pH Calculations in IGCSE CCEA Chemistry | IGCSE CCEA 化学：pH计算考点精讲

📚 pH Calculations in IGCSE CCEA Chemistry | IGCSE CCEA 化学：pH计算考点精讲

pH calculations form a cornerstone of the acids and bases topic in the CCEA IGCSE Chemistry specification. Understanding how to quantify acidity using the logarithmic pH scale is vital, as is being able to move confidently between hydrogen ion concentration and pH for both strong acids and strong bases. This revision guide covers all the essential theory, worked methods, and common exam-style calculations you will need, from the definition of pH right through to the role of the ionic product of water in finding the pH of alkaline solutions.

pH 计算是 CCEA IGCSE 化学课程中酸碱部分的核心内容。掌握如何使用对数形式的 pH 标度来量化酸性至关重要，而能够熟练地在氢离子浓度与 pH 值之间进行换算，无论是针对强酸还是强碱，同样是必考技能。本复习精讲涵盖了你需要掌握的所有基本理论、计算方法以及常见考试题型，从 pH 的定义一直到利用水的离子积来计算碱性溶液的 pH 值。

1. The Concept of Acidity and the Need for pH | 酸度的概念与引入pH的必要性

Acidity is fundamentally about the presence of hydrogen ions, H⁺(aq), in aqueous solution. When an acid dissolves in water, it dissociates to release H⁺ ions. However, the concentrations of H⁺ in typical laboratory acids can vary enormously, from several mol/dm³ down to 0.0000001 mol/dm³ or even lower. Handling such a wide range of tiny numbers with standard decimal notation is awkward and error-prone, which is why chemists adopted the pH scale – a compact, logarithmic way of expressing hydrogen ion concentration.

酸性从根本上讲与水溶液中氢离子 H⁺(aq) 的存在有关。当酸溶于水时，会解离出 H⁺ 离子。然而，实验室常见酸中 H⁺ 的浓度变化幅度极大，可能从几 mol/dm³ 降至 0.0000001 mol/dm³ 甚至更低。用标准十进制小数来处理如此宽且微小的数字既麻烦又容易出错，因此化学家采用了 pH 标度——一种用对数形式简洁表示氢离子浓度的方式。

The pH scale was introduced by the Danish chemist Soren Sorensen in 1909. Rather than saying ‘the hydrogen ion concentration is 1 × 10⁻³ mol/dm³’, we can simply say ‘the pH is 3’. This compression of information is not only convenient but also makes trends in acidity far easier to visualise and compare.

pH 标度由丹麦化学家 Soren Sorensen 于 1909 年提出。我们不必再说“氢离子浓度为 1 × 10⁻³ mol/dm³”，而是可以简单地说“pH 为 3”。这种信息的压缩不仅方便，而且使酸性的变化趋势更容易直观地呈现和比较。

2. The pH Scale from 0 to 14 | 0 到 14 的 pH 标度

The pH scale typically runs from 0 to 14, although values outside this range are possible in extremely concentrated solutions. A neutral solution, such as pure water at 25 °C, has a pH of exactly 7. Acidic solutions have pH values less than 7, with lower numbers indicating stronger acidity. Alkaline (basic) solutions have pH values greater than 7, with higher numbers indicating stronger alkalinity.

pH 标度通常范围为 0 到 14，尽管在极浓溶液中也能出现超出该范围的值。中性溶液（如 25 °C 时的纯水）的 pH 恰好为 7。酸性溶液的 pH 小于 7，数值越低表示酸性越强。碱性溶液的 pH 大于 7，数值越高表示碱性越强。

Description	描述	pH Range
Strongly acidic	强酸性	0–2
Weakly acidic	弱酸性	3–6
Neutral	中性	7
Weakly alkaline	弱碱性	8–11
Strongly alkaline	强碱性	12–14

It is crucial to remember that because the pH scale is logarithmic, a change of one pH unit corresponds to a tenfold change in H⁺ ion concentration. For example, a solution of pH 3 has ten times the concentration of H⁺ ions as a solution of pH 4, and one hundred times that of pH 5.

