Calculate the pH of a 0.200 M HCN Solution
Use this interactive weak-acid calculator to find pH, hydrogen ion concentration, percent ionization, and equilibrium concentrations for hydrocyanic acid using either the exact quadratic solution or the common weak-acid approximation.
HCN pH Calculator
Ka expression: Ka = ([H3O+][CN–]) / [HCN]
Exact setup: Ka = x2 / (C – x)
Quadratic form: x2 + Ka x – Ka C = 0
Then: pH = -log10(x)
Results
Click Calculate pH to see the equilibrium results for the 0.200 M HCN solution.
Expert Guide: How to Calculate the pH of a 0.200 M HCN Solution
Hydrocyanic acid, HCN, is a classic example of a weak acid used in general chemistry to teach equilibrium, acid dissociation constants, ICE tables, and the relationship between concentration and pH. If your task is to calculate the pH of a 0.200 M HCN solution, the key idea is that HCN does not fully dissociate in water. Unlike strong acids such as HCl, nitric acid, or perchloric acid, only a small fraction of HCN molecules donate a proton to water. That means you cannot simply say the hydrogen ion concentration is 0.200 M. Instead, you must use the acid dissociation constant, Ka, and solve the weak-acid equilibrium.
At 25 C, a commonly used Ka value for HCN is about 6.2 × 10-10. That number is very small, which immediately tells you two important things. First, HCN is a weak acid. Second, the equilibrium concentration of hydrogen ions will be much smaller than the initial concentration of HCN. In practice, this means the pH will be acidic, but not nearly as low as a strong acid at the same formal concentration.
Step 1: Write the acid dissociation reaction
In water, hydrocyanic acid donates a proton according to the equilibrium:
- HCN + H2O ⇌ H3O+ + CN–
For many textbook and calculator problems, the solvent water is omitted from the equilibrium expression because it is a pure liquid with effectively constant activity. The equilibrium constant is therefore written as:
- Ka = ([H3O+][CN–]) / [HCN]
Step 2: Set up an ICE table
An ICE table organizes the concentrations at the start and at equilibrium:
| Species | Initial (M) | Change (M) | Equilibrium (M) |
|---|---|---|---|
| HCN | 0.200 | -x | 0.200 – x |
| H3O+ | 0 | +x | x |
| CN– | 0 | +x | x |
Substituting these equilibrium values into the Ka expression gives:
Ka = x2 / (0.200 – x)
Using Ka = 6.2 × 10-10, you get:
6.2 × 10-10 = x2 / (0.200 – x)
Step 3: Solve for x, which equals [H3O+]
Because HCN is a weak acid, x will be tiny compared with 0.200. In many classroom settings, you are allowed to use the weak-acid approximation:
- 0.200 – x ≈ 0.200
That simplifies the equation to:
- x2 = Ka × C
- x = √(Ka × C)
Insert the values:
- x = √((6.2 × 10-10) × 0.200)
- x = √(1.24 × 10-10)
- x ≈ 1.114 × 10-5 M
Since x is the hydronium ion concentration, [H3O+] ≈ 1.114 × 10-5 M. Now calculate pH:
- pH = -log10(1.114 × 10-5)
- pH ≈ 4.953
So the pH of a 0.200 M HCN solution is about 4.95.
Step 4: Check whether the approximation is valid
The standard rule of thumb is that the approximation is acceptable if x is less than 5 percent of the initial concentration. Here:
- Percent ionization = (x / 0.200) × 100
- Percent ionization ≈ (1.114 × 10-5 / 0.200) × 100
- Percent ionization ≈ 0.00557%
That is far below 5 percent, so the approximation is excellent. If you use the exact quadratic equation, the answer changes by a negligible amount for ordinary coursework and routine calculations.
Exact quadratic solution
If your instructor, exam, or lab requires an exact answer, rearrange the equilibrium expression into quadratic form:
- x2 + Ka x – KaC = 0
Then solve using the quadratic formula:
- x = [-Ka + √(Ka2 + 4KaC)] / 2
For Ka = 6.2 × 10-10 and C = 0.200 M, the exact x remains approximately 1.114 × 10-5 M, giving a pH of about 4.953. This confirms the shortcut result.
