Problem 29 - Algebra and Trigonometry

Question

\(29-43\) . These exercises deal with logarithmic scales. Finding pH The hydrogen ion concentration of a sample of each substance is given. Calculate the pH of the substance. (a) Lemon juice: \(\left[\mathrm{H}^{+}\right]=5.0 \times 10^{-3} \mathrm{M}\) (b) Tomato juice: \(\left[\mathrm{H}^{+}\right]=3.2 \times 10^{-4} \mathrm{M}\) (c) Seawater: \(\left[\mathrm{H}^{+}\right]=5.0 \times 10^{-9} \mathrm{M}\)

Step-by-Step Solution

Answer

Lemon juice: pH 2.30, Tomato juice: pH 3.49, Seawater: pH 8.30.

Step 1: Understanding the pH Formula

The pH of a solution is calculated using the formula: \( \text{pH} = -\log_{10}[\text{H}^+] \), where \([\text{H}^+]\) is the hydrogen ion concentration.

Step 2: Calculate pH for Lemon Juice

For lemon juice, the hydrogen ion concentration is \(5.0 \times 10^{-3} \text{ M}\). Substitute into the formula: \( \text{pH} = -\log_{10}(5.0 \times 10^{-3}) \). Calculate to find the pH.

Step 3: Solve the Logarithm for Lemon Juice

Using the calculator, \(\log_{10}(5.0 \times 10^{-3})\) is approximately -2.30. Therefore, \( \text{pH} = -(-2.30) = 2.30 \).

Step 4: Calculate pH for Tomato Juice

For tomato juice, the hydrogen ion concentration is \(3.2 \times 10^{-4} \text{ M}\). Substitute into the formula: \( \text{pH} = -\log_{10}(3.2 \times 10^{-4}) \). Calculate the pH.

Step 5: Solve the Logarithm for Tomato Juice

Using the calculator, \(\log_{10}(3.2 \times 10^{-4})\) is approximately -3.49. Therefore, \( \text{pH} = -(-3.49) = 3.49 \).

Step 6: Calculate pH for Seawater

For seawater, the hydrogen ion concentration is \(5.0 \times 10^{-9} \text{ M}\). Substitute into the formula: \( \text{pH} = -\log_{10}(5.0 \times 10^{-9}) \). Calculate to find the pH.

Step 7: Solve the Logarithm for Seawater

Using the calculator, \(\log_{10}(5.0 \times 10^{-9})\) is approximately -8.30. Therefore, \( \text{pH} = -(-8.30) = 8.30 \).

Key Concepts

Logarithmic Scale

The logarithmic scale is a useful mathematical tool, particularly in the context of pH calculations. Unlike a linear scale, where each step is equal in size, a logarithmic scale increases by multiples. This means that each step on a logarithmic scale represents a tenfold change. This is why it is widely used in scientific fields like Chemistry. When we calculate the pH of a solution, we're using a base-10 logarithm, represented as \( \log_{10} \).

Hence, a small change in pH reflects a large change in hydrogen ion concentration. For example, a pH of 4 means a solution is ten times more acidic than a pH of 5. This non-linear scale is perfect for representing data over a wide range of values, as is common in chemical concentrations.

• Base-10 logarithms simplify calculations of hydrogen ion concentration.
• Logarithmic scales help to easily compare wide ranges of values.

Hydrogen Ion Concentration

The hydrogen ion concentration, denoted as \([\text{H}^+]\), plays a central role in determining the acidity or basicity of a solution. It is expressed in units of molarity (M), which represents the number of moles of hydrogen ions per liter of solution. In the context of calculating pH, a higher concentration of hydrogen ions indicates a higher level of acidity.

When considering various substances, such as lemon juice or seawater, the concentration of hydrogen ions can vary remarkably, which directly influences the pH level. For example, lemon juice has a high concentration of hydrogen ions, roughly \(5.0 \times 10^{-3} \text{ M}\), resulting in a low pH (high acidity). In contrast, seawater has a much lower concentration of hydrogen ions, \(5.0 \times 10^{-9} \text{ M}\), reflecting its basic nature (higher pH).

Understanding this concentration helps us predict the chemical behavior and properties of a solution in various contexts. This straightforward relationship is crucial for making accurate pH calculations.

Acid-Base Chemistry

Acid-base chemistry is fundamental in understanding the behavior of acids and bases in solution. Acids are substances that donate hydrogen ions \( (\text{H}^+) \) when dissolved in water, whereas bases accept hydrogen ions. The strength of an acid or base depends on its ability to donate or accept hydrogen ions quickly and efficiently.

The pH scale is a convenient way to express acidity or basicity, ranging from 0 (very acidic) to 14 (very basic), with a neutral point at 7. Substances with a pH less than 7 are considered acidic, while those with a pH greater than 7 are basic. Lemon juice, having a pH of 2.30, falls in the acidic range, while seawater, with a pH of 8.30, is on the basic side.

• Acidity is determined by the concentration of hydrogen ions.
• A pH lower than 7 implies more acidic, and higher than 7 implies more basic.
• Neutral solutions, like pure water, have a pH close to 7.

Understanding acid-base chemistry helps in predicting how substances will interact in an aqueous environment. This affects everything from culinary applications to biological systems, where pH balance is crucial for function and survival.