Calculate the pH in 0.100 m Hippuric Acid
Use this premium weak-acid calculator to determine the pH of a hippuric acid solution from concentration and dissociation data. The default setup is preloaded for 0.100 m hippuric acid, commonly approximated as 0.100 M for dilute aqueous calculations, and it solves the equilibrium using the exact quadratic method.
How to Calculate the pH in 0.100 m Hippuric Acid
To calculate the pH in 0.100 m hippuric acid, you treat hippuric acid as a weak monoprotic acid and solve its acid dissociation equilibrium in water. The key relation is the acid dissociation constant, Ka, or equivalently its logarithmic form, pKa. For hippuric acid, a commonly cited value near room temperature is a pKa of about 3.62, corresponding to Ka of approximately 2.40 × 10-4. If the concentration is 0.100 and the solution is dilute enough that molality and molarity are close, the exact equilibrium treatment gives a pH near 2.32.
Many students first try to solve weak-acid problems by assuming complete dissociation, which would be incorrect here. Hippuric acid is not a strong acid. It ionizes only partially, so the hydrogen ion concentration must be found from equilibrium, not simple stoichiometry. That is why an exact weak-acid calculator is useful. It bridges the gap between the chemical identity of the solute and the actual pH observed in solution.
What Is Hippuric Acid?
Hippuric acid is an aromatic carboxylic acid derived from benzoic acid chemistry and amino acid metabolism. Structurally, it contains a carboxylic acid group, and that acidic proton is what participates in the equilibrium with water. In aqueous solution, the acid dissociation can be written as:
HA + H2O ⇌ H3O+ + A–
Here, HA stands for hippuric acid and A– is the hippurate ion. Because this equilibrium lies mostly to the left, only a fraction of the dissolved hippuric acid donates its proton. The pH is therefore determined by the small but important concentration of hydronium produced at equilibrium.
Step-by-Step Weak Acid Setup
- Start with the formal concentration of hippuric acid. In this problem, it is 0.100 m, which is often approximated as 0.100 M for routine aqueous pH calculations.
- Use the acid dissociation constant. If pKa = 3.62, then Ka = 10-3.62 ≈ 2.40 × 10-4.
- Set up an ICE table for the equilibrium.
- Let x = [H+] formed at equilibrium.
- Write the Ka expression: Ka = x2 / (C – x), where C is the initial acid concentration.
- Solve either by approximation or by the exact quadratic equation.
- Compute pH = -log10[H+].
ICE Table for 0.100 Hippuric Acid
For a monoprotic weak acid with initial concentration C = 0.100:
- Initial: [HA] = 0.100, [H+] = 0, [A–] = 0
- Change: [HA] decreases by x, [H+] increases by x, [A–] increases by x
- Equilibrium: [HA] = 0.100 – x, [H+] = x, [A–] = x
The dissociation constant expression is:
Ka = x2 / (0.100 – x)
Substituting Ka = 2.40 × 10-4 gives:
2.40 × 10-4 = x2 / (0.100 – x)
Rearranging into quadratic form:
x2 + (2.40 × 10-4)x – (2.40 × 10-5) = 0
Solving for the positive root yields x ≈ 4.78 × 10-3 M. Therefore:
pH = -log(4.78 × 10-3) ≈ 2.32
Quick answer: If you calculate the pH in 0.100 m hippuric acid using pKa = 3.62 and the exact weak-acid equation, the pH is approximately 2.32.
Why the Approximation Is Close but Not Perfect
In many introductory chemistry courses, weak acids are estimated with the approximation x << C, so the denominator becomes just the initial concentration. That gives:
x ≈ √(KaC)
For hippuric acid at 0.100 concentration:
x ≈ √[(2.40 × 10-4)(0.100)] ≈ 4.90 × 10-3
This produces a pH of about 2.31, which is very close to the exact answer. However, the exact solution is still better because the percent ionization is several percent, not vanishingly small. In serious analytical work, the exact value should be preferred.
Percent Ionization
Percent ionization is another helpful check:
% ionization = (x / C) × 100
Using x ≈ 4.78 × 10-3 and C = 0.100:
% ionization ≈ 4.78%
That value is low enough that hippuric acid is clearly weak, but high enough that exact treatment is a good practice.
