Calculate the pH of a 0.2 M Solution of HCN
Use this interactive weak acid calculator to find the pH, hydronium concentration, cyanide ion concentration, percent ionization, and equilibrium composition of a 0.2 M hydrogen cyanide solution. The tool supports both the exact quadratic solution and the common weak-acid approximation.
HCN pH Calculator
Result Summary
Enter your values and click Calculate pH to solve for the pH of the HCN solution.
Expert Guide: How to Calculate the pH of a 0.2 M Solution of HCN
Calculating the pH of a 0.2 M solution of HCN is a classic weak-acid equilibrium problem. Hydrogen cyanide, HCN, is not a strong acid, so it does not fully dissociate in water. That single fact changes the entire approach. Instead of assuming that the hydronium ion concentration is equal to the initial acid concentration, we have to use an equilibrium expression based on the acid dissociation constant, Ka.
If you are solving this in general chemistry, analytical chemistry, or a lab course, the key idea is simple: weak acids establish an equilibrium between undissociated acid and its ions. For HCN, the equilibrium is:
HCN + H2O ⇌ H3O+ + CN-
Because HCN is weak, only a very small fraction of the original 0.2 M solution ionizes. That means the pH is acidic, but not nearly as low as the pH of a 0.2 M strong acid such as HCl. In practice, the pH of a 0.2 M HCN solution is close to 5, not close to 1.
Step 1: Write the acid dissociation expression
The acid dissociation constant for HCN at 25 degrees Celsius is commonly taken as approximately 6.2 × 10-10. The equilibrium expression is:
Ka = [H3O+][CN-] / [HCN]
For an initial HCN concentration of 0.2 M, we set up an ICE table:
- Initial: [HCN] = 0.2, [H3O+] = 0, [CN-] = 0
- Change: [HCN] decreases by x, [H3O+] increases by x, [CN-] increases by x
- Equilibrium: [HCN] = 0.2 – x, [H3O+] = x, [CN-] = x
Substituting into the Ka expression gives:
6.2 × 10-10 = x2 / (0.2 – x)
Step 2: Solve for x
Since HCN is a very weak acid, x will be much smaller than 0.2. That allows the common approximation:
0.2 – x ≈ 0.2
Now the equation becomes:
6.2 × 10-10 = x2 / 0.2
Rearranging:
x2 = (6.2 × 10-10)(0.2) = 1.24 × 10-10
Taking the square root:
x = 1.11 × 10-5 M
This x value is the hydronium ion concentration:
[H3O+] = 1.11 × 10-5 M
Now calculate pH:
pH = -log(1.11 × 10-5) ≈ 4.95
Step 3: Check whether the approximation is valid
One of the most important habits in acid-base chemistry is validating approximations. The 5 percent rule says that if x is less than 5 percent of the initial concentration, the approximation is acceptable.
Here, percent ionization is:
(1.11 × 10-5 / 0.2) × 100 = 0.00555 percent
That is far below 5 percent, so the approximation is excellent. In fact, the exact quadratic solution and the approximation give essentially the same pH at this concentration.
Why HCN gives a relatively high pH compared with strong acids
Students often look at 0.2 M and expect a very low pH, but concentration alone is not enough. Acid strength matters just as much. Strong acids ionize almost completely. Weak acids like HCN ionize only slightly. HCN has a very small Ka, meaning the equilibrium strongly favors the undissociated form.
That is why a 0.2 M HCl solution would have a pH near 0.70, while a 0.2 M HCN solution has a pH near 4.95. The difference is dramatic, even though the formal concentration is the same.
| Acid | Type | Typical Ka at 25 degrees Celsius | Estimated pH at 0.2 M | Key takeaway |
|---|---|---|---|---|
| HCl | Strong acid | Very large, effectively complete dissociation | 0.70 | Nearly all acid molecules ionize. |
| HF | Weak acid | 6.8 × 10-4 | 2.44 | Much stronger than HCN, but still not complete. |
| CH3COOH | Weak acid | 1.8 × 10-5 | 2.72 | Common benchmark weak acid. |
| HCN | Weak acid | 6.2 × 10-10 | 4.95 | Very weak acid with minimal ionization. |
Exact quadratic solution for HCN
Although the approximation works well here, some instructors want the exact method. Start with:
Ka = x2 / (C – x)
Multiply both sides:
Ka(C – x) = x2
KaC – Kax = x2
x2 + Kax – KaC = 0
Using the quadratic formula:
x = [-Ka + √(Ka2 + 4KaC)] / 2
Substitute Ka = 6.2 × 10-10 and C = 0.2:
x ≈ 1.11 × 10-5 M
Then:
- [H3O+] = 1.11 × 10-5 M
- [CN-] = 1.11 × 10-5 M
- [HCN] ≈ 0.1999889 M
- pH ≈ 4.95
For this particular problem, the exact and approximate methods agree to several significant figures. That is useful because it confirms that the chemistry is behaving exactly as expected for a very weak acid.
