Calculate the pH of a 0.400 M HCN Solution
Use this interactive weak acid calculator to determine the pH of hydrocyanic acid, also written as HCN, at 0.400 M or any custom concentration and acid dissociation constant. The tool applies the exact quadratic solution and also shows the common square root approximation for comparison.
HCN pH Calculator
Equilibrium Visualization
This chart compares the initial HCN concentration to the equilibrium concentrations of HCN, H+, and CN–. Because HCN is a weak acid, only a very small fraction ionizes in water.
- Reaction: HCN ⇌ H+ + CN–
- Weak acid behavior: Ka is much less than 1
- Expected pH: acidic, but not nearly as low as a strong acid of the same concentration
How to Calculate the pH of a 0.400 M HCN Solution
To calculate the pH of a 0.400 M hydrocyanic acid solution, you need to recognize that HCN is a weak acid. That matters because weak acids do not dissociate completely in water. Unlike a strong acid such as hydrochloric acid, where the hydrogen ion concentration is essentially equal to the initial acid concentration, hydrocyanic acid establishes an equilibrium. The pH must therefore be found from the acid dissociation expression rather than from a simple one step conversion.
Hydrocyanic acid dissociates according to the equilibrium:
The acid dissociation constant expression is:
For HCN, a commonly used value is Ka = 6.2 × 10-10 near room temperature. Since the problem asks for the pH of a 0.400 M HCN solution, we begin with an ICE table, which stands for Initial, Change, Equilibrium.
Step 1: Set Up the ICE Table
Let the amount of HCN that dissociates be x.
Change: [HCN] = -x, [H+] = +x, [CN–] = +x
Equilibrium: [HCN] = 0.400 – x, [H+] = x, [CN–] = x
Substitute these into the equilibrium expression:
Step 2: Solve for x
Because HCN is a weak acid and the Ka is very small, x will be tiny compared with 0.400. In classroom chemistry, the common approximation is to assume that 0.400 – x ≈ 0.400. That simplifies the equation to:
Now solve for x:
x = √(2.48 × 10-10) ≈ 1.575 × 10-5
Since x = [H+], the hydrogen ion concentration is approximately 1.575 × 10-5 M.
Step 3: Convert [H+] to pH
pH = -log(1.575 × 10-5) ≈ 4.80
So the pH of a 0.400 M HCN solution is approximately 4.80.
Step 4: Check the Approximation
It is always good chemistry practice to verify the approximation. The 5 percent rule says that if x is less than 5 percent of the initial concentration, then replacing 0.400 – x with 0.400 is acceptable.
Percent ionization = (1.575 × 10-5 / 0.400) × 100 ≈ 0.00394%
This is far below 5 percent, so the approximation is excellent. In fact, the exact quadratic solution and the simplified method give nearly the same pH to the reported precision.
Exact Quadratic Solution for Better Precision
Although the square root approximation is completely valid here, some instructors, exams, or lab reports may require the exact method. Starting from:
Rearrange into standard quadratic form:
Solving this quadratic gives a positive root extremely close to 1.575 × 10-5 M. The resulting pH is essentially the same, about 4.80. The calculator above uses the exact quadratic method first, then shows the approximation for comparison.
Why HCN Has a Higher pH Than a Strong Acid at the Same Concentration
A 0.400 M strong acid would fully dissociate and produce [H+] ≈ 0.400 M, leading to a pH around 0.40. HCN behaves very differently because its Ka is extremely small. Most of the acid remains in the molecular HCN form, and only a tiny amount converts to H+ and CN–. That is why a 0.400 M HCN solution still has an acidic pH, but not an intensely acidic one.
| Acid | Typical Ka or Strength Note | Initial Concentration | Approximate [H+] | Approximate pH |
|---|---|---|---|---|
| HCN | 6.2 × 10-10 | 0.400 M | 1.58 × 10-5 M | 4.80 |
| Acetic acid | 1.8 × 10-5 | 0.400 M | 2.68 × 10-3 M | 2.57 |
| Hydrochloric acid | Strong acid, nearly complete dissociation | 0.400 M | 0.400 M | 0.40 |
This comparison makes the chemistry very clear. HCN is much weaker than acetic acid and dramatically weaker than hydrochloric acid. Even at the same formal concentration, weak acids can differ enormously in the amount of hydrogen ion they produce.
