Library / Solution chemistry, equilibria and kinetics / SCH-19
Solution chemistry, equilibria and kinetics
Differential capacity of a monoprotic buffer
Buffer capacity: the amount of strong base per litre that raises the pH by one unit, at fixed volume.
- Ref
- SCH-19
- Rev
- v1
- Field
- Solution chemistry, equilibria and kinetics
- Result
- β [mol/L per pH]
The buffer capacity β is the amount of strong base per litre that raises the pH by one unit, β = dCb/dpH. For a monoprotic buffer, β = ln 10 [h + Kw/h + Ct Ka h/(Ka + h)²]: the first two terms come from water and strong acid or base, and the third from the buffer pair. The buffer term peaks at pH = pKa, where it equals ln 10 × Ct/4 ≈ 0.576 Ct.
Buffer capacity: the amount of strong base per litre that raises the pH by one unit, at fixed volume. It has contributions from water and from the buffer pair, and the buffer term is greatest when pH = pKa.
Calculator
- Water and free-ion contribution2.303×10−5 mol/L per pH
- Buffer-pair contribution0.05756 mol/L per pH
- Largest buffer-pair contribution (at pH = pKa), ln 10 · Ct/40.05756 mol/L per pH
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The buffer takes 0.05759 mol of strong base per litre for each pH unit, of which 100 % comes from the buffer pair.
- Away from pH = pKa the capacity falls: at one pH unit either side the pair term is about a third of its peak value.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mol/L per pH) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Hydronium concentration at the operating pH | 1e-12 to 1 mol/L | 5.101e-5 | 2.303 | 2.303 | +10% | −0.223 % |
| Ct Total buffer concentration | 0.001 to 2 mol/L | 5.987e-4 | 1.151 | 1.151 | +10% | +10 % |
| Ka Acid dissociation constant | 1e-10 to 1 | 2.533e-5 | 0.05759 | 0.05756 | +10% | −0.227 % |
| Kw Water ion product | 1e-15 to 1e-12 | 0.05759 | 0.05759 | 2.3e-7 | +10% | +4e-7 % |
Sweep of Hydronium concentration at the operating pH
| Hydronium concentration at the operating pH (mol/L) | Buffer capacity (mol/L per pH) |
|---|---|
| 1e-12 | 0.02303 |
| 3.162e-11 | 7.289e-4 |
| 1e-9 | 4.605e-5 |
| 3.162e-8 | 7.244e-4 |
| 1e-6 | 0.01903 |
| 3.162e-5 | 0.0421 |
| 0.001 | 0.00456 |
| 0.03162 | 0.07289 |
| 1 | 2.303 |
Sweep one variable
Variables and how each one contributes
Buffer capacity
What the equation gives you. Everything else feeds it.
Hydronium concentration at the operating pH
The hydronium concentration at the working pH, in mol/L. Raises or lowers the capacity depending on its position relative to Ka. Raising it by 10 % (to 1.1×10−5) moves the result by −0.223 %, from 0.05759 to 0.05746 mol/L per pH.
Water ion product
The water ion product. Raises the water term, which matters only at extreme pH. Raising it by 10 % (to 1.1×10−14) moves the result by +4×10−7 %, from 0.05759 to 0.05759 mol/L per pH.
Total buffer concentration
The total buffer concentration (acid plus base), in mol/L. Raises the buffer term in direct proportion. Raising it by 10 % (to 0.11) moves the result by +10 %, from 0.05759 to 0.06334 mol/L per pH.
Acid dissociation constant
The acid dissociation constant of the buffer. The capacity is greatest when Ka is close to h. Raising it by 10 % (to 1.1×10−5) moves the result by −0.227 %, from 0.05759 to 0.05746 mol/L per pH.
Worked example with the values above
- Water terms: h + Kw/h = 1×10−5 + 1×10−9 = 1.0001×10−5.
- Buffer term: Ct Ka h/(Ka + h)² = 0.025.
- β = ln 10 × (0.02501) = 0.057588 mol/L per pH.
Assumptions and limits
- A monoprotic weak acid and its conjugate base, both at appreciable concentration, in dilute solution.
- Constant volume, with concentration-based constants and negligible activity effects.
Common mistakes
- Adding the strong acid or base to the wrong component.
- Using a buffer more than about one pH unit from its pKa.
- Forgetting that the buffer ratio is of amounts in the same final volume.
Statistics and uncertainty
Buffer pH shifts with temperature, ionic strength and dilution, typically by 0.01 to 0.05 pH units, and the pKa itself carries an uncertainty of about 0.02 to 0.05.
Where it is used
- Preparing buffers and predicting their response to added acid or base.
- Choosing an indicator for a given pH range.
Maths you need
- Henderson–Hasselbalch equation: pH = pKa + log₁₀([A⁻]/[HA])
- Conjugate pair: a weak acid and its base
- Buffer capacity: the base needed per unit pH change