Calculate the pH of a 0.63 m sodium benzonate solution
This calculator estimates the pH of sodium benzoate in water by treating benzoate as a weak base, using the hydrolysis equilibrium C6H5COO– + H2O ⇌ C6H5COOH + OH–. For the common textbook case, 0.63 m is often approximated as 0.63 M when density is not supplied.
How to calculate the pH of a 0.63 m sodium benzonate solution
In most chemistry contexts, the phrase “sodium benzonate” is intended to mean sodium benzoate, the sodium salt of benzoic acid. When sodium benzoate dissolves in water, it separates into sodium ions and benzoate ions. The sodium ion is essentially neutral in this acid-base calculation, but the benzoate ion is the conjugate base of a weak acid, so it reacts slightly with water to form hydroxide. That reaction makes the solution basic, not neutral.
The key equilibrium is:
C6H5COO– + H2O ⇌ C6H5COOH + OH–
Since hydroxide is produced, the pH rises above 7. To calculate the pH correctly, you need the acid dissociation constant of benzoic acid, or equivalently its pKa. A standard textbook value is pKa = 4.20, which corresponds to Ka ≈ 6.31 × 10-5. Once you know Ka, you can find the base dissociation constant of benzoate using:
Kb = Kw / Ka = 1.00 × 10-14 / 6.31 × 10-5 ≈ 1.58 × 10-10
If the concentration is taken as 0.63 M, the weak-base approximation gives:
[OH–] ≈ √(KbC) = √((1.58 × 10-10)(0.63)) ≈ 1.00 × 10-5 M
That leads to:
- pOH = -log(1.00 × 10-5) ≈ 5.00
- pH = 14.00 – 5.00 = 9.00
So the expected pH of a 0.63 m sodium benzoate solution is approximately 9.00 under the common classroom assumption that 0.63 m can be treated as about 0.63 M. The calculator above also lets you refine that estimate by converting molality to molarity if you know the solution density.
Why sodium benzoate gives a basic pH
The chemistry is driven by conjugate acid-base behavior. Benzoic acid is a weak acid, meaning it does not fully ionize in water. Its conjugate base, benzoate, therefore has enough basic character to grab a proton from water. Every time that happens, a hydroxide ion is generated. In a concentrated solution like 0.63 m, there are many benzoate ions present, but because Kb is still very small, only a tiny fraction hydrolyzes. Even so, that tiny fraction is enough to push the pH to about 9.
- Sodium benzoate is a salt of a weak acid and strong base.
- Salts of weak acids usually make aqueous solutions basic.
- The stronger the conjugate base, the higher the pH at the same concentration.
- The higher the concentration, the greater the hydroxide concentration and the higher the pH.
Step by step derivation
A careful pH calculation starts from an ICE table. Let the initial benzoate concentration be C. If x mol/L of benzoate reacts with water, then at equilibrium:
- [C6H5COO–] = C – x
- [C6H5COOH] = x
- [OH–] = x
Substituting into the equilibrium expression:
Kb = x2 / (C – x)
Rearranging gives:
x2 + Kb x – KbC = 0
The exact quadratic solution is:
x = (-Kb + √(Kb2 + 4KbC)) / 2
For this problem, the value of x is so small relative to 0.63 that the shortcut x ≈ √(KbC) is excellent. In fact, the exact and approximate pH values agree to essentially the same reported pH, 9.00.
Exact answer for 0.63 m sodium benzoate
Using pKa = 4.20, we first convert to Ka and Kb:
- Ka = 10-4.20 = 6.31 × 10-5
- Kb = 10-14 / 6.31 × 10-5 = 1.58 × 10-10
Taking C = 0.63 and solving with the exact quadratic:
x = (-1.58 × 10-10 + √((1.58 × 10-10)2 + 4(1.58 × 10-10)(0.63))) / 2
This gives:
- [OH–] ≈ 9.99 × 10-6 M
- pOH ≈ 5.0003
- pH ≈ 8.9997
Rounded to normal reporting precision, the answer is pH = 9.00. This is why many worked examples show a neat pH of 9 for sodium benzoate near this concentration range.
What if 0.63 m really means molality, not molarity?