必须牢记，由于 pH 标度是对数标度，每变化一个 pH 单位，对应的 H⁺ 离子浓度变化十倍。例如，pH 为 3 的溶液，其 H⁺ 浓度是 pH 为 4 溶液的十倍，是 pH 为 5 溶液的一百倍。

3. The Mathematical Definition of pH | pH 的数学定义

The pH of a solution is defined by the equation:

溶液的 pH 由下式定义：

pH = –log₁₀[H⁺]

Here [H⁺] represents the concentration of hydrogen ions in mol/dm³, written using square brackets to denote ‘concentration of’. The logarithm used is to base 10. The negative sign ensures that most everyday pH values are positive numbers. This equation is the key to all pH calculations involving strong acids.

其中 [H⁺] 表示氢离子浓度，单位为 mol/dm³，方括号表示“某物质的浓度”。所用对数为以 10 为底的对数。负号使得日常生活中的 pH 值大多为正数。这个方程是所有涉及强酸的 pH 计算的关键。

On your calculator, you will typically use the ‘log’ button (which is log₁₀) to find pH. For example, if [H⁺] = 0.001 mol/dm³, you would enter 0.001, press log, and then change the sign to get pH = 3. Always be careful with the sign: the negative sign in the formula is part of the definition, not an indication that pH is automatically negative.

在计算器上，你通常会使用 “log” 键（即 log₁₀）来求 pH。例如，若 [H⁺] = 0.001 mol/dm³，你应键入 0.001，按 log，然后改变符号得到 pH = 3。务必注意符号：公式中的负号是定义的一部分，并不表示 pH 值本身为负。

4. Calculating the pH of Strong Acids | 计算强酸的 pH

Strong acids, such as hydrochloric acid (HCl), sulfuric acid (H₂SO₄) and nitric acid (HNO₃), are assumed to dissociate completely in aqueous solution. This means that the concentration of H⁺ ions is directly related to the concentration of the acid, taking account of stoichiometry.

强酸，如盐酸 (HCl)、硫酸 (H₂SO₄) 和硝酸 (HNO₃)，在水溶液中被认为完全解离。这意味着 H⁺ 离子的浓度直接与酸的浓度相关，同时需要考虑化学计量数。

Monoprotic acids like HCl and HNO₃ release one H⁺ per molecule, so [H⁺] = concentration of the acid.
一元酸如 HCl 和 HNO₃，每分子释放一个 H⁺，因此 [H⁺] = 酸的浓度。
Diprotic acids like H₂SO₄ release two H⁺ per molecule, so [H⁺] = 2 × concentration of the acid, assuming both protons dissociate completely.
二元酸如 H₂SO₄，每分子释放两个 H⁺，因此假设两个质子都完全解离，则 [H⁺] = 2 × 酸的浓度。

To calculate pH, simply determine [H⁺] from the acid concentration and stoichiometry, then apply pH = –log[H⁺]. For example, for 0.005 mol/dm³ HCl, [H⁺] = 0.005, and pH = –log(0.005) ≈ 2.3. For 0.005 mol/dm³ H₂SO₄, [H⁺] = 2 × 0.005 = 0.010 mol/dm³, so pH = –log(0.010) = 2.0.

要计算 pH，只需根据酸的浓度和化学计量数确定 [H⁺]，然后应用 pH = –log[H⁺] 即可。例如，对于 0.005 mol/dm³ 的 HCl，[H⁺] = 0.005，pH = –log(0.005) ≈ 2.3。对于 0.005 mol/dm³ 的 H₂SO₄，[H⁺] = 2 × 0.005 = 0.010 mol/dm³，因此 pH = –log(0.010) = 2.0。

Always check that your answer makes sense. A higher H⁺ concentration must give a lower pH. If you get a pH above 7 for an acid, re-check your working.

