Foundations — pH, pOH, pKa, pKb scales

Before you can read a single formula in pH, pOh, pKa, pKb scales, you need to own every squiggle it uses. This page builds them one at a time, from nothing. Never trust a symbol you cannot picture.

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1. What "concentration" and the square brackets $[]$ mean

Before any acid talk, we need the raw thing being measured: how crowded a dissolved particle is in water.

Look at the figure: same-size boxes, different numbers of orange dots. The left box is crowded (high $[{\text{H}}^{+}]$, acidic); the right box is sparse (low $[{\text{H}}^{+}]$). A "mole" is just a fixed huge count of particles (like "a dozen" but $6\times 10^{23}$) — you never need the exact number here, only that bigger number = more crowded.

Recall Why not just count individual particles?

Because the counts are astronomically large. We bundle them into moles so the numbers stay human-sized. ::: Moles are a convenient bundle so we write $0.001$ instead of $6\times 10^{20}$.

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2. The hydrogen ion ${\text{H}}^{+}$ and the hydroxide ion ${\text{OH}}^{-}$

Why the topic needs these two: acidity is measured by $[{\text{H}}^{+}]$, basicity by $[{\text{OH}}^{-}]$. They are the two lead characters of the entire chapter. You met them first in 2.6.1-Arrhenius-acids-and-bases.

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3. Water splitting itself: the $\rightleftharpoons$ arrow and $K_w$

Water quietly splits a tiny bit of itself:

${\text{H}}_2\text{O}\rightleftharpoons {\text{H}}^{+}+{\text{OH}}^{-}$

In pure water neither ion is favoured, so they share equally: $[{\text{H}}^{+}]=[{\text{OH}}^{-}]=\sqrt{10^{-14}}=10^{-7}\text{ M}.$

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4. Powers of ten and scientific notation $10^{-n}$

Everything above used numbers like $10^{-7}$. You must be fluent in what that means before logs make sense.

The figure lays $10^0$ down to $10^{-14}$ on a line. Notice how the actual values crush together near zero — that squashing is exactly why a plain ruler fails and why we need the log trick next.

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5. The logarithm ${\log }_{10}$ — the star tool

This is the single most important tool of the topic, so we build it slowly.

Why this tool and not, say, division? Because our numbers span 14 powers of ten. Division would give ratios, but log turns each ×10 jump into a simple +1 step — converting a wild exponential range into an even, walkable ladder.

Left panel: raw concentrations bunch up uselessly near zero. Right panel: after $-{\log }_{10}$, the same values spread into a tidy 0–14 line. The log un-squashes the scale. That is the whole reason p-scales exist.

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6. The minus sign and the p-operator $\text{pX}=-{\log }_{10}X$

Apply it three ways and you have three quarters of the topic:

• $\text{pH}=-{\log }_{10}[{\text{H}}^{+}]$
• $\text{pOH}=-{\log }_{10}[{\text{OH}}^{-}]$
• $\text{pKa}=-{\log }_{10}(K_a)$, $\text{pKb}=-{\log }_{10}(K_b)$

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7. Equilibrium constants $K_a$ and $K_b$

The last new symbols. These measure strength of weak acids and bases.

Why the topic needs $K_a,K_b$: applying the p-operator to them gives pKa and pKb, the strength scales. You will use these constantly in 2.6.7-Buffer-solutions and the 2.6.8-Henderson-Hasselbalch-equation.

Recall What does "conjugate" mean in one line?

A conjugate acid–base pair differ by exactly one ${\text{H}}^{+}$. ::: $\text{HA}$ and ${\text{A}}^{-}$ are a pair — remove one hydrogen ion to go from acid to its conjugate base.

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8. Two "+14" bridges the log tool builds

Once you have the p-operator and the log product law, two of the topic's headline formulas fall out in a few lines. We derive them here so no symbol on the map is left unearned.

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How these foundations feed the topic

Every arrow says "you need the left box before the right box makes sense." The two "Bridge" boxes are derived in Section 8 above. Read the parent note pH, pOh, pKa, pKb scales only once every box on the left of this map feels obvious.

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Equipment checklist

Test yourself — cover the right side and try to answer before revealing.

• What does $[{\text{H}}^{+}]$ mean, with units? ::: Concentration of hydrogen ions, in mol/L (M).
• What is the difference between a subscript and a superscript? ::: Subscript = how many atoms; superscript = electric charge.
• What does $\rightleftharpoons$ tell you about a reaction? ::: It runs both directions and settles at equilibrium (steady amounts).
• What is $K_w$ at $25^{\circ }\text{C}$ and what stays fixed? ::: $K_w=[{\text{H}}^{+}][{\text{OH}}^{-}]=10^{-14}$ (dilute solutions); the product is pinned.
• When can $K_w$ drift from $10^{-14}$? ::: If the temperature changes or the solution is very concentrated (water activity drops).
• What question does ${\log }_{10}(x)$ answer? ::: "10 to what power gives $x$?"
• Why do p-scales use base 10 and not $\ln$? ::: Because concentrations vary in powers of ten; base 10 makes each ×10 a +1 step.
• What does the minus in $\text{pX}=-{\log }_{10}X$ accomplish? ::: It flips tiny negative logs into friendly positive numbers and makes higher pX mean lower X.
• Can pH ever be below 0 or above 14? ::: Yes — very concentrated strong acids give negative pH, very concentrated strong bases give pH above 14.
• Show in one line why $\text{pH}+\text{pOH}=14$. ::: Take $-{\log }_{10}$ of $K_w=[{\text{H}}^{+}][{\text{OH}}^{-}]=10^{-14}$; the product law splits it into pH + pOH = 14.
• Show in one line why ${\text{pK}}_a+{\text{pK}}_b=14$. ::: $K_a\cdot K_b=K_w=10^{-14}$, so $-{\log }_{10}$ gives pKa + pKb = 14.
• What makes $K_a$ large? ::: The acid dissociating a lot (more products on top of the fraction) — a strong acid.
• What is the conjugate base of $\text{HA}$? ::: ${\text{A}}^{-}$ — the acid minus one ${\text{H}}^{+}$.
• What is $[{\text{H}}^{+}]$ in pure water at $25^{\circ }\text{C}$? ::: $10^{-7}$ M, giving pH 7.