Crash Course Chemistry Notes: pH, pOH, Kw, and Acid-Base Equilibria

pH balance is connected to the equilibrium state of reversible reactions and the familiar pH scale
The term pH is explained as follows:
- The exact origin of the lowercase p is unknown; the Danish chemist behind the term did not explain the reasoning. Some theories:
- From power (perhaps in French or Latin pondus)
- A common chemist’s habit of differentiating a test solution labeled p from a reference solution Q
- The H stands for hydrogen because hydrogen ions (protons) are central to acid-base behavior
- A handy mnemonic is thinking of pH as the "power of hydrogen" in a solution, i.e., the strength of the acid/base character
pH’s alter ego is pOH (p o h), the negative logarithm of hydroxide concentration
pH is mainly defined through water chemistry; water is special because it can act as both an acid and a base (autoprotolysis)
Water can release protons to form hydronium (H₃O⁺) and can accept protons to form hydroxide (OH⁻)
The key reversible reaction (water autoprotolysis):
2\,\mathrm{H2O} \rightleftharpoons \mathrm{H3O^+} + \mathrm{OH^-}
The pH scale conceptually centers on water neutral behavior; neutral water has equal tendencies to form H₃O⁺ and OH⁻

In chemistry, pH represents the power of hydrogen in a solution, i.e., the acidity/basicity
Mathematically, pH is defined as the negative base-10 logarithm of the hydrogen ion concentration:
\mathrm{pH} = -\log [\mathrm{H_3O^+}]
Note: historically, some use H⁺ instead of H₃O⁺; hydronium ion concentration is the intended quantity, but H⁺ is commonly used in simplified reactions
The logarithm used here is base 10; the base-10 log is often written simply as "log" in chemistry shorthand
Example to build intuition:
- If the hydronium concentration is roughly $1 \times 10^{-5}$ M, then
  \mathrm{pH} = -\log(1 \times 10^{-5}) = 5
Why logarithms? They compress very large or very small numbers into manageable scales and make multiplicative changes additive on the pH scale

In pure water, autoprotolysis produces equal amounts of H₃O⁺ and OH⁻
The water dissociation constant is defined as:
Kw = [\mathrm{H3O^+}][\mathrm{OH^-}] = 1.0 \times 10^{-14}
Because water is essentially pure and its concentration is so large compared with the ion concentrations, we simplify the expression by excluding water in the denominator, leading to the product form only
In pure water at equilibrium, the concentrations are equal:
- Let $x = [\mathrm{H_3O^+}] = [\mathrm{OH^-}]$ at equilibrium
- Then Kw = x^2 \quad \Rightarrow \quad x = \sqrt{Kw} = \sqrt{1.0 \times 10^{-14}} = 1.0 \times 10^{-7}\,\text{M}
- Therefore, neutral water has [\mathrm{H_3O^+}] = [\mathrm{OH^-}] = 1.0 \times 10^{-7}\,\text{M} and
  \mathrm{pH} = -\log(1.0 \times 10^{-7}) = 7

Litmus paper is an indicator that changes color with pH:
- Acids turn litmus pink
- Bases turn litmus blue
- Neutral solutions yield a purple-ish color
Indicators provide qualitative color-based guidance; there are many indicators with different color ranges; more on indicators next week

pH scale typically runs from 0 to 14 (extremes outside this range occur in very strong solutions and are rarely encountered in everyday contexts)
Acids vs bases on the scale:
- Acids (0–6.9): higher hydrogen ion concentration
- Neutral at 7.0
- Bases (7.1–14): higher hydroxide concentration
Strong vs weak acids:
- Strong acids (e.g., HCl, HNO₃) dissociate completely, releasing large amounts of protons; they generally have very low pH
- Weak acids (e.g., citric acid) dissociate incompletely, releasing fewer protons and typically have higher pH values in the 4–6 range
Strong bases (e.g., NaOH) remove many protons, leading to very high pH values
Weak bases (e.g., NaHCO₃, baking soda) remove fewer protons, leading to pH values typically in the 8–11 range
Practical neutral pH (7.0) is often treated as 6–8 in everyday contexts due to measurement precision and environmental variations