务必验证你的答案是否合理。较高的 H⁺ 浓度必须对应较低的 pH 值。如果你为酸性溶液算得 pH 大于 7，请重新检查计算过程。

5. Finding Hydrogen Ion Concentration from pH | 由 pH 求氢离子浓度

Examiners often ask you to work backwards: given the pH of a strong acid solution, calculate its concentration. This requires rearranging the definition of pH. From pH = –log[H⁺], we can write:

考官经常要求你进行逆向计算：已知强酸溶液的 pH，求其浓度。这需要将 pH 的定义式进行变换。由 pH = –log[H⁺] 可以得到：

[H⁺] = 10⁻ᵖᴴ

On your calculator, you typically use the ’10ˣ’ or ‘antilog’ function, often accessed by pressing ‘shift’ or ‘2nd’ then ‘log’. Enter the pH, make it negative, and then apply the 10ˣ function to find [H⁺]. For example, if pH = 1.6, then [H⁺] = 10⁻¹·⁶ = 0.0251 mol/dm³. Once you have [H⁺], you relate it back to the acid concentration by dividing by the number of H⁺ per formula unit for polyprotic acids.

在计算器上，你通常使用 “10ˣ” 或 “antilog” 功能，一般通过按 “shift” 或 “2nd” 然后按 “log” 来调用。输入 pH 值，取其负值，然后应用 10ˣ 功能得出 [H⁺]。例如，若 pH = 1.6，则 [H⁺] = 10⁻¹·⁶ = 0.0251 mol/dm³。得到 [H⁺] 后，若为多元酸，需除以每个分子释放的 H⁺ 数，从而推算出酸的浓度。

For instance, a solution of sulfuric acid has a pH of 1.0. [H⁺] = 10⁻¹·⁰ = 0.10 mol/dm³. Since H₂SO₄ provides 2 H⁺ per molecule, the concentration of the acid is 0.10 / 2 = 0.050 mol/dm³. This type of multi-step reasoning is common in CCEA IGCSE questions.

例如，某硫酸溶液的 pH 为 1.0。则 [H⁺] = 10⁻¹·⁰ = 0.10 mol/dm³。由于每分子 H₂SO₄ 提供 2 个 H⁺，该酸的浓度为 0.10 / 2 = 0.050 mol/dm³。这类多步推理在 CCEA IGCSE 的考题中很常见。

6. Introducing pOH and the Ionic Product of Water | pOH 与水的离子积 Kw

To handle the pH of alkaline solutions, we need a link between H⁺ ions and OH⁻ ions in water. Water itself undergoes a very slight self-ionisation:

要处理碱性溶液的 pH，我们需要将水中的 H⁺ 离子与 OH⁻ 离子联系起来。水本身会发生极微弱的自电离：

H₂O(l) ⇌ H⁺(aq) + OH⁻(aq)

At 25 °C, the concentrations of these ions in pure water are each 1.0 × 10⁻⁷ mol/dm³. The ionic product of water, Kw, is defined as:

在 25 °C 下，纯水中这两种离子的浓度均为 1.0 × 10⁻⁷ mol/dm³。水的离子积 Kw 定义如下：

Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ mol²/dm⁶ (at 25 °C)

This relationship holds in any dilute aqueous solution at a given temperature. It means that if we know the concentration of OH⁻ ions, we can calculate [H⁺] and therefore pH. In parallel, the pOH is defined analogously to pH: pOH = –log[OH⁻]. A useful derived equation at 25 °C is:

该关系在任何给定温度下的稀水溶液中均成立。这意味着如果我们知道 OH⁻ 的浓度，便可以计算出 [H⁺]，进而求出 pH。与此类似，pOH 的定义与 pH 相仿：pOH = –log[OH⁻]。在 25 °C 下，一个有用的推导公式为：

pH + pOH = 14

This equation is extremely helpful for quickly finding the pH of a base once pOH is known, or vice versa.

