Stomach acid has about ten million times more H+ in it than pure water does. Writing numbers like that gets ugly fast, so chemists squash them onto a short, tidy scale using logs. Once you see what the log is actually doing, pH stops being a button on your calculator and starts making sense.
📘 What you need to know
How acidic a solution is depends on its concentration of H+ (or H3O+) ions, measured in mol dm−3.
pH = −log10[H+], and rearranged, [H+] = 10−pH.
The scale is logarithmic to base 10: every whole pH unit is a factor of 10 in [H+].
A lower pH means more H+ and a more acidic solution.
At 298 K: below 7 is acidic, exactly 7 is neutral, above 7 is alkaline.
Give pH answers to 2 decimal places unless the question says otherwise.
Measure it accurately with a pH meter and electrode; universal indicator only gives a rough value.
Why we bother with logs
Real solutions have hydrogen ion concentrations spread over an enormous range. A strong acid might be at 1 mol dm−3 while pure water sits at 0.0000001 mol dm−3. Comparing those two on a graph would be hopeless.
Taking a log to base 10 answers a simple question: how many tens is this? The log of 0.0000001 is −7, because that number is 10−7. Since concentrations below 1 always give negative logs, we stick a minus sign in front to keep the everyday numbers positive. That is the entire trick.
The two forms you must know
pH = −log10[H+] and [H+] = 10−pH
These two are the same equation, just rearranged. If a question gives you a concentration, use the first. If it gives you a pH, use the second. Deciding which one you need before touching the calculator saves a lot of wrong answers.
The 0 to 14 range is a convenience, not a rule. Very concentrated acids can measure below 0 and very concentrated alkalis above 14.
What “ten times” really looks like
Students often read “pH 3 is more acidic than pH 5” and picture a small difference, because 3 and 5 are close together. They are not close at all. Two pH units means a factor of 10 × 10 = 100.
The diagram below shows the same idea with dots. Each dot means the same number of hydrogen ions, and the panels are only one pH unit apart.
Try covering the labels and guessing which panel is stomach acid and which is rainwater. The gap between them is far bigger than the numbers suggest.
Reading the numbers straight off
When the concentration is a neat power of ten, you do not need a calculator at all. The pH is just the power with the sign flipped.
[H+] / mol dm−3
Written as a power of 10
pH
1.0
100
0
0.1
10−1
1
0.01
10−2
2
0.001
10−3
3
0.0000001
10−7
7
0.00000000001
10−11
11
Quick sense check. If [H+] is between 10−3 and 10−4, the pH must land between 3 and 4. Do that estimate before you trust your calculator — it catches typing mistakes instantly.
Measuring pH in the lab
Two methods come up, and the IB wants you to know which is which.
pH meter with a glass electrode. The electrode is dipped into the solution and the meter reads the pH straight off, usually to 2 decimal places. This is the accurate method, and it needs calibrating with buffer solutions first.
Universal indicator paper or solution. The paper changes colour and you match it against a printed chart. Cheap and quick, but you can only read it to the nearest whole number, and coloured solutions ruin it.
If a question asks why a pH meter is better, the answer is not “because it is electronic”. It is because it gives a continuous numerical reading rather than a colour you have to judge by eye, so it is far more precise and it removes personal opinion from the measurement.
Worked examples
WORKED EXAMPLE
A solution has [H+] = 2.50 × 10−4 mol dm−3. Calculate its pH.
Step 1: pick the right form of the equation
We have the concentration and want the pH, so use pH = −log10[H+].
Step 2: estimate first
2.50 × 10−4 sits between 10−4 and 10−3, so the answer must be between 3 and 4.
Step 3: put it in the calculatorpH = −log10(2.50 × 10−4) = 3.6020…pH = 3.60 (2 d.p.)it landed between 3 and 4 exactly as predicted, so the answer is safe
WORKED EXAMPLE
A cleaning solution has pH 11.20. Calculate its hydrogen ion concentration.
Step 1: pick the right form
We have the pH and want the concentration, so use [H+] = 10−pH.
Step 2: estimate first
pH is between 11 and 12, so the answer must be between 10−12 and 10−11.
Step 3: calculate[H+] = 10−11.20 = 6.3095… × 10−12[H+] = 6.31 × 10−12 mol dm−3find the 10ⁿ key on your calculator now, not in the exam hall
WORKED EXAMPLE
25.0 cm3 of hydrochloric acid of pH 2.00 is diluted with water to a total volume of 250.0 cm3. What is the new pH?
Step 1: turn the starting pH into a concentration[H+] = 10−2.00 = 0.0100 mol dm−3Step 2: work out the dilution factor250.0 ÷ 25.0 = 10, so the solution is 10 times more dilute.Step 3: find the new concentration[H+] = 0.0100 ÷ 10 = 1.00 × 10−3 mol dm−3Step 4: convert back to pHpH = −log10(1.00 × 10−3) = 3.00pH = 3.00diluting by 10 always raises the pH of a strong acid by exactly 1
💡 Exam tip
Quote pH to 2 decimal places. Only the digits after the decimal point count as significant figures in a log, which surprises people.
Estimate before you calculate. Knowing the answer sits between 3 and 4 catches a mistyped exponent every time.
For dilution questions, always go pH → concentration → new concentration → pH. Never try to divide the pH itself.
Dilute a strong acid ten times and the pH goes up by 1. A hundred times, up by 2. Learn this shortcut.
Watch your units. Concentration must be in mol dm−3, so convert cm3 to dm3 by dividing by 1000.
If a question mentions temperature, be careful with “pH 7 is neutral”. That is only true at 298 K, and the next page explains why.
⚠ Common mix-up
Forgetting the minus sign. log10(10−3) is −3, so the pH is +3. Miss the minus and every answer comes out negative.
Thinking pH 4 is twice as acidic as pH 2. It is a hundred times less acidic, and the direction is the wrong way round too.
Halving the pH when you halve the concentration. pH is a log, so it does not scale like that at all.
Believing pH cannot go below 0 or above 14. It can. The scale is a convention, not a physical limit.
Using [HCl] instead of [H+]. They are only equal because HCl is strong and gives one proton per molecule. For H2SO4 or a weak acid, they are not.
Reading universal indicator to two decimal places. A colour chart gives you a whole number at best.
Up next: The Ionic Product of Water. Pure water is not quite as inert as it looks — a tiny fraction of it splits into ions, and that single fact is what fixes 7 as the neutral point.
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