Problem 93 - Precalculus

Question

The pH of a solution is a measure of the molar concentration of hydrogen ions, \(H^{+},\) in moles per liter, in the solution, which means that it is a measure of the acidity or basicity of the solution. The letters pH stand for "power of hydrogen," and the numerical value is defined as $$\mathrm{pH}=-\log _{10}\left[H^{+}\right]$$ Very acid corresponds to pH values near \(1,\) neutral corresponds to a pH near 7 (pure water), and very basic corresponds to values near \(14 .\) In the next six exercises you will be asked to calculate the pH value of wine, Pepto- Bismol, normal rainwater, bleach, and fruit. List these six liquids and use your intuition to classify them as neutral, acidic, very acidic, basic, or very basic before you calculate their actual pH values. If wine has an approximate hydrogen ion concentration of \(5.01 \times 10^{-4},\) calculate its pH value.

Step-by-Step Solution

Answer

The pH of wine is approximately 3.300, which is acidic.

Step 1: Understanding the Formula

The pH value is calculated using the formula \( \mathrm{pH} = -\log_{10} [H^+] \), where \([H^+]\) is the concentration of hydrogen ions in the solution. Our task is to find the pH based on the given hydrogen ion concentration.

Step 2: Substitute the Values

The concentration of hydrogen ions \([H^+]\) in the wine is given as \(5.01 \times 10^{-4}\). We will substitute this into the formula: \[ \mathrm{pH} = -\log_{10} (5.01 \times 10^{-4}) \]

Step 3: Calculate the Logarithm

Calculate the logarithm base 10 of \(5.01 \times 10^{-4}\) using a calculator or logarithm tables. The logarithm \( \log_{10} (5.01 \times 10^{-4}) \) is approximately \(-3.300\).

Step 4: Find the pH

Now, substitute the calculated logarithm value back into the formula to find the pH: \[ \mathrm{pH} = -(-3.300) = 3.300 \].

Step 5: Interpret the pH Value

A pH of 3.300 indicates that the wine is acidic, as it falls below the neutral pH of 7.

Key Concepts

Logarithm

A logarithm is a mathematical concept that helps us to work with large or small numbers in a more manageable way. It asks the question, "To what power do we need to raise a certain number, known as the base, to get another number?" In the context of pH calculations, the base of the logarithm is 10. This is because the pH scale is based on powers of 10.
For example:

• In the logarithmic equation \( \, \log_{10}(100) = 2, \) the number 10 must be raised to the second power to equal 100.
• Similarly, \( \, \log_{10}(0.01) = -2, \) because 10 raised to the power of -2 equals 0.01.

In the context of acidity and pH calculations, logarithms make it easier to express the very small hydrogen ion concentrations in a comprehensible way. Instead of dealing directly with small numbers like \( 5.01 \times 10^{-4}, \) we can work with the pH scale, which translates these into the more user-friendly numbers from 0 to 14.

Hydrogen Ion Concentration

Hydrogen ion concentration is crucial in determining the acidity or basicity of a solution. It is expressed in moles per liter (M), and its concentration affects the pH of a solution. The more hydrogen ions present, the more acidic the solution.

To find the hydrogen ion concentration, you often need to work with powers of 10 because these concentrations can be very small. For instance, a concentration of \(5.01 \times 10^{-4} M\) is quite small, representing a weak amount of hydrogen ions in each liter of solution.

• High hydrogen ion concentration: Significantly acidic and pH closer to 0.
• Low hydrogen ion concentration: More basic and pH closer to 14.

To convert hydrogen ion concentration to a pH value, you use the formula \( \mathrm{pH} = -\log_{10} [H^+] \). This formula calculates how many times you must multiply 10 to get to the concentration in question, effectively compressing a wide range of hydrogen ion concentrations into a more manageable pH scale.

Acid-Base Chemistry

Acid-base chemistry studies how acids and bases interact, dissolve, and react with each other. It plays a pivotal role in many biological and chemical processes. At its core, acid-base chemistry is about the transfer of protons (hydrogen ions).

Acids are substances that increase the concentration of hydrogen ions, \([H^+]\), in a solution. This results in a lower pH.

• Strong acids, like hydrochloric acid (HCl), fully dissociate in water to produce a high \([H^+]\).
• Weak acids, such as acetic acid, only partially dissociate, leading to a lower increase in \([H^+]\).

On the other hand, bases decrease the concentration of hydrogen ions by providing hydroxide ions \((OH^-),\) which combine with hydrogen ions to form water, thus increasing the pH.
Moreover, the pH scale is essential in acid-base chemistry as it helps to quantify the acidity or basicity of the solution ranging from 0 (very acidic) to 14 (very basic), with 7 being neutral. Understanding this interaction helps in managing chemical reactions, medical treatments, and even environmental pollution controls.