C1V1 = C2V2 is probably the most-used equation in any teaching lab. It is also the one most often applied where it does not hold, because it looks like arithmetic when it is actually a physical claim with four assumptions folded into it.
The clearest way to see the limits is to dilute two beakers by exactly the same factor and watch them disagree completely. Take a strong acid and a buffer, add nine parts water to one part of each, and the pH of the first moves by exactly one unit while the pH of the second does not move at all. Same dilution, same tenfold factor, opposite outcomes — and both are correct.
What the dilution equation actually assumes
The equation says the amount of solute is conserved: concentration times volume before equals concentration times volume after. That is true, and it is not the whole story, because using it assumes four further things.
First, it assumes the solute does not react with the diluent. Second, that volumes add — pour 50 mL into 50 mL and you get 100 mL. Third, that you are using the same concentration unit on both sides, which is less obvious than it sounds. Finally, it assumes that concentration is the quantity you care about, rather than a property derived from it.
The volume assumption fails more often than students are told. Concentrated sulfuric acid and water contract on mixing: the final volume is measurably less than the sum, which is why you make up to a mark in a volumetric flask rather than adding a calculated volume of water. The fourth assumption is the interesting one, and it is where the two beakers part company.
A strong acid, diluted tenfold
A strong acid is fully dissociated, so the hydrogen-ion concentration is the acid concentration, and pH is just its negative logarithm. Divide the concentration by ten and the logarithm shifts by one:
| Concentration | pH |
|---|---|
| 0.01 M HCl | 2.00 |
| 0.001 M HCl | 3.00 |
| 0.0001 M HCl | 4.00 |
Exactly one unit per tenfold dilution, because that is what a base-ten logarithm does. This predictable shift is what most people expect from dilution, which is why the buffer result is so surprising.
A buffer, diluted tenfold
A buffer is a weak acid and its conjugate base together, and its pH is given by the Henderson-Hasselbalch relation: pH equals the pKa plus the logarithm of the ratio of base to acid. The word doing the work is ratio.
Dilution divides both the base and the acid by the same number. A ratio of one to one stays one to one. So the logarithm term does not move, and neither does the pH:
The strong acid moved by exactly 1.00 pH units over the same dilution. The buffer moved by exactly 0.00. This explains why a recipe specifies a buffer's concentration when the pH is what you wanted: the concentration is not setting the pH.
Then what does the concentration set?
Capacity. A buffer resists pH change because there is a reservoir of acid and base to absorb whatever you add, and diluting the buffer drains that reservoir proportionally without touching the pH it holds. A tenfold dilution leaves the pH where it was and leaves the buffer roughly a tenth as able to defend it.
So the two numbers answer different questions. The ratio sets where the buffer sits. The concentration sets how hard it is to push off that spot. A dilute buffer at exactly the right pH can be worse than useless, because it reads correctly on a meter and then collapses the moment your reaction produces acid.
This is also why the equation has honest limits at both ends. Very concentrated buffers deviate because ions interfere with each other and concentrations stop behaving like activities. Very dilute ones deviate because water's own dissociation stops being negligible. This leads to a classic trap.
The trap at the bottom of the scale
Take hydrochloric acid at 0.00000001 molar — ten to the minus eight. Apply the rule that served us perfectly above, and the pH is 8.
An acid cannot make water alkaline. That answer is obviously wrong, which is what makes this a good test of whether someone is using a formula or merely applying it.
What went wrong is the first assumption of the strong-acid case: that the hydrogen ions come only from the acid. At ordinary concentrations that is a fine approximation, because water contributes about ten to the minus seven molar and the acid contributes far more. At ten to the minus eight the acid is contributing less than the water. Solve it properly, with water's own dissociation included, and the answer is pH 6.98 — very slightly acidic, which is exactly what adding a trace of acid to neutral water should do.
What this does not mean
It does not mean the dilution equation is unreliable. It is exact for what it describes, which is the amount of solute in the flask. Every failure above comes from asking it a question about something else — pH, capacity, or a volume that did not add.
It does not mean Henderson-Hasselbalch is wrong about dilution either. It reports the pH correctly and it is telling you something true and useful: within its working range, a buffer's pH genuinely does not depend on how concentrated it is.
And it does not mean you should stop using either. It means reading what the equation is a statement about before deciding what it can answer, which is a habit rather than a fact and generalises well past chemistry.
Before diluting, ask what needs to stay constant. A simple dilution preserves the amount of solute and a buffer's pH, and destroys its capacity. Make up to the mark in a volumetric flask rather than adding a calculated volume of water, which is what protects you when volumes do not add. Specify a buffer by its ratio AND its concentration: the ratio is the pH you are asking for, the concentration is whether it will hold. Keep the same unit on both sides of the equation, and treat any mix of molarity with percent as an error until proven otherwise. And sanity-check any pH near neutral — if adding acid ever produces a number above 7, the formula has been used outside its assumptions, not the chemistry gone strange.
What this audit found in our own tools
Every guide in this series checks the tools it links. Here the arithmetic came back clean, which is itself worth reporting.
Our pH calculator handles the ten-to-the-minus-eight case correctly, and it does so by scope rather than by cleverness. It is a converter between pH, pOH, hydrogen-ion concentration and hydroxide concentration — you enter one and it derives the other three. Entering a hydrogen-ion concentration of ten to the minus eight and getting pH 8 is not a bug; it is the definition, correctly applied. The trap only bites when someone treats the concentration of acid added as though it were the hydrogen-ion concentration, and the tool never invites that.
The page is missing that warning, however. It carries caveats about activities and ionic strength, but says nothing about the difference between acid concentration and hydrogen-ion concentration — the most common way its input is misread. This is a gap in the explanation, not an error in the arithmetic, and it is flagged for the tool's next review.
The buffer calculator was checked the same way and is likewise sound. It states its limits at high ionic strength and at extreme dilution. It does not mention that dilution leaves the pH alone — the most counterintuitive consequence of the equation it implements, and the reason for this guide.
Run your own numbers
Our molarity and dilution calculator solves C₁V₁ = C₂V₂ for whichever term you are missing, and the molarity calculator gets you there from a mass and a volume. For the two cases in this guide, the pH calculator converts between pH, pOH and the two ion concentrations, and the Henderson-Hasselbalch buffer calculator gives the buffer pH from the ratio — the one that does not move when you dilute it.
- Every figure in this guide was computed rather than quoted: the strong-acid table is the negative base-ten logarithm of the stated concentrations, the buffer figures are the Henderson-Hasselbalch relation at a pKa of 4.76 for acetic acid, and the pH 6.98 result solves the quadratic that includes water's own dissociation at 25 °C. The computations assert that the strong acid moves exactly 1.00 pH units per tenfold dilution, that the buffer moves exactly 0.00, and that the dilute-acid case lands just below 7 rather than above it.
- Definitions and constants follow the same references our pH tool cites: the IUPAC Gold Book definition of pH, NIST's ionisation constant for water, and the CRC Handbook of Chemistry and Physics. The ion product of water is taken as 1 × 10⁻¹⁴ at 25 °C, which is where every number here is quoted.
- One notation difference worth knowing if you are reading across curricula: Chinese-language chemistry writes the base-ten logarithm as "lg", so the definition appears as pH = −lg[H⁺] rather than pH = −log₁₀[H⁺]. Our tools carry both, and they mean the same thing.
This is general chemistry education, not laboratory safety guidance or a protocol. Always dilute concentrated acid by adding acid to water and never the reverse, and follow the procedures and risk assessments in force where you work.
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