该公式在已知 pOH 时快速求得碱的 pH 值非常有用，反之亦然。

7. Calculating the pH of Strong Bases | 计算强碱的 pH

Strong bases such as sodium hydroxide (NaOH) and potassium hydroxide (KOH) dissociate completely in water, releasing one OH⁻ per formula unit. For bases like barium hydroxide, Ba(OH)₂, which release two OH⁻ ions, the stoichiometric factor must be used, exactly as for diprotic acids.

强碱，如氢氧化钠 (NaOH) 和氢氧化钾 (KOH)，在水中完全解离，每单位分子式释放一个 OH⁻。对于像氢氧化钡 Ba(OH)₂ 这样释放两个 OH⁻ 的碱，必须像处理二元酸一样使用化学计量因子。

The method for finding the pH of a strong base solution involves two main steps. First, determine [OH⁻] from the base concentration and the number of OH⁻ ions per formula unit. Second, use the Kw expression to find [H⁺]:

求算强碱溶液 pH 的方法主要包含两个步骤。首先，根据碱的浓度以及单位分子式所含 OH⁻ 数目确定 [OH⁻]。其次，利用 Kw 表达式求出 [H⁺]：

[H⁺] = Kw / [OH⁻]

Then apply pH = –log[H⁺]. Alternatively, you can calculate pOH = –log[OH⁻] and then use pH = 14 – pOH. Both approaches give the same answer; choose the one you find most logical.

然后应用 pH = –log[H⁺]。或者，你可以先计算 pOH = –log[OH⁻]，再通过 pH = 14 – pOH 得出 pH。两种方法结果相同，选择你认为逻辑更清晰的一种即可。

Example: Calculate the pH of 0.02 mol/dm³ NaOH solution. [OH⁻] = 0.02 mol/dm³. Using Kw: [H⁺] = (1.0 × 10⁻¹⁴) / 0.02 = 5.0 × 10⁻¹³ mol/dm³. pH = –log(5.0 × 10⁻¹³) ≈ 12.3. This high value makes sense for a strong base.

示例：计算 0.02 mol/dm³ NaOH 溶液的 pH。[OH⁻] = 0.02 mol/dm³。利用 Kw：[H⁺] = (1.0 × 10⁻¹⁴) / 0.02 = 5.0 × 10⁻¹³ mol/dm³。pH = –log(5.0 × 10⁻¹³) ≈ 12.3。对于强碱而言，这个较高的数值是合理的。

8. The Effect of Temperature on Kw and pH | 温度对 Kw 和 pH 的影响

Kw is a temperature-dependent constant. As the temperature increases, the self-ionisation of water becomes more significant, and the value of Kw rises. This means that pure water at higher temperatures has a pH lower than 7, yet it remains neutral because [H⁺] = [OH⁻]. Neutrality is defined by the equality of the two ion concentrations, not by a pH of exactly 7.

Kw 是一个随温度变化的常数。温度升高时，水的自电离程度增强，Kw 值随之增大。这意味着较高温度下的纯水 pH 会低于 7，但由于 [H⁺] = [OH⁻] 依然成立，溶液仍为中性。中性由两种离子浓度相等来定义，而非 pH 恰好为 7。

In IGCSE examinations, calculations are nearly always assumed to take place at 25 °C, so Kw = 1.0 × 10⁻¹⁴. You should, however, be aware that stating ‘a solution with pH 6.5 must be acidic’ is only strictly true at 25 °C; at a higher temperature, pH 6.5 could represent a neutral sample. This is a classic conceptual question designed to test your understanding beyond routine calculation.

