left[ \mathrm{H}_3\mathrm{O}^+ \right] \leq 4.27 \times 10^{-8} \text{ M} \).

Key Concepts

Hydronium Ion Concentration

In chemistry, the concentration of hydronium ions (\( [\mathrm{H}_3\mathrm{O}^+] \)) plays a crucial role in determining the acidity or basicity of a solution. When dissolved in water, acids release protons (\( \mathrm{H}^+ \)) that quickly join with water molecules to form hydronium ions. This concentration is essential because it reflects the chemical activity of protons in the solution.
Calculating the hydronium ion concentration allows us to classify whether a solution is acidic, neutral, or basic:

• An acidic solution has a higher concentration of hydronium ions, making \( [\mathrm{H}_3\mathrm{O}^+] \) greater than \( 10^{-7} \text{ M} \)
• A neutral solution maintains a \( [\mathrm{H}_3\mathrm{O}^+] \) of approximately \( 10^{-7} \text{ M} \)
• A basic solution has a lower \( [\mathrm{H}_3\mathrm{O}^+] \), implying values less than \( 10^{-7} \text{ M} \)

This indicates how acid or base changes affect chemical reactions, biological activities, and various industrial processes. Therefore, understanding hydronium ion concentration is key to grasping how substances can alter pH levels.

pH Scale

The pH scale is a measure used to determine the acidity or basicity of a solution. It ranges from 0 to 14 and is a critical tool in both scientific and everyday contexts. A neutral pH is 7, which indicates a balance between hydrogen ions and hydroxide ions in solution.
Acidic solutions have a pH less than 7, and basic solutions present a pH greater than 7. Here's how the pH scale operates:

• Values from 0 to 7 signify increasing acidity.
• Exactly 7 indicates a neutral environment, such as pure water.
• Values from 7 to 14 indicate increasing basicity.

This scale is logarithmic, meaning each whole number change reflects a tenfold change in hydronium ion concentration. How does this happen? If the pH drops by 1 unit, it implies the hydronium ion concentration has increased tenfold.
Utilizing the pH scale in science provides insights into natural processes such as ocean chemistry, enzymatic reactions, and human blood regulation. Each application gives us profound insights into the workings of chemical balances in everyday life.

Logarithmic Function

The concept of logarithmic functions is foundational in understanding how the pH scale operates. A logarithmic function, specifically base ten, is used in calculating pH, reflecting how small changes in hydronium ion concentration can result in more significant changes in pH value.
Logarithmic functions transform multiplication into addition, making them handy in scaling measurements like the pH scale. With this approach, the equation \( \text{pH} = -\log_{10} [\mathrm{H}_3\mathrm{O}^+] \) gives a simplified way to manage wide-ranging concentrations of hydronium ions that occur in various solutions.
This nature of logarithmic functions is useful:

• They help simplify complex equations and align measurements with standard scales, such as the pH scale.
• Provide a framework for transforming exponential changes into linear ones, making analyses more manageable.
• Allow calculation of pH through direct reading of hydronium concentration, enhancing our understanding of their impact on solution characteristics.

Overall, logarithmic functions make sense of how tiny shifts in hydrogen ion concentration can lead to broader chemical impacts. It's a mathematical tool that clarifies the links between concentration levels and pH magnitude.