Calculate pH of 10-7 M NaOH
Use this premium chemistry calculator to find the exact pH, pOH, hydroxide concentration, and hydrogen ion concentration for dilute sodium hydroxide solutions. For very dilute bases like 10-7 M NaOH, the exact answer is slightly above 7 because water autoionization matters.
Result preview
Enter your values and click Calculate pH. For the standard case of 1 × 10-7 M NaOH at 25 C, the exact pH is about 7.21, not exactly 7.00.
How to calculate pH of 10-7 M NaOH correctly
If you want to calculate pH of 10-7 M NaOH, the most important idea is this: the usual quick shortcut for strong bases is not accurate enough at such a low concentration. Many students first assume that sodium hydroxide is a strong base, so 10-7 M NaOH should provide 10-7 M OH–, which would give pOH = 7 and therefore pH = 7 at 25 C. That seems tidy, but it misses a major source of hydroxide and hydrogen ions already present in water.
Pure water at 25 C autoionizes slightly into H+ and OH–. In neutral water, each is present at 1.0 × 10-7 M. That means when your added NaOH concentration is also 1.0 × 10-7 M, the hydroxide from the base is on the same order of magnitude as the hydroxide generated by water. In other words, the approximation that ignores water autoionization is no longer reliable.
Bottom line: at 25 C, the exact pH of 1 × 10-7 M NaOH is about 7.21, not 7.00 and not 8.00. The solution is slightly basic, but only slightly.
The chemistry behind the calculation
NaOH is a strong base and dissociates essentially completely:
NaOH → Na+ + OH–
If the concentration is C = 1.0 × 10-7 M, then the sodium ion concentration is also 1.0 × 10-7 M. However, the total hydroxide concentration in solution is not simply equal to C. Water contributes additional ions through the equilibrium:
H2O ⇌ H+ + OH–
At 25 C, the ion product of water is:
Kw = [H+][OH–] = 1.0 × 10-14
Because NaOH adds hydroxide, charge balance tells us:
[OH–] = C + [H+]
Substitute that into the Kw expression:
[H+](C + [H+]) = Kw
Let h = [H+]. Then:
h2 + Ch – Kw = 0
Now use the quadratic formula with C = 1.0 × 10-7 and Kw = 1.0 × 10-14:
h = (-C + √(C2 + 4Kw)) / 2
h = (-(1.0 × 10-7) + √(1.0 × 10-14 + 4.0 × 10-14)) / 2
h = (-(1.0 × 10-7) + √(5.0 × 10-14)) / 2
h ≈ 6.18 × 10-8 M
Then:
pH = -log(6.18 × 10-8) ≈ 7.21
The corresponding hydroxide concentration is:
[OH–] = Kw / [H+] ≈ 1.62 × 10-7 M
So the solution is definitely basic, but not nearly as basic as the rough approximation would suggest.
Why the common shortcuts fail
There are two common wrong answers for this problem. The first wrong answer is pH = 8.00. That comes from assuming [OH–] = 1.0 × 10-7 M from NaOH, then using pOH = 7 and pH = 14 – 7 = 7. Actually, that arithmetic itself would lead to pH = 7, not 8, so students often mix up the acid and base relationships while already using a weak assumption. Another frequent mistake is to say the pH must be exactly 7.00 because the concentration is 10-7. That also ignores the fact that adding NaOH shifts the water equilibrium.
The proper conclusion is subtle but important: because the added base is very dilute, the pH increases only slightly above neutral. The exact pH must be above the neutral pH for that temperature, but only by a modest amount.
When you can safely ignore water autoionization
In many textbook problems, the concentration of strong acid or strong base is high enough that water contributes a negligible amount. For example, with 1.0 × 10-3 M NaOH at 25 C, the water contribution of about 1.0 × 10-7 M is only 0.01 percent of the added hydroxide. In that case, the approximation works beautifully. But for 1.0 × 10-7 M NaOH, the added OH– is on the same scale as water itself, so the approximation breaks down badly.
| NaOH concentration at 25 C | Approximate pH if water is ignored | Exact pH including water | Difference |
|---|---|---|---|
| 1 × 10^-3 M | 11.00 | 11.00 | Less than 0.01 pH unit |
| 1 × 10^-5 M | 9.00 | 9.00 | About 0.00 pH unit |
| 1 × 10^-6 M | 8.00 | 8.00 | Small |
| 1 × 10^-7 M | 8.00 by the simple pOH shortcut, but conceptually inconsistent | 7.21 | Large and important |
| 1 × 10^-8 M | 7.00 to 8.00 depending on mistaken shortcut | 7.02 | Very large relative error |
This table shows the key pattern. At higher concentrations, the exact and approximate methods converge. Near 10-7 M and below, the exact calculation becomes essential.