在 IGCSE 考试中，几乎所有计算都默认在 25 °C 下进行，故 Kw = 1.0 × 10⁻¹⁴。不过你应该意识到，诸如“pH 为 6.5 的溶液一定是酸性的”这种说法仅在 25 °C 下严格成立；在更高温度下，pH 6.5 可能代表一个中性样品。这是一道经典的概念题，目的在于检验你是否超越了机械的计算理解。

9. Dilution and Its Impact on pH | 稀释对 pH 的影响

When an acid solution is diluted by adding water, the concentration of H⁺ ions decreases, causing the pH to rise towards 7. However, the pH can never become alkaline (above 7) just by diluting an acid; it approaches 7 asymptotically. For a strong acid, a tenfold dilution (i.e. reducing concentration by a factor of 10) will increase the pH by exactly 1 unit. Diluting by a factor of 100 increases pH by 2 units, and so on.

当酸溶液加水稀释时，H⁺ 离子浓度降低，导致 pH 升高并趋向 7。然而，仅靠稀释酸溶液，pH 永远无法变成碱性（超过 7）；它只会无限接近 7。对于强酸，稀释十倍（即浓度降低为原来的十分之一）将使 pH 恰好增加 1 个单位。稀释 100 倍则 pH 增加 2 个单位，以此类推。

For bases, dilution causes [OH⁻] to decrease and [H⁺] to increase, thus pH falls towards 7. Again, the pH will not drop below 7 on dilution alone. The quantitative effect can be predicted using the formulas we have already covered: find the new concentration, calculate new [H⁺] or [OH⁻], and then determine the new pH.

对碱而言，稀释导致 [OH⁻] 降低，[H⁺] 升高，因此 pH 下降并趋向 7。同样，仅靠稀释，pH 不会降至 7 以下。可利用我们已经掌握的公式进行定量预测：求出新浓度，计算新的 [H⁺] 或 [OH⁻]，再确定新的 pH。

Common exam question: A 0.1 mol/dm³ HCl solution has pH = 1. What is the pH after a 1 in 100 dilution? After dilution, [HCl] = 0.001 mol/dm³, [H⁺] = 0.001 mol/dm³, pH = 3. The change is +2 units, exactly as expected.

常见考题：0.1 mol/dm³ HCl 溶液的 pH = 1。稀释 100 倍后 pH 为多少？稀释后，[HCl] = 0.001 mol/dm³，[H⁺] = 0.001 mol/dm³，pH = 3。变化恰好为 +2 个单位，符合预期。

10. pH of Weak Acids – Qualitative Treatment | 弱酸的 pH——定性处理

Weak acids, such as ethanoic acid (CH₃COOH), only partially dissociate in water. The equilibrium lies far to the left, so the [H⁺] is always much lower than the stoichiometric concentration of the acid. For a weak acid of the same concentration as a strong acid, the pH will be higher (less acidic) because fewer H⁺ ions are released.

弱酸，如乙酸 (CH₃COOH)，在水中仅部分解离。平衡位置强烈偏向左侧，因此 [H⁺] 总是远低于酸的分析浓度。对于相同浓度的弱酸与强酸，弱酸的 pH 更高（酸性更弱），因为释放出的 H⁺ 离子更少。

At IGCSE level, you are not required to perform pH calculations for weak acids using Ka, but you may be asked to compare experimental pH values or to explain why 0.1 mol/dm³ ethanoic acid has a higher pH than 0.1 mol/dm³ hydrochloric acid. The explanation always rests on incomplete dissociation versus complete dissociation.

在 IGCSE 阶段，你不需要用 Ka 进行弱酸的 pH 计算，但可能会被要求比较实验测得的 pH 值，或解释为何 0.1 mol/dm³ 的乙酸比 0.1 mol/dm³ 的盐酸 pH 更高。解释的关键始终在于不完全解离与完全解离的对比。

You should also be able to connect this with electrical conductivity: a weak acid of the same concentration will have a lower conductivity because fewer mobile ions are present in solution. Similarly, the rate of reaction with metals or carbonates is slower for a weak acid, despite having the same concentration as a strong acid. pH measurements provide the key evidence for differing degrees of ionisation.

