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A better understanding of pH starts with the behavior of water molecules.

Each is made of two hydrogen atoms bound to one oxygen atom and is symbolized by the familiar formula H₂O.

The following discussion deals with pure water -- no dissolved solids, no dissolved gasses, no fish, no shrimp, no plants...only H₂O, the foundation of water-quality management. (Salinity effects are mentioned in a few places below, but that topic and the others are taken up in other sections.)

Becoming familiar with this material will give you a much better understanding of important features of pH discussed in the next section -- such as how small pH changes represent large changes in acidity, the proper way to average pH values, and which pH scales must be used when working with seawater.

This document was developed with Idyll [1], a “toolkit for creating data-driven stories and explorable explanations”. Custom components were built with React and D3.

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Two water molecules react to yield one H₃O⁺ and one OH^-:

H₃O⁺ is a hydronium ion (also called a hydroxonium or oxonium ion).

OH^- is a hydroxide ion.

For our purposes, no important details are lost by representing...

• ...H₃O⁺ as H⁺ (a hydron or, more commonly, a proton)
• ...self-dissociation of one H₂O molecule instead of the reaction of two

That simplifies the reaction to

[FYI, this is only for notational convenience: No bare protons are swimming around in solution. For that matter, even single hydronium ions quickly react with others to form hydrate complexes.]

The double arrows ( $\leftrightarrows$ ) indicate that the reaction is reversible: water molecules dissociate to produce ions and ions combine to form water molecules.

When temperature is constant, the reaction reaches a dynamic equilibrium:

• dynamic (as opposed to static), because the reaction doesn’t stop -- water molecules still break up, driving the reaction to the right while ions still combine, driving the reaction to the left
• equilibrium (or steady state), because the reaction rates are constant -- and that means that the concentration of each component is constant

An equilibrium constant is the ratio of the products of a chemical reaction to the reactants after the reaction has reached dynamic equilibrium.

In the general case, if reactants A and B combine to produce C and D

then the equilibrium constant K is

In our case, the reaction is

so K_w, the equilibrium constant of water (or its dissociation constant) is the ratio of the ions produced, H⁺ and OH^-, to the sole reactant, H₂O

The braces ({}) are one way to represent activity or effective concentration.

In chemistry, activity is not the same as concentration. Two simplifications will take us quickly from generally unfamiliar activity to familiar concentration.

First, the activity of water is given as 1.0, so we can simplify the formula to

Next, the difference between activity and concentration generally is small enough in dilute solutions -- those in which we’re interested -- so that we can use concentration without introducing unacceptable error. This leads to

The “wavy equals sign”, $\approx$ , means “approximately equals”.

A popcorn analogy

A new bag of microwave popcorn might contain 750 un-popped kernels.

Heat that bag in the microwave and, in the ideal case, all 750 kernels will pop. If they do, then we can say that the concentration of popcorn is 750/bag.

But in the real world maybe only 700 of the 750 kernels pop. If so, then the effective concentration of popcorn is lower: only 700/bag.

Chemistry makes a similar distinction for dissolved substances.

Concentration in the chemical sense

Image by Gerd Altmann from Pixabay

We’re all familiar with concentration: Measure a quantity of a substance (maybe 5.0 g), dissolve it in a volume of a solvent (e.g., 2 L of water), and then divide the added amount by the volume -- 5.0 g divided by 2L, or 2.5 g/L.

That’s such a simple, straighforward procedure that it’s natural to wonder why we’d need any other concept.

The key is that concentration, as the term is used in Chemistry, doesn’t account for interactions that the substance has with the solvent (water, in our case) and any other dissolved solids in the solution.

Concentration thus represents the ideal case -- like expecting all the popcorn to pop.

Activity in the chemical sense

Activity -- or effective concentration -- does take those interactions into account. It thus represents the more complicated non-ideal case that describes how dissolved substances really behave in solution.

The interactions in question commonly involve ion-ion pairing: i.e., a certain fraction of the ions from the dissolved substance react with other ions in the solution instead of remaining free (as we expect in the ideal case). This makes it appear that the solution is less concentrated than expected based on the amount of substance added.

Using the example above, imagine that 8% of the dissolved ions from the added 2.5 g/L interact in some way with other ions already in solution. That 8% then is the popcorn that didn’t pop.

The rest of the added substance -- 92% of 2.5, or 2.3 g/L -- is the effective concentration or activity. That lesser amount is the popcorn that did pop.

The bottomline

Just like the actual number of popped kernels is less than the total number of kernels added to the bag, so the activity of a dissolved substance is less than the amount of substance in the solution.

Concentration is close enough to activity in dilute solutions of the type we run in aquaculture, aquaponics, pool, hydroponics, and aquarium work that we generally can use concentration without unacceptable loss of accuracy or precision.

We still need to be aware that the chemical properties of real solutions are better described by activity than by concentration, and that activity is...

• ...more important in concentrated solutions
• ...more important in solutions with high ionic strength (like seawater)
• ...essential for any precise chemical work

We’ll mention activity again shortly when we define pH because pH strictly is defined in terms of activity, not concentration; and we go into a bit more detail farther along when we discuss pH scales and measuring pH in seawater.

