Training Chemistry Math The pH Scale and Logarithms
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The pH Scale and Logarithms

24 min Chemistry Math

The pH scale provides a compact way to express the acidity or basicity of a solution using logarithms. Defined as pH = −log₁₀[H⁺], it compresses the enormous range of hydrogen-ion concentrations (from about 10⁰ to 10⁻¹⁴ M) into a manageable 0-to-14 scale. Understanding pH is essential for fields ranging from biochemistry and environmental science to water treatment and food science. The complementary quantity pOH, together with the autoionization constant of water (Kᵂ = 10⁻¹⁴), lets you move fluidly between hydrogen and hydroxide ion concentrations.

The pH Scale and Logarithms

The acidity of every solution is measured by a logarithm — one of the most practical math functions in science.

pH Definition

$$\text{pH} = -\log_{10}[\text{H}^+]$$

where $[\text{H}^+]$ is the hydrogen-ion concentration in mol/L (M).

Key Properties
  • pH 7 is neutral (pure water).
  • pH < 7 is acidic; pH > 7 is basic.
  • Each pH unit represents a 10-fold change in $[\text{H}^+]$.
  • $[\text{H}^+] = 10^{-\text{pH}}$ (the inverse).
Example 1

Find the pH if $[\text{H}^+] = 1.0 \times 10^{-4}$ M.

  1. $\text{pH} = -\log(10^{-4}) = 4.0$
Example 2

A solution has pH 3.5. Find $[\text{H}^+]$.

  1. $[\text{H}^+] = 10^{-3.5} \approx 3.16 \times 10^{-4}$ M.
Example 3

How many times more acidic is pH 2 than pH 5?

  1. The difference is 3 pH units
  2. so $10^3 = 1{,}000$ times more acidic.
  3. Each unit is a factor of 10.
pOH and the Ion Product

$$\text{pH} + \text{pOH} = 14 \quad \text{(at 25°C)}$$

$$[\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14}$$

Practice Problems

1. Find the pH of a solution with $[\text{H}^+] = 10^{-9}$ M.
2. Find $[\text{H}^+]$ for pH = 6.2.
3. Is pH 8.5 acidic or basic?
4. How many times more acidic is pH 1 than pH 4?
5. If pH = 3, what is pOH?
6. Orange juice has $[\text{H}^+] \approx 3.2 \times 10^{-4}$ M. Find its pH.
Show Answer Key

1. pH = 9

2. $10^{-6.2} \approx 6.31 \times 10^{-7}$ M

3. Basic

4. $10^3 = 1{,}000$ times

5. $14 - 3 = 11$

6. $-\log(3.2 \times 10^{-4}) \approx 3.5$

🧪 pH / pOH Calculator
[H⁺] concentration
pOH
[OH⁻] concentration
Classification
Stoichiometry and the Mole Radioactive Decay and Half-Life