Important Concepts Behind This Calculation
1. Molarity Sets the Starting Point
The 0.400 M value tells you the formal concentration of HCN placed into solution. It does not tell you the hydrogen ion concentration directly. For weak acids, the hydrogen ion concentration comes from equilibrium and is usually much smaller than the starting concentration.
2. Ka Determines How Far Dissociation Proceeds
The size of Ka indicates acid strength. A very small Ka means the equilibrium lies mostly to the left, favoring undissociated HCN. That is the key reason the pH of the solution is only moderately acidic.
3. Percent Ionization Is Often Tiny for Weak Acids
For this solution, only about 0.00394 percent of the HCN molecules ionize. This tiny fraction validates the approximation and confirms that the weak acid assumption is chemically consistent.
4. Water Autoionization Is Negligible Here
Pure water contributes about 1.0 × 10-7 M H+ at 25 degrees Celsius. Because the HCN solution produces around 1.575 × 10-5 M H+, the contribution from water is negligible in this context.
Worked Summary in Ordered Form
- Write the dissociation reaction: HCN ⇌ H+ + CN–.
- Set up the ICE table using an initial concentration of 0.400 M.
- Use Ka = 6.2 × 10-10.
- Write the equilibrium expression: Ka = x2 / (0.400 – x).
- Apply the weak acid approximation: 0.400 – x ≈ 0.400.
- Solve x = √(KaC) = √[(6.2 × 10-10)(0.400)] ≈ 1.575 × 10-5.
- Compute pH = -log(1.575 × 10-5) ≈ 4.80.
- Check percent ionization and confirm the approximation is valid.
Common Mistakes Students Make
- Treating HCN as a strong acid. This gives a wildly incorrect pH near 0.40 instead of 4.80.
- Using the wrong Ka. Different acids require different constants, and even small numerical mistakes can affect the answer.
- Forgetting the square root. From x2 = KaC, you must take the square root to solve for x.
- Skipping the approximation check. The 5 percent rule should be verified when using a simplified equilibrium expression.
- Using natural log instead of base 10 log. pH uses the common logarithm, log base 10.
Data Table: How pH Changes with HCN Concentration
The concentration of a weak acid affects pH, but not in a simple one to one linear way. For HCN with Ka = 6.2 × 10-10, the approximate pH values below show how pH changes as concentration increases.
| HCN Concentration, M | Approximate [H+], 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 | 0.00787% |
| 0.400 | 1.58 × 10-5 | 4.80 | 0.00394% |
| 1.000 | 2.49 × 10-5 | 4.60 | 0.00249% |
Notice two useful trends. First, pH drops as concentration rises because more acid is present overall. Second, percent ionization actually decreases as concentration increases, which is a classic equilibrium behavior for weak acids.
When to Use the Quadratic Formula
For this specific 0.400 M HCN problem, the weak acid approximation is excellent. However, if the acid were stronger, or if the concentration were much lower, x might not be negligible compared with the starting concentration. In such cases, using the exact quadratic formula is the safest and most defensible method. Many advanced chemistry courses, analytical chemistry settings, and rigorous lab writeups prefer the exact solution, especially when precision matters.
Authoritative Chemistry References
If you want to verify acid strength concepts, pH fundamentals, and equilibrium methods, these sources are excellent starting points:
- U.S. Environmental Protection Agency, pH overview
- Chemistry LibreTexts, university hosted chemistry explanations
- University of Wisconsin acid base tutorial
Final Answer
Using Ka = 6.2 × 10-10 for HCN and an initial concentration of 0.400 M, the calculated hydrogen ion concentration is about 1.575 × 10-5 M, which gives a pH of 4.80. Because the percent ionization is far below 5 percent, the approximation used in the weak acid calculation is fully justified.