What Is pH Actually Measuring?

pH is measuring:

How much hydrogen ion concentration exists?

Concentrations can get REALLY small.

Examples:

• 0.1
• 0.01
• 0.001
• 0.0000001

Scientists hated writing all those zeros. So they noticed:

0.1 = 10⁻¹

0.01 = 10⁻²

0.001 = 10⁻³

0.0000001 = 10⁻⁷

And then they asked:

What exponent is attached to the 10?

Sound familiar?

That's a logarithm question.

Can you translate the formulas using Logarithmic knowledge?

Scientists realized if they knew the pH they could find the amount of hydrogen ions present in their sample OR if they knew the amount of hydrogen ions present in their sample they could find the pH, they are connected inversely.

$pH=-\log[H^+]$ and $[H^+]=10^{-pH}$

Remember that logarithms are used to find the exponent (x in this case) when we know the base (8 in this case) and the result (64 in this case):

$64 = 8^x$ what power gives us the result of 64.

also written as:

$x = \log_{8}{64}$ what power gives us the result of 64. (it means the same thing)

If we look at $[H^+]=10^{-pH}$ scientifically speaking, it is telling us the formula for finding hydrogen ions—verbatim it is saying “the amount of hydrogen ions equals 10 to the power of negative pH”. Mathematically speaking it wants us to find the exponent (-pH in this case) when we know the base (10 in this case) and the result (H+ in this case):

$[H^+]=10^{-pH}$ what power gives us the result of $[H^+]$

**keeping in mind, the questions you may solve for this will provide pH or H+**

Lets also remember that logs work in base 10 when the base isn’t specified or visible in the problem (i.e. without a specific base, it is assumed the base is 10, like the base was 8 in the exponent example above $64 = 8^x$ ).

So if we were to read this second formula mathematically:

$pH=-\log[H^+]$

we would say: 10 (base is 10 since base isn’t specified) to what power (pH) equals $[H^+]$ (our hydrogen ion concentration).

Notice something. These are inverse operations. The first formula finds pH using H+ concentration. The second formula finds H+ concentration using pH. They're undoing each other. Which then takes me back to my first statement:

Scientists realized if they knew the pH they could find the amount of hydrogen ions present in their sample OR if they knew the amount of hydrogen ions present in their sample they could find the pH, they are connected inversely.

Let’s Test This Theory:

Suppose:

$[H^+]=10^{-3}$

*I am giving you the hydrogen ions present in our sample. We have $10^{-3}$ hydrogen ions or .001 hydrogen ions.

Plug into pH formula:

(original formula): $pH=-\log[H^+]$

(how we plug in our given hydrogen ions): $pH=-\log(10^{-3})$

The log is asking:

10 raised to what power equals $10^{-3}$?

$10^{x} = 10^{-3}$

Answer: -3

The leave us with our right side of the equation almost solved:

$pH = -(-3)$

Apply the negative sign outside the logarithm.

So:

pH = 3

Notice what happened. The logarithm recovered the exponent because it acted just like an inverse operation should — it undid the math around our unknow value and released it to us.

The Steps Quickly:

Hydrogen concentration $10^{-3}$ contains the growth/decay exponent.

The logarithm extracts it.

The negative sign makes the answer positive.

That’s it.

The Trick Most Students Memorize

Because the log is simply recovering the exponent.

Practice Example 1

Given:

$[H^+]=10^{-5}$

Find pH.

Practice Example 2

Given:

pH = 4

Find hydrogen concentration.

pOH Is The Same Process

$pOH=-\log[OH^-]$ and $[OH^-] = 10^{-pOH}$

Same exact idea.

Just hydroxide concentration instead of hydrogen concentration.

The Magic Relationship

$pH+pOH=14$

This is a great relationship to use when solving:

ex: If you know pH = 3 then pOH = 11 and since pH is less than 7, the solution would be acidic.

Remember:

Low pH = More H⁺ = More Acidic

High pH = Less H⁺ = More Basic

How to Practice

Start here:

Skill 1

Convert:

10⁻³ ↔ pH 3

10⁻⁶ ↔ pH 6

10⁻⁹ ↔ pH 9

until it feels automatic.

Skill 2

Use:

pH + pOH = 14

quickly.

Skill 3

Only then use a calculator for weird values:

pH = -log(0.0032)

or

$[H^+] = 10^{-4.27}$

More Practice Problems:

H⁺ Problems

1. Find the pH

Given:

$[H^+]=1\times10^{-4}$

Find:

• pH
• pOH
• Acidic, Basic, or Neutral?

2. Find the Hydrogen Ion Concentration

Given:

pH = 2

Find:

• $[H^+]$
• pOH

3. Multi-Step H⁺ Problem

Given:

pH = 9

Find:

• $[H^+]$
• pOH
• Acidic or Basic?

OH⁻ Problems

4. Find the pOH

Given:

$[OH^-]=1\times10^{-5}$

Find:

• pOH
• pH
• Acidic, Basic, or Neutral?

5. Find the Hydroxide Ion Concentration

Given:

pOH = 3

Find:

• $[OH^-]$
• pH

6. Multi-Step OH⁻ Problem

Given:

pOH = 10

Find:

• $[OH^-]$
• pH
• Acidic or Basic?

Answer Key

1

$[H^+]$ = $10^{-4}$

pH = 4

pOH = 10

Acidic

2

pH = 2

$[H^+]$ = $10^{-2}$

pOH = 12

Acidic

3

pH = 9

$[H^+]$ = $10^{-9}$

pOH = 5

Basic

4

$[OH^-]$= $10^{-5}$

pOH = 5

pH = 9

Basic

5

pOH = 3

$[OH^-]$ = $10^{-3}$

pH = 11

Basic

6

pOH = 10

$[OH^-]$= $10^{-10}$

pH = 4

Acidic

Bonus Challenge (Calculator Required)

7

Given:

$[H^+]=3.2\times10^{-4}$

Find:

• pH
• pOH
• acidic or basic

8

Given:

$[OH^-]=4.5\times10^{-9}$

Find:

• pOH
• pH
• acidic or basic

Bonus Challenge Answer Key

7. Given [H⁺] = 3.2 × 10⁻⁴

pH = -log(3.2 × 10⁻⁴)

pH ≈ 3.49

pOH = 14 - 3.49

pOH ≈ 10.51

acidic because pH < 7

8. Given [OH⁻] = 4.5 × 10⁻⁹

pOH = -log(4.5 × 10⁻⁹)

pOH ≈ 8.35

pH = 14 - 8.35

pH ≈ 5.65

acidic because pH < 7.