Problem 77 - Precalculus

Question

We introduce three widely used measurement scales which involve common logarithms: the Richter scale, the decibel scale and the pH scale. The computations involved in all three scales are nearly identical so pay attention to the subtle differences. The \(\mathrm{pH}\) of a solution is a measure of its acidity or alkalinity. Specifically, \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]\) where \(\left[\mathrm{H}^{+}\right]\) is the hydrogen ion concentration in moles per liter. A solution with a pH less than 7 is an acid, one with a pH greater than 7 is a base (alkaline) and a pH of 7 is regarded as neutral. (a) The hydrogen ion concentration of pure water is \(\left[\mathrm{H}^{+}\right]=10^{-7} .\) Find its \(\mathrm{pH}\). (b) Find the pH of a solution with \(\left[\mathrm{H}^{+}\right]=6.3 \times 10^{-13}\) (c) The pH of gastric acid (the acid in your stomach) is about \(0.7 .\) What is the corresponding hydrogen ion concentration?

Step-by-Step Solution

Answer

(a) pH = 7 (b) pH ≈ 12.2 (c) \( [\mathrm{H}^{+}] \approx 0.2 \) M

Step 1: Calculate the pH of Pure Water

To find the pH of pure water, we use the formula \( \mathrm{pH} = -\log \left[ \mathrm{H}^{+} \right] \). Since \( \left[ \mathrm{H}^{+} \right] = 10^{-7} \) for pure water, the pH is \( -\log (10^{-7}) \). Using properties of logarithms, this simplifies to \( -(-7) = 7 \). Therefore, the pH of pure water is 7.

Step 2: Calculate the pH of the Given Solution

For the given solution with \( \left[ \mathrm{H}^{+} \right] = 6.3 \times 10^{-13} \), we apply the pH formula: \( \mathrm{pH} = -\log (6.3 \times 10^{-13}) \). Splitting the logarithm, \( \mathrm{pH} = - ( \log 6.3 + \log 10^{-13} ) = -\log 6.3 + 13 \). Approximating \( \log 6.3 \approx 0.8 \), we get \( \mathrm{pH} \approx 13 - 0.8 = 12.2 \). Therefore, the pH of the solution is approximately 12.2.

Step 3: Calculate Hydrogen Ion Concentration from pH

For gastric acid with a pH of 0.7, we reverse the pH equation: \( \mathrm{pH} = -\log \left[ \mathrm{H}^{+} \right] \) implies \( \left[ \mathrm{H}^{+} \right] = 10^{-\mathrm{pH}} \). Hence, \( \left[ \mathrm{H}^{+} \right] = 10^{-0.7} \). To approximate, \( 10^{-0.7} \approx 0.2 \). Thus, the hydrogen ion concentration is approximately \( 0.2 \) moles per liter.

Key Concepts

Logarithms

Logarithms are mathematical tools used to handle exponential numbers more easily. When we talk about the pH scale, we are employing a type of logarithm called a common logarithm, which has a base of 10. Here's how it works. For a number expressed as 10 raised to a power, the logarithm helps us find that power. For instance, if we have 10 raised to the power of \(-2\), the logarithm would simply be \(-2\). This reverses the process of exponentiation.

• Common logarithms express numbers as powers of 10.
• The logarithm of a number essentially tells you the exponent needed to reach that number from base 10.
• In the context of pH, logarithms help to determine how concentrated hydrogen ions are in a solution.

The formula \(\mathrm{pH} = -\log [\mathrm{H}^{+}]\)uses a logarithmic scale to make it simpler by converting tiny numbers like \(10^{-7}\) into manageable whole numbers like 7. Using logarithms in science helps us work across wide ranges of values efficiently, making phenomena more understandable rather than dealing with cumbersome decimals.

Hydrogen Ion Concentration

Hydrogen ion concentration \([\mathrm{H}^{+}]\)is a crucial component in understanding acidity and the pH scale. It refers to the amount of hydrogen ions present in a solution, measured in moles per liter. In any aqueous solution, water naturally dissociates into hydrogen \([\mathrm{H}^{+}]\)and hydroxide ions \([\mathrm{OH}^{-}]\).

• Pure water has equal amounts of \([\mathrm{H}^{+}]\) and \([\mathrm{OH}^{-}]\),making it neutral.
• An acidic solution has higher \([\mathrm{H}^{+}]\)than \([\mathrm{OH}^{-}]\).
• An alkaline or basic solution features more \([\mathrm{OH}^{-}]\)compared to \([\mathrm{H}^{+}]\).

As seen in the exercise, the concentration of hydrogen ions \([\mathrm{H}^{+}]\) serves as a direct measure to determine the pH of a solution. For instance, if\([\mathrm{H}^{+}]\) is \(6.3 \times 10^{-13}\),applying the pH formula reveals how acidic or alkaline this solution is by calculating \(\mathrm{pH} = -\log(6.3 \times 10^{-13})\),achieving a pH around 12.2, indicating an alkaline solution.

Acidity and Alkalinity

The concept of acidity and alkalinity in chemistry helps us understand how substances react in various environments. Measured by the pH scale, these concepts are crucial in evaluating the characteristics of solutions. Acidity refers to a higher concentration of hydrogen ions \([\mathrm{H}^{+}]\). Alkalinity, on the other hand, means more hydroxide ions \([\mathrm{OH}^{-}]\) prevalently.

• A pH less than 7 indicates acidity, such as the hydrochloric acid in your gastric juices, which typically has a pH around 0.7.
• A neutral solution, like pure water, has a pH close to 7.
• A pH greater than 7 signals alkalinity; for example, household ammonia can exhibit pH values between 11 to 13.

Understanding and calculating the pH is essential in fields like medicine, where the acidity level of gastric acid is significant for digestion. For a solution with pH 0.7, the formula \(\mathrm{pH} = -\log [\mathrm{H}^{+}]\) rewritten as \([\mathrm{H}^{+}] = 10^{-\mathrm{pH}}\)reveals that \([\mathrm{H}^{+}]\) approximates to 0.2. This knowledge aids in figuring out how a solution might interact chemically or biologically.