Calculate the pH of 1.0 x 10-10 M NaOH
Use this premium calculator to find the correct pH of an extremely dilute sodium hydroxide solution. It accounts for water autoionization, which is essential at very low concentrations.
How to calculate the pH of 1.0 x 10-10 M NaOH correctly
At first glance, this chemistry problem looks simple. Sodium hydroxide is a strong base, so many students immediately assume that its hydroxide concentration equals the stated molarity. If that shortcut were always valid, then for a 1.0 x 10-10 M NaOH solution you would set [OH–] = 1.0 x 10-10 M, calculate pOH = 10, and conclude that pH = 4. That answer is clearly impossible for a sodium hydroxide solution because NaOH is basic, not acidic. The reason the shortcut fails is that the base concentration is so tiny that the contribution of pure water can no longer be ignored.
Pure water self-ionizes slightly to produce H+ and OH–. At 25 C, the ion-product constant for water is Kw = 1.0 x 10-14, and in pure water both [H+] and [OH–] are 1.0 x 10-7 M. That means neutral water already contains one thousand times more hydroxide than the NaOH added in this problem. Because the added base is much smaller than 1.0 x 10-7 M, the calculation must combine the hydroxide from NaOH with the ionization equilibrium of water.
Step by step derivation
Let the formal concentration of NaOH be C = 1.0 x 10-10 M. Because NaOH dissociates completely, it contributes 1.0 x 10-10 M Na+. The total hydroxide concentration is not just C, because water also contributes ions. We therefore need a charge-balance and the equilibrium expression for water.
Since [Na+] = C, this becomes:
Now apply the water equilibrium relation:
Substitute [OH–] = C + [H+] into the Kw expression:
This gives a quadratic equation:
Using the quadratic formula, the physically meaningful root is:
Plug in C = 1.0 x 10-10 and Kw = 1.0 x 10-14:
Evaluating this gives:
Then:
So the correct answer is that the pH of 1.0 x 10-10 M NaOH is approximately 7.02 at 25 C. The solution is only slightly basic, not strongly basic. This makes chemical sense because the base added is tiny compared with the natural ion concentrations in water.
Why the shortcut fails
In many introductory chemistry problems, teachers train students to use the strong-base shortcut:
- Assume complete dissociation.
- Set [OH–] equal to the base concentration.
- Compute pOH = -log[OH–].
- Find pH = 14 – pOH.
This method works well when the base concentration is much larger than 1.0 x 10-7 M. For example, 1.0 x 10-3 M NaOH gives [OH–] dominated by the solute, so water contributes almost nothing in comparison. But at 1.0 x 10-10 M, water contributes far more hydroxide than the solute does. The shortcut effectively ignores the main source of ions in the system, so the result becomes nonsense.
Rule of thumb
- If strong acid or strong base concentration is far above 1.0 x 10-6 M, the shortcut is usually fine.
- If the concentration is near 1.0 x 10-7 M, be cautious.
- If the concentration is below about 1.0 x 10-7 M, include water autoionization explicitly.
Comparison table: naive vs correct method
| Method | Assumed [OH–] | Calculated pOH | Calculated pH | Interpretation |
|---|---|---|---|---|
| Naive shortcut | 1.0 x 10-10 M | 10.00 | 4.00 | Impossible for NaOH because it predicts acidity |
| Corrected equilibrium method | 1.051 x 10-7 M total OH– | 6.98 | 7.02 | Physically meaningful, slightly basic solution |
Data table: pH of dilute NaOH at 25 C
The trend below shows why pH approaches 7 as NaOH becomes extremely dilute. These values are based on the corrected equilibrium treatment using Kw = 1.0 x 10-14 at 25 C.
| NaOH concentration (M) | Naive pH | Corrected pH | Difference |
|---|---|---|---|
| 1.0 x 10-3 | 11.00 | 11.00 | ~0.00 |
| 1.0 x 10-6 | 8.00 | 8.00 | Very small |
| 1.0 x 10-7 | 7.00 | 7.21 | 0.21 |
| 1.0 x 10-8 | 6.00 | 7.02 | 1.02 |
| 1.0 x 10-10 | 4.00 | 7.02 | 3.02 |
Conceptual interpretation of the answer
A corrected pH near 7.02 can surprise learners because they expect every NaOH solution to be obviously basic. The important point is that pH measures total hydrogen ion activity in solution, not just the amount of NaOH added. When the concentration is extremely small, the solution behaves almost like pure water. Adding a minute amount of hydroxide shifts the balance only slightly toward basic conditions. That is why the pH moves just above 7 rather than jumping to a large basic value.
This result also teaches a broader lesson in equilibrium chemistry: approximations have domains of validity. The statement “strong bases fully dissociate” is true, but it is not enough by itself. Full dissociation tells you the amount of sodium ion delivered by the solute, yet it does not automatically imply that all hydroxide in the solution comes from the solute. At ultradilute concentrations, the solvent matters.
Common mistakes students make
- Ignoring water autoionization: This is the biggest error and leads to pH values below 7 for a base.
- Using pH + pOH = 14 without checking assumptions: At 25 C this relation is valid, but only after [H+] and [OH–] are found correctly.
- Equating total OH– only with solute OH–: In very dilute systems, solvent-generated ions matter significantly.
- Forgetting units and exponents: An error in scientific notation can produce answers off by many orders of magnitude.
When this type of problem appears
Questions about 1.0 x 10-10 M NaOH often show up in general chemistry, AP Chemistry, first-year university chemistry, MCAT-style practice, and analytical chemistry introductions. Instructors use them to test whether students understand the limits of idealized shortcuts. These are not trick questions. They are realistic examples of why equilibrium concepts must be applied thoughtfully.
Practical contexts
Ultradilute acid-base calculations matter in environmental chemistry, water purification, analytical blank solutions, and high-purity reagent systems. In such settings, the background contribution of water and dissolved atmospheric carbon dioxide may become relevant. In real laboratory practice, the measured pH of extremely dilute NaOH can deviate from the idealized value because dissolved CO2, instrument limitations, ionic strength effects, and contamination all become significant. Still, for textbook calculations at 25 C, using Kw and the quadratic expression gives the accepted answer.
Authoritative chemistry references
If you want to verify the principles behind this calculation, review authoritative educational and government resources on water chemistry and equilibrium constants:
- LibreTexts Chemistry for acid-base and water autoionization explanations.
- U.S. Environmental Protection Agency for pH fundamentals in water quality.
- NIST Chemistry WebBook for trusted chemistry data resources.
- University of Wisconsin Chemistry for academic acid-base materials.
Final answer
For a textbook problem asking you to calculate the pH of 1.0 x 10-10 M NaOH at 25 C, the correct approach is to include water autoionization and solve the quadratic equilibrium expression. Doing so gives:
This means the solution is slightly basic, which is chemically consistent with sodium hydroxide, even though the concentration is far too small for the usual strong-base shortcut to work.