Problem 30 - Algebra and Trigonometry

Question

The pH reading of a glass of liquid is given. Find the hydrogen ion concentration of the liquid. (a) Beer: \(\mathrm{pH}=4.6\) (b) Water: \(\mathrm{pH}=7.3\)

Step-by-Step Solution

Answer

(a) Beer: \(2.51 \times 10^{-5}\, \mathrm{mol/L}\); (b) Water: \(5.01 \times 10^{-8}\, \mathrm{mol/L}\).

Step 1: Understanding the pH Concept

The pH scale is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. The pH is defined as the negative logarithm (base 10) of the hydrogen ion concentration: \( \mathrm{pH} = -\log_{10}(\left[ \mathrm{H}^+ \right]) \). Therefore, to find the hydrogen ion concentration, we can rearrange the formula: \( \left[ \mathrm{H}^+ \right] = 10^{-\mathrm{pH}} \).

Step 2: Calculating Hydrogen Ion Concentration for Beer

For beer with a \( \mathrm{pH} = 4.6 \), substitute this value into the formula for hydrogen ion concentration: \( \left[ \mathrm{H}^+ \right] = 10^{-4.6} \). Using a calculator, this evaluates to \( \left[ \mathrm{H}^+ \right] \approx 2.51 \times 10^{-5} \) moles per liter.

Step 3: Calculating Hydrogen Ion Concentration for Water

For water with a \( \mathrm{pH} = 7.3 \), substitute this pH value into the formula: \( \left[ \mathrm{H}^+ \right] = 10^{-7.3} \). Using a calculator, this gives \( \left[ \mathrm{H}^+ \right] \approx 5.01 \times 10^{-8} \) moles per liter.

Key Concepts

Logarithmic Scale

The concept of a logarithmic scale might initially be intimidating, but it's actually quite straightforward when broken down. A logarithmic scale is a type of scale used for a range of quantities that covers large variations in numbers. In simpler terms, instead of increasing by addition (like 1, 2, 3), this scale increases by multiplication (like 10, 100, 1000).
For pH and hydrogen ion concentration, the logarithmic scale is used to measure the concentration levels efficiently because these can vary greatly. This makes it particularly useful in chemistry and biology, where these measurements often range over several magnitudes.

• The pH scale is a logarithmic representation, specifically base 10. This means every whole number change on the pH scale represents a tenfold change in hydrogen ion concentration.
• For example, a change from a pH of 4 to 3 indicates a hydrogen ion concentration that is ten times greater.

Acidity and Basicity

Acidity and basicity describe the nature of a substance concerning its hydrogen ion concentration. On the pH scale:

• Substances with a pH less than 7 are considered acidic, meaning they have a higher concentration of hydrogen ions (H⁺).
• Those with a pH greater than 7 are basic (or alkaline), having a lower hydrogen ion concentration compared to pure water.
• A substance with a pH of 7, like pure water, is neutral, meaning its hydrogen ion concentration is balanced.

Understanding whether a solution is acidic or basic helps predict its reactivity and compatibility with other substances. A strong acidic solution (like gastric acid in your stomach) will behave differently from a weak acidic solution (like coffee), based on these hydrogen ion concentrations. Knowing these differences is crucial in many scientific and everyday applications.

Aqueous Solution

An aqueous solution is a solution where water is the solvent. This is an important concept because water is a universal solvent, meaning it dissolves more substances than any other liquid.
Most biological and chemical reactions occur in aqueous environments, which is why understanding pH in these solutions is so essential.

• In an aqueous solution, pH measures the hydrogen ion concentration. Thus, the higher the amount of H⁺, the lower the pH and the more acidic the solution is.
• Conversely, as hydrogen ion concentration decreases, the solution becomes more basic (higher pH).

When dealing with aqueous solutions, we also consider the role of water in both dissolving the solute and in facilitating chemical reactions. Water's polarity allows it to break down ionic compounds and interact with molecules, affecting the pH significantly.