Calculate the \(\mathrm{pH}\) of (a) black coffee, \(5.0 \times 10^{-5} \mathrm{MH}^{+}\) (b) limewater, \(3.4 \times 10^{-11} \mathrm{M} \mathrm{H}^{+}\) (c) fruit punch, \(2.1 \times 10^{-4} \mathrm{MH}^{+}\) (d) cranberry apple drink, \(1.3 \times 10^{-3} \mathrm{M} \mathrm{H}^{+}\)

Understanding acid-base chemistry is fundamental for calculating the \(\text{pH}\) of solutions. This branch of chemistry studies the properties of acids and bases and their reactions with each other. An acid is a substance that donates hydrogen ions (\(\text{H}^+\)) in a solution, while a base accepts them. Key concepts within acid-base chemistry include the strength of acids and bases, and the concept of conjugate acid-base pairs.
The \(\text{pH}\) of a solution is a measure of its acidity or basicity. A lower \(\text{pH}\) indicates an acidic solution, and a higher \(\text{pH}\) indicates a basic or alkaline solution. Neutral solutions have a \(\text{pH}\) of around 7.

• Acidic: \(\text{pH} < 7\)
• Neutral: \(\text{pH} = 7\)
• Basic: \(\text{pH} > 7\)

For example, hydrochloric acid (HCl) is a strong acid and dissociates completely in water, producing a high concentration of \(\text{H}^+\) ions. Conversely, acetic acid (vinegar) is a weak acid and only partially dissociates in water, resulting in a lower concentration of \(\text{H}^+\) ions.

Logarithms are mathematical functions that are essential for understanding \(\text{pH}\) calculations. The logarithm of a number is the power to which a given base must be raised to produce that number. In \(\text{pH}\) calculations, we use the base 10 logarithm, often written as \(-\text{log}(x)\).
The \(\text{pH}\) formula: \(\text{pH} = -\log[\text{H}^+]\), means we take the negative logarithm of the hydrogen ion concentration.
To calculate logarithms, use the following steps: \[-\text{log}(a \times 10^b) = -\text{log}(a) - \text{log}(10^b) = -\text{log}(a) + (-b)\] For example, for black coffee: \( \text{pH} = -\log(5.0 \times 10^{-5}) = -\log(5.0) - (-5) = -0.6990 + 5 = 4.301\).
Logarithms simplify complex multiplications and divisions involving large numbers by transforming them into easier additions and subtractions.

Hydrogen ion concentration, represented as \(\text{H}^+\) or \([\text{H}^+]\), is a crucial part of calculating \(\text{pH}\). It's a measure of the amount of hydrogen ions present in a solution.
High \(\text{H}^+\) concentration means the solution is more acidic, while low \(\text{H}^+\) concentration indicates a more basic or alkaline solution. This concentration is usually expressed in moles per liter (M).
In practice, when provided with \(\text{H}^+\) concentration:

• If \(\text{H}^+ = 1 \times 10^{-3} \text{M}, then \text{pH} = 3\)
• If \(\text{H}^+ = 1 \times 10^{-7} \text{M}, then \text{pH} = 7\)

For example, in calculating the \(\text{pH}\) of cranberry apple drink with \(\text{H}^+ = 1.3 \times 10^{-3} \text{M}\), you find the \(\text{pH}\) as \(\text{pH} = -\log(1.3 \times 10^{-3}) = -\log(1.3) - \log(10^{-3}) = -0.1139 + 3 \), resulting in a \(\text{pH}\) of 2.8861.

Solution acidity refers to how acidic a solution is, based on its \(\text{pH}\). The term 'acidity' encompasses the concentration of hydrogen ions within a solution. A more acidic solution has a higher concentration of \(\text{H}^+\), resulting in a lower \(\text{pH}\).
Different solutions have varying levels of acidity:

• Black coffee: \(\text{H}^+ = 5.0 \times 10^{-5} \text{M}, \text{pH} = 4.301\)
• Limewater: \(\text{H}^+ = 3.4 \times 10^{-11} \text{M}, \text{pH} = 10.4685\)
• Fruit punch: \(\text{H}^+ = 2.1 \times 10^{-4} \text{M}, \text{pH} = 3.6778\)
• Cranberry apple drink: \(\text{H}^+ = 1.3 \times 10^{-3} \text{M}, \text{pH} = 2.8861\)

In essence, the higher the concentration of hydrogen ions, the more acidic the solution, and the lower its \(\text{pH}\).