Why HCN has a higher pH than strong acids at the same concentration
Compare HCN to a strong acid like HCl. A 0.200 M HCl solution fully dissociates, so [H3O+] is essentially 0.200 M and the pH is about 0.699. In contrast, the 0.200 M HCN solution produces only about 1.114 × 10-5 M hydronium ions, so the pH is much higher at around 4.95. That large difference exists because Ka for HCN is extremely small.
| Solution | Formal concentration | Acid strength behavior | Approximate [H3O+] | Approximate pH |
|---|---|---|---|---|
| HCN | 0.200 M | Weak acid, Ka ≈ 6.2 × 10-10 | 1.114 × 10-5 M | 4.95 |
| Acetic acid | 0.200 M | Weak acid, Ka ≈ 1.8 × 10-5 | 1.90 × 10-3 M | 2.72 |
| HCl | 0.200 M | Strong acid, essentially complete dissociation | 0.200 M | 0.70 |
This table highlights an important chemical principle: concentration alone does not determine pH. The strength of the acid matters enormously. Two solutions can have the same formal concentration but vastly different pH values if one acid dissociates extensively and the other barely dissociates at all.
Is 0.200 m the same as 0.200 M?
Students often notice that some problems use a lowercase m and others use an uppercase M. Strictly speaking, M means molarity, which is moles of solute per liter of solution, while m means molality, which is moles of solute per kilogram of solvent. pH equilibrium calculations are usually set up using molarity. However, in dilute aqueous problems at introductory level, a stated value such as 0.200 m is often treated as effectively similar to 0.200 M if density information is not provided. In professional analytical chemistry, the distinction can matter, but in many textbook contexts the expected path is still the weak-acid equilibrium calculation with concentration near 0.200 mol/L.
Key statistics and reference values relevant to HCN
The chemistry of HCN is important not only in classrooms but also in toxicology, environmental chemistry, and industrial safety. The following reference data help place the calculation in context.
| Property or reference metric | Typical value | Why it matters |
|---|---|---|
| Ka of HCN at 25 C | About 6.2 × 10-10 | Determines the extent of acid dissociation and pH |
| pKa of HCN at 25 C | About 9.21 | Useful for buffer calculations and acid-base comparisons |
| Approximate pH of 0.200 M HCN | About 4.95 | Shows weak acidity despite substantial formal concentration |
| Percent ionization at 0.200 M | About 0.0056% | Confirms that only a tiny fraction dissociates |
Common mistakes when solving this problem
- Treating HCN as a strong acid. If you assume complete dissociation, you would calculate a pH near 0.70, which is completely wrong for HCN.
- Using pKa instead of Ka without conversion. If you are given pKa, you must convert using Ka = 10-pKa.
- Forgetting the square root in the approximation. From x2 = KaC, you must take the square root to get x.
- Reporting too many digits. Chemistry answers should reflect appropriate significant figures.
- Ignoring the 5 percent check in borderline cases. HCN passes this check easily here, but stronger weak acids or more dilute solutions may require the quadratic formula.
How concentration affects pH for weak acids
For a weak acid with fixed Ka, increasing the formal concentration usually lowers pH because more acid is available to dissociate. However, the relationship is not linear the way it is for a strong acid. For weak acids, [H3O+] often scales roughly with the square root of concentration under the usual approximation. That is why changing concentration by a factor of 100 changes the hydronium concentration by about a factor of 10, not 100, for many weak-acid systems.
This behavior explains why HCN can have a relatively moderate pH even when the formal concentration is not small. Its Ka is so low that dissociation remains minimal. The equilibrium strongly favors undissociated HCN over H3O+ and CN–.
Authority sources for chemistry data and safety context
- PubChem, National Institutes of Health (.gov): Hydrogen cyanide compound record
- CDC NIOSH Pocket Guide (.gov): Hydrogen cyanide
- LibreTexts Chemistry (.edu-hosted educational resource): acid-base equilibrium tutorials
Final answer
Using Ka = 6.2 × 10-10 for hydrocyanic acid and an initial concentration of 0.200 M, the hydronium ion concentration is approximately 1.114 × 10-5 M. Therefore:
- pH ≈ 4.95
If your instructor asks for an exact quadratic solution, you will get essentially the same result. If your course allows the weak-acid approximation, it is fully justified here because the percent ionization is only about 0.0056 percent. In other words, almost all HCN remains undissociated at equilibrium.
The calculator above automates this process. You can change the concentration, enter a different Ka value, and compare the exact and approximate methods. That makes it a useful tool not only for this single problem, but also for understanding how weak-acid equilibria behave more broadly across acid-base chemistry.