Percent ionization of a 0.2 M HCN solution
Percent ionization tells you how much of the original acid actually dissociated. It is defined as:
Percent ionization = ([H3O+] at equilibrium / initial acid concentration) × 100
Using the HCN values:
Percent ionization = (1.11 × 10-5 / 0.2) × 100 ≈ 0.0056 percent
That number is tiny. It means more than 99.994 percent of the acid remains as HCN molecules at equilibrium. In other words, the acid is present mostly in its molecular form, not as ions.
| Initial HCN concentration (M) | [H3O+] from approximation (M) | Approximate pH | Percent ionization |
|---|---|---|---|
| 0.010 | 2.49 × 10-6 | 5.60 | 0.0249% |
| 0.050 | 5.57 × 10-6 | 5.25 | 0.0111% |
| 0.100 | 7.87 × 10-6 | 5.10 | 0.0079% |
| 0.200 | 1.11 × 10-5 | 4.95 | 0.0056% |
| 0.500 | 1.76 × 10-5 | 4.75 | 0.0035% |
The table shows two useful trends. First, as concentration increases, pH decreases because more hydronium ions are present. Second, percent ionization decreases as the acid becomes more concentrated. That is a typical equilibrium effect for weak acids.
Common mistakes when solving HCN pH problems
- Treating HCN as a strong acid. This leads to a wildly incorrect pH around 0.70 for a 0.2 M solution.
- Using pKa incorrectly. If you are given pKa instead of Ka, convert using Ka = 10-pKa.
- Forgetting the ICE table. Without tracking changes, it is easy to misplace x terms.
- Skipping the approximation check. Even if the approximation is likely valid, it should be justified.
- Rounding too early. Keep extra digits until the final pH step.
When to use the approximation and when to use the quadratic formula
For weak acids with very small Ka values and moderate concentrations, the approximation is usually fine. HCN at 0.2 M is a perfect example. However, if the acid is less weak, or if the concentration is very low, then x may no longer be negligible compared with the initial concentration. In those cases, the exact quadratic solution is safer.
The interactive calculator above lets you switch between the two methods instantly. For the 0.2 M HCN problem, you will notice that the pH barely changes, which is evidence that the weak-acid shortcut is justified.
Real-world chemistry context for hydrogen cyanide
Hydrogen cyanide is important in both industrial chemistry and toxicology. In water, it behaves as a weak acid, but its danger in real life is not primarily due to acidity. Its major hazard comes from cyanide toxicity. That means a pH calculation is useful for equilibrium chemistry and solution speciation, but it should not be confused with a safety assessment.
For reliable safety and chemical background information, see these authoritative resources:
- PubChem, U.S. National Library of Medicine (.gov)
- U.S. Environmental Protection Agency cyanide information (.gov)
- CDC and NIOSH cyanide resources (.gov)
Short answer for exams and homework
If you need a clean, compact write-up, you can present it like this:
- Write the dissociation: HCN + H2O ⇌ H3O+ + CN-
- Use Ka = [H3O+][CN-]/[HCN] = 6.2 × 10-10
- Let x = [H3O+] produced, so Ka = x2/(0.2 – x)
- Since x is small, use 0.2 – x ≈ 0.2
- Solve x = √[(6.2 × 10-10)(0.2)] = 1.11 × 10-5
- Compute pH = -log(1.11 × 10-5) = 4.95
Final conclusion
To calculate the pH of a 0.2 M solution of HCN, you must treat HCN as a weak acid and use its dissociation constant rather than assuming complete ionization. With Ka = 6.2 × 10-10, the hydronium concentration is about 1.11 × 10-5 M, giving a pH of approximately 4.95. The percent ionization is only about 0.0056 percent, confirming that HCN ionizes only slightly in water.
That result makes chemical sense, matches the equilibrium math, and illustrates one of the most important distinctions in acid-base chemistry: concentration tells you how much acid is present, but Ka tells you how strongly that acid actually donates protons.