📚 pH Curves in Acid-Base Titrations | 酸碱滴定中的pH曲线
A pH curve is a graphical representation of pH versus the volume of titrant added during an acid-base titration. It provides a clear view of the neutralisation process and helps identify key features such as the equivalence point, buffer regions, and suitable indicators. In IB Chemistry, interpreting and sketching pH curves is essential for understanding acid-base equilibria, weak acid/base behaviour, and quantitative analysis.
pH曲线是酸碱滴定中pH值随滴定剂加入体积变化的图像表示。它清晰地展示了中和过程，并有助于识别等当点、缓冲区域和合适的指示剂等关键特征。在IB化学中，解读和绘制pH曲线对于理解酸碱平衡、弱酸/弱碱行为以及定量分析至关重要。
1. Understanding pH and Its Logarithmic Nature | 理解pH及其对数性质
pH is defined as the negative logarithm (base 10) of the hydrogen ion concentration:
pH = –log₁₀[H⁺]
This means a tenfold decrease in [H⁺] raises the pH by one unit. The ionic product of water at 298 K is Kw e is what gives the pH curve its characteristic steep sections near the equivalence point.
pH定义为氢离子浓度的以10为底的负对数。这意味着[H⁺]每减小到原来的十分之一，pH增加一个单位。298 K时水的离子积常数Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴，因此在纯水中[H⁺] = [OH⁻] = 1.0 × 10⁻⁷ mol dm⁻³，pH = 7。正是这种对数关系使pH曲线在等当点附近呈现出特征性的陡峭部分。
2. Titration Setup and the Equivalence Point | 滴定装置与等当点
In a typical acid-base titration, a solution of accurately known concentration (the titrant) is delivered from a burette into a known volume of the analyte in a conical flask. The equivalence point is reached when the amount of titrant added is stoichiometrically equal to the amount of analyte. At this stage, the neutralisation reaction is complete, but the pH of the solution is not necessarily 7 — it depends on the nature of the acid and base.
在典型的酸碱滴定中，将准确已知浓度的溶液（滴定剂）从滴定管滴加到锥形瓶中已知体积的被测溶液中。当加入滴定剂的物质的量与被测物的物质的量按化学计量完全相等时，即达到等当点。此时中和反应恰好完成，但溶液的pH值不一定为7——它取决于酸和碱的性质。
3. Strong Acid – Strong Base Titration Curve | 强酸-强碱滴定曲线
When a strong acid such as HCl is titrated with a strong base like NaOH, the pH starts low (close to 1 for 0.1 M HCl). Adding base causes a slow initial rise in pH. Very close to the equivalence point, the pH increases dramatically — often by 6–8 units with the addition of a single drop. After the equivalence point, the pH rises more gradually as excess strong base accumulates. The equivalence point lies at exactly pH 7 because the salt (NaCl) does not hydrolyse.
当用强碱（如NaOH）滴定强酸（如HCl）时，起始pH很低（0.1 M HCl的pH约为1）。在滴定初期，pH缓慢上升。接近等当点时，pH急剧上升——通常一滴之差pH就会突变6–8个单位。等当点之后，由于过量的强碱积累，pH上升又趋于平缓。等当点恰好位于pH 7，因为生成的盐（NaCl）不发生水解。
A typical data set and curve shape are summarised below:
典型的滴定数据和曲线形状总结如下：
| Volume of NaOH added (cm³) | pH |
|---|---|
| 0.0 | 1.0 |
| 10.0 | 1.2 |
| 24.9 | 3.0 |
| 25.0 (eq. pt.) | 7.0 |
| 25.1 | 10.0 |
| 30.0 | 12.0 |
Table: Approximate pH values for titration of 25 cm³ of 0.1 M HCl with 0.1 M NaOH.
表：用0.1 M NaOH滴定25 cm³ 0.1 M HCl的近似pH值。
4. Weak Acid – Strong Base Titration Curve | 弱酸-强碱滴定曲线
d and its conjugate base forms, causing the pH to change only gradually. The centre of this buffer region is the half-equivalence point, where [HA] = [A⁻] and pH = pKₐ.
当弱酸（如乙酸CH₃COOH）与强碱滴定时，曲线有几个显著区域。由于弱酸仅部分电离，其起始pH比同浓度的强酸高。随着碱的加入，弱酸与其共轭碱形成缓冲混合物，使pH变化平缓。该缓冲区域的中心是半等当点，此时[HA] = [A⁻]且pH = pKₐ。
The Henderson-Hasselbalch equation for a buffer made from a weak acid and its conjugate base is:
pH = pKₐ + log₁₀([A⁻]/[HA])
At the equivalence point, all the weak acid has been converted to the conjugate base, which hydrolyses water to produce OH⁻ ions. Therefore, the pH at the equivalence point is greater than 7. The vertical jump is less steep than in a strong acid–strong base titration, narrowing the range for indicator selection.
在等当点，所有弱酸都已转化为共轭碱，共轭碱水解水产生OH⁻离子，因此等当点的pH大于7。垂直突跃不及强酸-强碱滴定陡峭，指示剂选择范围较窄。
5. Weak Base – Strong Acid Titration Curve | 弱碱-强酸滴定曲线
a strong acid like HCl yields a curve that is essentially the mirror image of a weak acid– NH₄⁺. The equivalence point occurs at a pH below 7 because NH₄⁺ is acidic and hydrolyses to give H⁺. The steep drop in pH marks the end point.
