Calculate the \(\mathrm{pH}\) of solutions having the following \(\left[\mathrm{H}^{+}\right]\): (a) \(0.0020 \mathrm{M}\) (b) \(7.0 \times 10^{-8} M\) (c) \(3.0 \mathrm{M}\)

Acid-base chemistry deals with the balance between acids and bases in a solution. An acid is a substance that increases the hydrogen ion concentration \(\text{[H}^{+}]\) when dissolved in water. Conversely, a base is a substance that decreases \(\text{[H}^{+}]\) in the solution. Understanding pH is crucial in this field as it gives a numerical value to how acidic or basic a solution is.

The pH scale typically ranges from 0 to 14:

• A pH less than 7 is acidic.
• A pH of 7 is neutral.
• A pH greater than 7 is basic.

For example, if we look at three solutions with different hydrogen ion concentrations, we can calculate their pH to determine their acidity:
(a) 0.0020 M \(\text{H}^{+}\) results in pH \(\text{≈ 2.70}\), indicating an acidic solution.
(b) 7.0 \(\times 10^{-8}\) M \(\text{H}^{+}\) results in pH \(\text{≈ 7.15}\), which is slightly basic.
(c) 3.0 M \(\text{H}^{+}\) results in a negative pH \(\text{≈ -0.48}\), an extremely acidic solution.

Logarithmic functions are critical in pH calculations. The pH of a solution is calculated using the formula \(\text{pH} = -\log [\text{H}^{+}]\). Here, \(\log\) is the logarithm base 10, which helps transform multiplicative relationships into additive ones.

Logarithms work as follows:

• \(\log (1) = 0\)
• \(\log (10) = 1\)
• \(\log(100) = 2\)

For tiny values of \([\text{H}^{+}]\), scientific notation simplifies the numbers, making calculations easier. For instance, calculating pH for \(7.0 \times 10^{-8}\) M involves:

\(-\log(7.0 \times 10^{-8})\) simplifies to \(-\log 7.0 - \log 10^{-8}\) because of the logarithmic property \(\log (a \times b) = \log a + \log b\). This gives us an approximate pH value of 7.15.

The concentration of hydrogen ions \([\text{H}^{+}]\) in a solution is integral in determining its acidity or basicity. In pH calculations, we use \([\text{H}^{+}]\) to find the pH value.

For example, for a \([\text{H}^{+}]\) of:

• 0.0020 M, the calculation is \(\text{pH} = -\log(0.0020)\) which results in a pH ≈ 2.70.
• 7.0 \times 10^{-8} M, the calculation is \(\text{pH} = -\log(7.0 \times 10^{-8})\), resulting in pH ≈ 7.15.
• 3.0 M, the calculation is \(\text{pH} = -\log(3.0)\), resulting in a pH ≈ -0.48.

The pH provides a way to easily understand the hydrogen ion concentration without dealing with many decimal places or exponents. Higher \([\text{H}^{+}]\) means a lower pH and a more acidic solution, while lower \([\text{H}^{+}]\) means a higher pH and a more basic solution.