Molality and molarity are not identical. Molality is moles of solute per kilogram of solvent, while molarity is moles of solute per liter of solution. pH expressions are formally written in terms of concentration or activity, so molarity is often used in introductory equilibrium problems. If density is not known, instructors commonly approximate 0.63 m as 0.63 M. This is generally acceptable for a textbook calculation unless the problem explicitly asks for high-precision conversion.
If you do know solution density, the calculator above converts molality to an estimated molarity using:
M = (1000 × d × m) / (1000 + m × MW)
where d is density in g/mL and MW for sodium benzoate is 144.11 g/mol. This gives a more realistic concentration for the equilibrium calculation.
Comparison table: concentration versus pH
| Sodium benzoate concentration | Kb used | Approximate [OH-] | Approximate pH | Exact pH |
|---|---|---|---|---|
| 0.010 M | 1.58 × 10-10 | 1.26 × 10-6 M | 8.10 | 8.10 |
| 0.10 M | 1.58 × 10-10 | 3.98 × 10-6 M | 8.60 | 8.60 |
| 0.63 M | 1.58 × 10-10 | 9.99 × 10-6 M | 9.00 | 9.00 |
| 1.00 M | 1.58 × 10-10 | 1.26 × 10-5 M | 9.10 | 9.10 |
The table shows an important pattern: increasing the salt concentration raises pH, but not dramatically, because weak-base hydrolysis depends on the square root of concentration. Doubling concentration does not double pH. It changes hydroxide concentration by a square-root factor, so pH shifts gradually.
Comparison table: approximation quality
| Case | Approximation used | Exact [OH-] | Approximate [OH-] | Percent hydrolysis |
|---|---|---|---|---|
| 0.63 M sodium benzoate | x ≈ √(KbC) | 9.99 × 10-6 M | 9.99 × 10-6 M | 0.00159% |
| 0.10 M sodium benzoate | x ≈ √(KbC) | 3.98 × 10-6 M | 3.98 × 10-6 M | 0.00398% |
| 0.010 M sodium benzoate | x ≈ √(KbC) | 1.26 × 10-6 M | 1.26 × 10-6 M | 0.0126% |
These values confirm why the weak-base shortcut is so reliable here. The percent hydrolysis is far below 5%, so subtracting x from the initial concentration makes almost no difference.
Practical interpretation of the result
A pH around 9.00 means the solution is mildly basic. It is nowhere near as alkaline as a strong base like sodium hydroxide, but it is clearly above neutral. In practical chemistry, sodium benzoate solutions are often discussed in food chemistry, buffering systems, and preservative behavior. The equilibrium pH matters because the acid and base forms of benzoic acid participate differently in solubility, preservation efficiency, and analytical chemistry.
Here are the most important takeaways:
- The benzoate ion is a weak base because it is the conjugate base of benzoic acid.
- The pH is determined by Kb and concentration, not by sodium ion chemistry.
- For a 0.63-level concentration, the pH comes out almost exactly 9.00 using standard constants.
- Using the exact quadratic equation changes the answer only in the fourth decimal place.
Common mistakes students make
- Treating sodium benzoate as neutral. It is not neutral because benzoate is the conjugate base of a weak acid.
- Using Ka directly without converting to Kb. For a salt of a weak acid, the hydrolysis is a base reaction, so you need Kb = Kw/Ka.
- Confusing pOH and pH. Once [OH-] is found, calculate pOH first, then convert to pH.
- Mixing up m and M. If the problem gives molality but no density, note the approximation clearly.
- Overcomplicating the algebra. Because Kb is tiny, the square-root shortcut usually works very well here.
When to use the exact quadratic method
The exact quadratic method is best when concentration is low, the equilibrium constant is larger, or your instructor specifically requires exact treatment. For sodium benzoate at 0.63, the approximation is already excellent. Still, advanced courses often prefer the quadratic route because it demonstrates full control over the equilibrium model and avoids relying blindly on the 5% rule.
Authoritative references for pH and weak-acid chemistry
For additional reading on pH, acid-base behavior, and chemical property data, consult these sources:
Final answer summary
For a 0.63 m sodium benzoate solution, using pKa = 4.20 for benzoic acid and the standard approximation that 0.63 m is about 0.63 M, the solution pH is approximately 9.00.
If you want a more refined answer, use the calculator at the top of the page and provide solution density so the molality can be converted to molarity. In normal instructional chemistry settings, however, pH = 9.00 is the accepted and chemically correct result.