Step by step method for students
- Write the strong base dissociation: NaOH → Na+ + OH–.
- Set the analytical concentration of NaOH equal to C.
- Use the water equilibrium: Kw = [H+][OH–].
- Use charge balance for a dilute strong base: [OH–] = C + [H+].
- Substitute into Kw and solve the quadratic h2 + Ch – Kw = 0.
- Take the physically meaningful positive root for [H+].
- Calculate pH = -log[H+].
That sequence works not only for 10-7 M NaOH, but also for any very dilute strong acid or strong base where water autoionization is not negligible.
Exact answer at 25 C
- NaOH concentration, C = 1.0 × 10-7 M
- Kw = 1.0 × 10-14
- [H+] ≈ 6.18 × 10-8 M
- [OH–] ≈ 1.62 × 10-7 M
- pOH ≈ 6.79
- pH ≈ 7.21
Effect of temperature on the answer
Another advanced point is temperature. Neutral pH is not always 7.00. The value 7.00 is specific to 25 C because Kw changes with temperature. As Kw increases, the neutral pH decreases. A solution can still be neutral even when its pH is below 7 if the temperature is high enough.
That matters because the exact pH of 10-7 M NaOH depends on the selected temperature. The calculator above includes several common Kw values so you can compare the result across temperatures.
| Temperature | Kw | Neutral pH | Exact pH of 1 × 10^-7 M NaOH |
|---|---|---|---|
| 0 C | 1.15 × 10^-15 | 7.47 | 7.64 |
| 25 C | 1.00 × 10^-14 | 7.00 | 7.21 |
| 50 C | 5.48 × 10^-14 | 6.63 | 6.79 |
| 75 C | 2.95 × 10^-13 | 6.27 | 6.41 |
The trend is clear. As temperature rises, the neutral pH shifts downward because Kw increases. The solution remains slightly basic relative to neutrality at that temperature, but the absolute pH number is lower.
Exam strategy and conceptual checks
If this problem appears on homework, an AP chemistry worksheet, a general chemistry exam, or an engineering fundamentals course, it often tests whether you can recognize when assumptions fail. Here are practical checks you can use:
- If the strong acid or strong base concentration is near 10-7 M at 25 C, think about water autoionization immediately.
- If your answer says 10-7 M NaOH is exactly neutral, that is physically suspicious because NaOH is a base.
- If your answer says the solution is strongly basic, that is also suspicious because the concentration is extremely small.
- The exact result should be only a little above the neutral pH.
For 10-7 M NaOH at 25 C, a reasonable answer should land near 7.2. That passes the chemistry sanity check: the solution is basic, but only slightly.
Common mistakes to avoid
- Using pH + pOH = 14 without noting temperature. At non-25 C, the correct relationship uses pKw, not always 14.
- Ignoring water autoionization at very low concentrations.
- Forgetting that NaOH is fully dissociated, so sodium is a spectator and hydroxide is the active base species.
- Choosing the wrong quadratic root. The physically meaningful hydrogen ion concentration must be positive.
Authoritative references for deeper study
If you want to verify the science behind this calculation, these sources are useful and trustworthy:
- USGS Water Science School: pH and Water
- National Institute of Standards and Technology, NIST
- University of Wisconsin Department of Chemistry
Final takeaway
To calculate pH of 10-7 M NaOH, do not rely on the simplest strong-base shortcut. Because the solution is so dilute, water autoionization contributes significantly to the ion balance. The exact method combines NaOH dissociation with Kw and solves a quadratic equation. At 25 C, the correct answer is approximately pH = 7.21. That is the value you should remember for the standard version of this problem.
If you want quick, accurate work for other dilute strong bases, use the calculator above. It shows both the exact and approximate interpretations, displays the numerical ion concentrations, and draws a visual comparison chart so you can immediately see how much the simplifying assumption changes the answer.