pOH is the negative log of the hydroxide ion concentration:
\mathrm{pOH} = -\log [\mathrm{OH^-}]
The Kw relationship ties hydrogen and hydroxide concentrations together:
Kw = [\mathrm{H3O^+}][\mathrm{OH^-}] = 1.0 \times 10^{-14}
Because $[\mathrm{H3O^+}]$ and $[\mathrm{OH^-}]$ multiply to a fixed value, one can compute the other via division: [\mathrm{OH^-}] = \frac{Kw}{[\mathrm{H_3O^+}]}
The pH and pOH are related by a simple additive rule (at standard conditions):
\mathrm{pH} + \mathrm{pOH} = 14
Example (from the transcript): if $[\mathrm{H_3O^+}] = 3.2 \times 10^{-4}$ M,
- \mathrm{pH} = -\log(3.2 \times 10^{-4}) \approx 3.5
- [\mathrm{OH^-}] = \frac{Kw}{[\mathrm{H3O^+}]} = \frac{1.0 \times 10^{-14}}{3.2 \times 10^{-4}} \approx 3.1 \times 10^{-11}\,\text{M}
- \mathrm{pOH} = -\log(3.1 \times 10^{-11}) \approx 10.5
- Confirm \mathrm{pH} + \mathrm{pOH} \approx 14
A quick orange juice example from the transcript:
- If the orange juice has $[\mathrm{H_3O^+}] = 3.2 \times 10^{-4}$ M, then pH ≈ 3.5 and pOH ≈ 10.5; the hydroxide concentration is $3.1 \times 10^{-11}$ M
Quick practical note: because pH is a logarithmic scale, large changes in hydrogen ion concentration produce relatively modest changes in pH

Pure water (neutral) at equilibrium: [\mathrm{H_3O^+}] = [\mathrm{OH^-}] = 1.0 \times 10^{-7}\,\text{M} and \mathrm{pH} = 7
Adding a strong acid increases [H₃O⁺] and decreases pH; the pH drops quickly due to the log relationship
Adding a strong base decreases [H₃O⁺] and increases pH; the pH rises toward 14 as [OH⁻] grows
The pH scale is designed so that equal, opposite changes in proton/hydroxide concentrations reflect in the pH and pOH values in a way that balances out on the 14-point scale

Definition of pH:
\mathrm{pH} = -\log [\mathrm{H_3O^+}]
Water autoprotolysis and Kw:
2\,\mathrm{H2O} \rightleftharpoons \mathrm{H3O^+} + \mathrm{OH^-}
Kw = [\mathrm{H3O^+}][\mathrm{OH^-}] = 1.0 \times 10^{-14}
Neutral water concentrations and pH:
[\mathrm{H_3O^+}] = [\mathrm{OH^-}] = 1.0 \times 10^{-7}\,\text{M}
\mathrm{pH} = 7
pOH and its definition:
\mathrm{pOH} = -\log [\mathrm{OH^-}]
Relationship between pH and pOH:
\mathrm{pH} + \mathrm{pOH} = 14
Example conversions:
- If $[\mathrm{H_3O^+}] = 3.2 \times 10^{-4}$ M, then
  \mathrm{pH} \approx 3.5, \quad [\mathrm{OH^-}] \approx 3.1 \times 10^{-11}\,\text{M}, \quad \mathrm{pOH} \approx 10.5
Indicators and practical notes:
- Litmus: acids pink, bases blue, neutral purple
- 0–14 is the typical range; extremes exist in very strong or highly concentrated solutions
Strong vs weak species (qualitative):
- Strong acids/bases dissociate or react completely; weak acids/bases dissociate/react only partially
Important caveat: pH is temperature-dependent in the real world; the 14-point sum is specifically for standard conditions (the transcript emphasizes the 14-rule as stated)

pH is tied to equilibrium concepts and the idea that water can self-ionize and re-form
Kw provides a bridge between micro-level ion concentrations and macro-level pH measurements
The logarithmic nature of pH explains why ion concentration changes spanning many orders of magnitude map to a relatively compact scale
Real-world relevance includes calibration of consumer products (soaps, shampoos, deodorants, makeup) and understanding why certain formulations are designed to maintain specific pH levels
The pH/pOH framework lays groundwork for buffers and stabilization strategies covered in subsequent discussions