用强酸（如HCl）滴定弱碱（如NH₃氨水）得到的曲线基本上是弱酸-强碱滴定曲线的镜像。起始pH呈中等碱性。体系中形成由弱碱及其共轭酸（NH₄⁺）构成的缓冲区域。在半等当点处，pH = NH₄⁺的pKₐ。等当点出现在pH小于7的位置，因为NH₄⁺呈酸性并水解生成H⁺。pH陡降标志着滴定终点。
6. Buffer Regions and the Half-Equivalence Point | 缓冲区域与半等当点
ong base or acid. Maximum buffer capacity occurs when the concentrations of the weak acid and its conjugate base (or weak base and conjugate acid) are equal — imental determination of pKₐ values directly from a titration curve by noting the pH when half the acid has been neutralised.
pH曲线上的缓冲区域是指加入少量强碱或强酸时pH抵抗大幅变化的区域。当弱酸与其共轭碱（或弱碱与共轭酸）浓度相等时，缓冲容量最大——这正是半等当点。此时log₁₀([A⁻]/[HA]) = 0，故pH = pKₐ。因此，可以通过记录酸被中和一半时的pH值，直接从滴定曲线上实验测定pKₐ。
7. Selecting the Right Indicator | 选择合适的指示剂
The colour change occurs over a range of about pH = pKᵢₙ ± 1, where pKᵢₙ is the indicator' . To obtain an accurate endpoint that closely matches the equivalence point, the indicator' lour change interval must fall entirely within the steep vertical section of the pH curve.
酸碱指示剂本身是一种弱酸或弱碱，其质子化形态和去质子化形态具有不同颜色。变色范围约为pH = pKᵢₙ ± 1，其中pKᵢₙ是指示剂的解离常数。为了获得与等当点吻合的准确终点，指示剂的变色区间必须完全位于pH曲线的陡峭垂直段内。
Common indicators and their suitability are summarised below:
常见指示剂及其适用性总结如下：
| Indicator | pH range | Colour change | Suitable for |
|---|---|---|---|
| Methyl orange | 3.1 – 4.4 | Red to yellow | Strong acid – strong base; Weak base – strong acid |
| Bromothymol blue | 6.0 – 7.6 | Yellow to blue | Strong acid – strong base |
| Phenolphthalein | 8.2 – 10.0 | Colourless to pink | Strong acid – strong base; Weak acid – strong base |
8. Polyprotic Acids and Multiple Equivalence Points | 多元酸与多个等当点
he pH curve then shows multiple steep rises. For H₃PO₄ (pKₐ₁ ≈ 2.15, pKₐ₂ ≈ 7.20, pKₐ₃ ≈ 12.35), titration with NaOH typically reveals two distinct jumps corresponding to the first and second equivalence points. The third proton is very weakly acidic and its equivalence point is often difficult to detect in aqueous solution.
多元酸如磷酸（H₃PO₄）或硫酸（H₂SO₄）具有多个可电离质子。在滴定过程中，只要各级酸解离常数相差足够大（ΔpKₐ ≥ 4），质子会依次被中和，产生独立的等当点，pH曲线因此呈现多个陡升。对于H₃PO₄（pKₐ₁ ≈ 2.15，pKₐ₂ ≈ 7.20，pKₐ₃ ≈ 12.35），用NaOH滴定时通常可观察到两个明显的突跃，分别对应第一和第二等当点。第三个质子酸性极弱，其等当点在水溶液中往往难以检测。
9. Sketching pH Curves from Experimental Data | 根据实验数据绘制pH曲线
To sketch a pH curve, begin by estimating the initial pH from the nature of the analyte. Add titrant in regular increments (e.g., 1.0 cm³), and record the pH after each addition. The region near the expected equivalence point should be investigated with smaller increments (0.1 cm³) to accurately capture the steep change. After plotting volume on the x-axis and pH on the y-axis, draw a smooth curve through the points. The equivalence point can be located by finding the midpoint of the vertical portion or by using a derivative plot (ΔpH/ΔV vs. average volume).
绘制pH曲线时，首先根据被测物质的性质估算起始pH。以固定增量（例如1.0 cm³）加入滴定剂，记录每次加入后的pH。在预期等当点附近应以更小的增量（0.1 cm³）进行滴定，以准确捕捉陡峭变化。将体积作为x轴、pH作为y轴描点，然后用光滑曲线连接各点。可通过找出垂直部分的中点或利用导数图（ΔpH/ΔV对平均体积作图）来确定等当点。
10. Comparison of Titration Curve Features | 滴定曲线特征比较
The table below contrasts the key features of the three main types of monoprotic titration curves.
下表对比了三类主要一元酸碱滴定曲线的关键特征。
| Feature | Strong Acid + Strong Base | Weak Acid + Strong Base | Weak Base + Strong Acid |
|---|---|---|---|
| Initial pH | Low (∼1–2) | Higher (∼3–5) | Moderately high (∼9–11) |
| Buffer region | None significant | Before equivalence; pH = pKₐ at half-equivalence | Before equivalence; pH = pKₐ of conjugate acid at half-equivalence |
| pH at equivalence | = 7 | >
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