[FYI, a similar distinction is made with gasses: partial pressure represents the ideal case; fugacity (effective partial pressure), the non-ideal case. For our purposes, we’re usually good-to-go using partial pressure.]

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After those simplifications, the dissociation constant of water is

The brackets ([]) denote concentration in moles per kg of water (mol/kg or m, termed molality) or moles per liter of water (mol/L or M, called molarity).

[FYI, oceanographers simplify certain analyses by using per kg instead of per liter because a mass of water (in kg) doesn’t change with temperature or pressure while its volume (e.g. in liters) does.]

It’s important to be comfortable with the concept of moles because you’ll encounter it very frequently in water quality work.

As explained in the following panel, a mole is a measure of abundance; it’s the number of “things” in a collection. Just like one dozen refers to a collection of 12 things, one mole refers to a collection of a very, very, very large number of things.

Image by Dirk (Beeki®) Schumacher from Pixabay

Why do we need moles?

Chemical reactions describe how substances interact in terms of the numbers of their reactants and products.

For example, one molecule of sodium carbonate dissolved in water produces two sodium ions and one carbonate ion:

$\color{blue}{1}\color{black}Na_2CO_3 \leftrightarrows {\color{blue}{2}\color{black}Na^+} + \color{blue}{1}\color{black}{CO_3^{-2}}$

Similarly, one molecule of H₂O produces one H⁺ and one OH^-:

$\color{green}{1}\color{black}H_2O \leftrightarrows \color{green}{1}\color{black}{H^+} + \color{green}{1}\color{black}{OH^-}$

And 5 dozen (60) H₂O molecules produce 5 dozen H⁺ and 5 dozen OH^-:

$\color{green}{60}\color{black}H_2O \leftrightarrows \color{green}{60}\color{black}{H^+} + \color{green}{60}\color{black}{OH^-}$

But molecules and ions are so physically small that such small quantities aren’t practical in water-quality management.

To adjust pH in a pool or alkalinity in a hydroponics tank, we need to work with much larger amounts.

A mole is such an amount.

6.02214076 x 10²³...things

A pair is a collection of two; a dozen, 12; the archaic score (”...four score and seven years ago...”), 20; a ream (of paper), 500. A mole is also a collection -- a collection of...

602,214,076,000,000,000,000,000

In words:

six hundred two sextillion, two hundred fourteen quintillion, seventy-six quadrillion

And written compactly in scientific notation:

6.02214076 x 10²³

How big is really big?

A mole is too big of a number to grasp; it’s just well outside the realm of our experience. Intro Chem classes approach this matter in fanciful ways.

Image by Larry White from Pixabay

• one mole of donuts would cover the Earth to a depth of about 8 km (5 miles)
• a human contains an average of ~3.72 x 10¹³ cells [2], and with 7,800,000,000 (7.8 billion, 2020) people on Earth, that’s less than 1/2 mole of human cells
• if one mole of dollars were divided equally among each of the 7.8 billion people of Earth, and if a person counted their bills at the rate of one per second, it would take almost 2.5 million years to complete the task

Just too big to grasp.

The standard unit of abundance

The mole (mol) is one of the seven fundamental units in the International System of Units (Système international d’unités or SI).

It previously was defined as the number of atoms in 0.012 kg of carbon-12 (Avogadro’s Number, named for the early 19th century Italian scientist Amedeo Avogadro).

But the kilogram was re-defined and so was the number of items in a mole.

On 20 May 2019, one mole was defined officially as the amount of a substance with exactly 6.02214076 × 10²³ things -- compounds, molecules, ions, donuts...

[FYI, 23 October, between 6:02 AM and 6:02 PM (6:02 10/23 in some time-date formats) is Mole Day.]

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[H⁺] determines a solution’s acidity and we can yank it out of the K_w formula:

We start with the fact that water is electrically neutral. In other words:

positve charges from H⁺ must balance negative charges from OH^-

That means the concentration of H⁺ ions equals the concentration of OH^- ions:

Because the ion concentrations are equal, we can substitute [H⁺] for [OH^-]:

Now some Algebra. Write the product of the [H⁺] concentrations as a square:

Exchange the two members, just because we’re used to the unknown on the left:

And take the square root of both sides to get an expression for [H⁺]:

[H⁺], the concentration of H⁺ ions in moles per liter (mol/L), measures a solution’s acidity. We’re close to turning that into pH, but let’s first get some K_w values to illustrate how this is done.

K_w can be measured experimentally. We don’t need the details of how that’s done, we just need some K_w values to play with.

K_w depends on temperature, pressure, and dissolved substances (i.e., salinity). The pressure effect is relatively small so we’ll stick with sea-level pressure.

Formulae that give K_w as a function of temperature and salinity have been developed. We’ll first focus on temperature dependence.

Get an idea of the temperature effect on K_w and [H⁺] by playing with the slider.

As temperature increases, note that K_w and [H⁺] increase.

The reason: Higher temperatures drive the reaction to the right, increasing the dissociation of water...

...and producing higher concentrations of H⁺ (and OH^-).

Fair enough, but those [H⁺] numbers are filled with so many zeros that, as-is, they’re impractical to work with.

We deal with that in the next section where we finally get to pH.