你还应能将此与电导率联系起来：相同浓度的弱酸具有更低的电导率，因为溶液中存在的离子较少。同样，弱酸与金属或碳酸盐的反应速率也较慢，尽管其浓度与强酸相同。pH 测量为电离度的差异提供了关键证据。

11. Experimental Determination of pH | pH 的实验测定

The pH of a solution can be estimated using universal indicator solution or paper, which displays a continuous range of colours from red (pH 0–1) through green (pH 7) to purple (pH 14). A more precise measurement requires a pH meter (pH probe) connected to a data logger or digital display. The pH meter measures the electrical potential difference generated by H⁺ ions and converts it directly into a pH reading. This method is accurate to 0.1 pH units or better and is not affected by the colour or turbidity of the solution.

溶液的 pH 可以用通用指示剂溶液或试纸进行估测，它们会显示从红色（pH 0–1）经过绿色（pH 7）到紫色（pH 14）的连续颜色变化。更精确的测量需要使用 pH 计（pH 探头），与数据记录仪或数字显示器相连。pH 计测量由 H⁺ 离子产生的电位差，并将其直接转换为 pH 读数。此方法精确到 0.1 个 pH 单位甚至更高，且不受溶液颜色或浑浊度的影响。

In a typical IGCSE practical context, you might be asked to describe how to measure the pH of a soil sample, a solution of an unknown acid, or the change in pH during a neutralisation titration. You should mention the need for calibration of the pH meter using standard buffer solutions of known pH (e.g., pH 4, pH 7, and pH 9). CCEA questions often value practical accuracy and procedural details.

在 IGCSE 典型的实验背景下，你可能需要描述如何测量土壤样品、未知酸溶液的 pH，或在中和滴定过程中 pH 的变化。你应该提及需要用已知 pH 的标准缓冲溶液（如 pH 4、pH 7、pH 9）对 pH 计进行校准。CCEA 考试往往重视实际操作的准确性和实验步骤细节。

12. Summary of Key Formulae and Exam Tips | 核心公式与应试技巧汇总

The essential algebraic toolkit for pH calculations is remarkably compact. Memorise it and practise applying it to a wide variety of numerical problems.

pH 计算所需的基本代数工具非常精简。请牢记这些公式，并通过大量数理问题的练习来熟练应用它们。

Key equations:

核心方程式：

pH = –log[H⁺]
[H⁺] = 10⁻ᵖᴴ
Kw = [H⁺][OH⁻] = 1.0 × 10⁻¹⁴ at 25 °C
pOH = –log[OH⁻]
pH + pOH = 14 (at 25 °C)

When tackling an exam question, first identify whether you are dealing with an acid or a base. For strong acids, go from concentration to [H⁺] to pH. For strong bases, go from concentration to [OH⁻] to [H⁺] (using Kw) to pH, or via the pOH route. Always show your steps clearly, as marks are often awarded for method even if the final answer has a minor arithmetic slip.

处理考试题目时，首先确认你面对的是酸还是碱。对于强酸，由浓度到 [H⁺] 再到 pH。对于强碱，由浓度到 [OH⁻]，利用 Kw 求得 [H⁺] 再到 pH，或者通过 pOH 路径。务必清晰地展示每一步计算，因为即使最终答案因些许计算错误而不正确，通常也会因方法正确而获得分数。

A final, crucial piece of advice: check that your final pH value is chemically sensible. An acid must have pH < 7 (at 25 °C), and a base must have pH > 7. If your arithmetic gives a pH of 9 for hydrochloric acid, you have made an error in the sign or the Kw conversion. Developing this self-check habit will greatly improve your examination performance.

最后一条至关重要的建议：检查你最终得到的 pH 值在化学上是否合理。酸必须满足 pH < 7（25 °C 条件下），碱必须满足 pH > 7。如果你的计算得出盐酸的 pH 为 9，那么你在符号或 Kw 换算上出了错。养成这种自我检验的习惯将极大提升你的考试表现。

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pH Calculations in IGCSE CCEA Chemistry | IGCSE CCEA 化学：pH计算 考点精讲