Comparison Table: Hippuric Acid Versus Other Common Weak Acids at 0.100 M
The table below gives a sense of where hippuric acid sits among familiar weak acids. These pH values are computed with the exact quadratic approach at 25 C using commonly cited pKa values.
| Weak acid | Typical pKa | Ka | Initial concentration | Exact [H+] | Calculated pH |
|---|---|---|---|---|---|
| Acetic acid | 4.76 | 1.74 × 10-5 | 0.100 M | 1.31 × 10-3 M | 2.88 |
| Benzoic acid | 4.20 | 6.31 × 10-5 | 0.100 M | 2.48 × 10-3 M | 2.61 |
| Hippuric acid | 3.62 | 2.40 × 10-4 | 0.100 M | 4.78 × 10-3 M | 2.32 |
| Formic acid | 3.75 | 1.78 × 10-4 | 0.100 M | 4.13 × 10-3 M | 2.38 |
This comparison shows that hippuric acid is stronger than acetic acid and benzoic acid, and of similar order to formic acid. That is why its pH at 0.100 concentration falls in the low 2 range rather than the upper 2 or low 3 range.
Effect of Concentration on pH
Because weak-acid ionization is concentration dependent, the pH changes noticeably when the formal concentration changes. At lower concentration, the acid dissociates to a greater fraction, but the total hydrogen ion concentration is still usually smaller. At higher concentration, the total [H+] rises, even though the percent ionization often falls slightly.
| Hippuric acid concentration | Ka used | Exact [H+] | Calculated pH | Percent ionization |
|---|---|---|---|---|
| 0.010 M | 2.40 × 10-4 | 1.44 × 10-3 M | 2.84 | 14.4% |
| 0.050 M | 2.40 × 10-4 | 3.35 × 10-3 M | 2.47 | 6.70% |
| 0.100 M | 2.40 × 10-4 | 4.78 × 10-3 M | 2.32 | 4.78% |
| 0.200 M | 2.40 × 10-4 | 6.81 × 10-3 M | 2.17 | 3.40% |
These numbers reveal two useful trends. First, pH drops as concentration rises. Second, percent ionization decreases as concentration rises. Those two ideas often seem contradictory when students first see them, but they are fully consistent with weak-acid equilibrium behavior.
0.100 m Versus 0.100 M: Does It Matter?
The original wording says 0.100 m, where lower-case m normally means molality. Strictly speaking, molality and molarity are different units. Molality is moles of solute per kilogram of solvent, while molarity is moles of solute per liter of solution. In very dilute aqueous solutions, especially around room temperature, the numerical difference between 0.100 m and 0.100 M is often small enough that classroom calculations treat them as approximately interchangeable. This calculator does the same by default, but it labels the input clearly so you can document the assumption.
In high-precision physical chemistry, this distinction matters more. Activity coefficients, ionic strength, density, and temperature can all influence the relationship between formal concentration and effective acidity. For most general chemistry and many analytical chemistry exercises, however, using 0.100 as the working concentration is acceptable unless the problem explicitly requires activities or density corrections.
Common Mistakes When Solving This Problem
- Using pKa directly in the equilibrium expression instead of converting it to Ka.
- Assuming the acid is strong and setting [H+] = 0.100.
- Forgetting that hippuric acid is monoprotic, so only one acidic proton is released in the weak-acid model.
- Mixing up molality and molarity without stating the approximation used.
- Rounding too early, which can shift the final pH by a few hundredths.
Best Practice Formula Summary
If you want the most reliable quick method for this problem, use the quadratic equation for a weak monoprotic acid:
x = [-Ka + √(Ka2 + 4KaC)] / 2
Then compute:
pH = -log10(x)
For the standard case:
- C = 0.100
- pKa = 3.62
- Ka = 2.40 × 10-4
- [H+] ≈ 4.78 × 10-3 M
- pH ≈ 2.32
Authoritative References for Further Study
If you want to cross-check acid-base fundamentals or learn more about pH and weak-acid behavior, consult these reliable sources:
- PubChem, National Institutes of Health: Hippuric Acid
- U.S. Environmental Protection Agency: pH Overview
- Purdue University Chemistry: Acids and Bases Review
Final Answer
When you calculate the pH in 0.100 m hippuric acid using a typical pKa of 3.62 and the exact weak-acid equilibrium equation, the solution pH is approximately 2.32. If you use the weak-acid approximation instead, you obtain about 2.31, which is close but slightly less accurate. For coursework, lab writeups, and polished scientific presentation, the exact value is the stronger result.