Problem 37 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Acidity is determined by the concentration of hydrogen ions in a solution. The pH scale, proposed by Sorensen in the early \(1900 \mathrm{~s}\), defines \(\mathrm{pH}\) to be \(-\log \left[H^{+}\right]\), where \(\left[H^{+}\right]\) is the concentration of hydrogen ions given in moles per liter. A pH of 7 is considered neutral; a pH greater than 7 means the solution is basic, while a pH of less than 7 indicates acidity. (a) If the concentration of hydrogen ions in a solution is increased tenfold, what happens to the \(\mathrm{pH}\) ? (b) If a blood sample has a hydrogen ion concentration of \(3.15 \times 10^{-8}\), what is the \(\mathrm{pH} ?\) (c) You'll find that the blood sample described in part (b) is mildly basic. Which has a higher concentration of hydrogen ions: the blood sample or something neutral? How many times greater is it?

Step-by-Step Solution

Answer

(a) The pH of the solution decreases by 1 unit. (b) Using the given formula, the pH of the blood sample comes out to be slightly above 7. (c) The neutral solution has a higher concentration of hydrogen ions, and the precise ratio works out to be approximately 3.17 times.

Step 1: Understanding the Relation Between pH and Hydrogen Ion Concentration

The equation \(pH = -log[H^+]\) describes the negative logarithmic relationship between pH and hydrogen ion concentration. This implies that as the concentration of hydrogen ions increases, the pH decreases, making the solution more acidic. Conversely, a decrease in hydrogen ion concentration increases the pH, shifting the nature towards basic.

Step 2: Effect of Increased Hydrogen Ion Concentration on pH

The increase in the hydrogen ion concentration by ten times, or an order of magnitude, decreases the pH by 1 unit. This is because a tenfold increase corresponds to moving one step on the logarithmic scale.

Step 3: Calculate the pH of the Blood Sample

The pH of the blood sample can be calculated by simply inserting the given hydrogen ion concentration into the pH formula: \(pH = -log(3.15 * 10^{-8})\).

Step 4: Comparison of Hydrogen Ion Concentration

Neutral solutions have a pH of 7. Being mildly basic, the blood sample would have a pH slightly above 7. Thus, the concentration of hydrogen ions in a neutral solution would be more than that in the mildly basic blood sample. The exact ratio can be calculated by taking the ratio of hydrogen ion concentration of a neutral solution (which is \(10^{-7}\) M) to the hydrogen ion concentration of the blood sample.

Key Concepts

Acid-Base Chemistry

At the heart of acid-base chemistry lies the interaction between hydrogen ions (H^+) and hydroxide ions (OH^-). Acids are substances that increase the concentration of hydrogen ions in a solution, often by donating H^+ ions when they dissolve in water, while bases decrease the concentration of hydrogen ions, typically by providing OH^- ions that combine with H^+ to form water.Acidic solutions have a higher concentration of hydrogen ions and are characterized by a PH less than 7. Basic or alkaline solutions, in contrast, have lower concentrations of H^+ and a PH above 7. The power of hydrogen, or PH, is a measure of the hydrogen ion concentration and helps determine the acidity or basicity of a solution.

• Acids: substances that release hydrogen ions (H^+) into solution, lowering PH
• Bases: substances that accept H^+ ions or release hydroxide ions (OH^-), raising PH
• Neutral: a balanced condition where H^+ and OH^- concentrations are equal, resulting in a PH of 7

The human body tightly regulates its PH levels, especially in the blood, where a slightly basic PH is maintained for optimal function. Any deviation from this can indicate or cause health issues, making the understanding of acid-base balance critical in physiology and medicine.

Logarithmic Relationships

Logarithmic relationships are pivotal in understanding how the pH scale operates. A logarithm tells us what exponent we need to raise a certain number (the base) to get another number. This relationship is particularly helpful when dealing with quantities that range over several orders of magnitude, such as hydrogen ion concentration levels.The equation connecting pH and hydrogen ion concentration is an excellent example of a logarithmic relationship: PH = -log([H^+]). The 'log' here is a base-10 logarithm, meaning it indicates the power to which 10 must be raised to yield the concentration of hydrogen ions. Because this relationship is negative, an increase in H^+ ion concentration results in a lower PH, indicating higher acidity, while a decrease leads to a higher PH, thereby showing increased alkalinity.

• Logarithms handle multiplicative relationships, ideal for pH which is exponentially related to H^+ concentration.
• The logarithmic pH scale compresses the wide range of hydrogen ion concentrations into a manageable scale of 0 to 14.
• Each whole number change on the pH scale represents a tenfold change in hydrogen ion concentration.

Understanding this concept helps to grasp how small changes in hydrogen ion concentration can lead to significant shifts in pH, and consequently, in the acidity or basicity of a solution.

PH Calculation

Calculating the pH of a solution is a straightforward process using the formula PH = -log([H^+]). By inputting the concentration of hydrogen ions in moles per liter, we can determine the pH value, which provides a clear indication of the solution's acidity or basicity.Let's consider an example: If we have a hydrogen ion concentration of 3.15 \times 10^{-8} M, we determine the pH as follows:\[\begin{equation}PH = -log(3.15 \times 10^{-8})\approx 7.5\end{equation}\]This result implies that the solution is mildly basic, as the pH is above 7. To compare hydrogen ion concentrations between this sample and a neutral solution with a pH of 7, we calculate the difference in their ion concentrations. For a neutral solution, the hydrogen ion concentration is 10^{-7} M. Therefore, our blood sample is ten times less concentrated in H^+ ions than a neutral solution (10^{-7} M / 3.15 \times 10^{-8} M = 3.17).This pH calculation is vital in various disciplines, from chemistry and environmental science to biology and medicine. It aids in determining the compatibility of a substance with living organisms or the environment, the progress of chemical reactions, and much more.