The Mole and Counting Particles by Mass | 摩尔与粒子质量计数

📚 The Mole and Counting Particles by Mass | 摩尔与粒子质量计数

In chemistry, we rarely work with individual atoms or molecules because they are far too small to see or weigh directly. Instead, we use the mole as a counting unit that connects the microscopic world of particles to the macroscopic world of grams and litres.

在化学中，我们很少直接处理单个原子或分子，因为它们太小而无法直接观察或称量。相反，我们用“摩尔”作为一个计数单位，把微观粒子世界与宏观的克和升联系起来。

1. What is a Mole? | 什么是摩尔

A mole (mol) is the amount of substance that contains exactly as many elementary entities as there are atoms in 12 grams of carbon-12. This number is known as the Avogadro constant.

摩尔（mol）是物质的量单位：1摩尔任何物质所含的基本粒子数，等于12克碳-12中所含的原子数。这个数值称为阿伏伽德罗常数。

One mole of any substance contains the same number of particles, whether they are atoms, molecules, ions, electrons, or formula units.

任何物质的一摩尔都含有相同数目的粒子，无论这些粒子是原子、分子、离子、电子还是式单元。

1 mol = 6.02 × 10²³ particles

In IB chemistry, you should use the value 6.02 × 10²³ mol⁻¹ (or sometimes 6.022 × 10²³ mol⁻¹) for the Avogadro constant.

在IB化学中，阿伏伽德罗常数应使用 6.02 × 10²³ mol⁻¹（有时也用 6.022 × 10²³ mol⁻¹）。

2. Avogadro’s Constant | 阿伏伽德罗常数

The Avogadro constant, symbol Nₐ, has units of per mole (mol⁻¹). It is defined as the number of atoms in exactly 12 g of carbon-12.

阿伏伽德罗常数，符号为 Nₐ，单位是每摩尔（mol⁻¹）。它定义为12克碳-12中所含的原子数目。

This constant is a conversion factor between the number of particles and the amount of substance in moles:

这个常数是粒子数目与物质的量（摩尔数）之间的转换因子：

N = n × Nₐ

where N is the number of particles, n is the amount in moles, and Nₐ = 6.02 × 10²³ mol⁻¹.

其中 N 是粒子数目，n 是物质的量（摩尔数），Nₐ = 6.02 × 10²³ mol⁻¹。

For example, 0.5 mol of water molecules contains 0.5 × 6.02 × 10²³ = 3.01 × 10²³ water molecules.

例如，0.5摩尔水分子含有 0.5 × 6.02 × 10²³ = 3.01 × 10²³ 个水分子。

3. Molar Mass | 摩尔质量

Molar mass (M) is the mass of one mole of a substance. Its SI unit is grams per mole (g mol⁻¹), but it is numerically equal to the relative atomic mass (Aᵣ) or relative molecular mass (Mᵣ).

摩尔质量（M）是一摩尔物质的质量。其国际单位是克每摩尔（g mol⁻¹），数值上等于相对原子质量（Aᵣ）或相对分子质量（Mᵣ）。

For elements, the molar mass is the atomic mass from the periodic table. For compounds, add the molar masses of all atoms in the formula.

对于单质，摩尔质量就是元素周期表上的原子质量。对于化合物，则把化学式中所有原子的摩尔质量相加。

M (compound) = Σ (number of atoms × atomic mass)

Example: M(H₂O) = 2 × 1.01 + 16.00 = 18.02 g mol⁻¹.

例如：M(H₂O) = 2 × 1.01 + 16.00 = 18.02 g mol⁻¹。

4. Converting Mass, Moles and Particles | 质量、摩尔与粒子数的换算

The central relationship is:

核心关系是：

n = m / M

where n is amount in mol, m is mass in grams, and M is molar mass in g mol⁻¹.

其中 n 是物质的量（摩尔），m 是质量（克），M 是摩尔质量（g mol⁻¹）。

To count particles, first convert mass to moles, then multiply by the Avogadro constant:

要计算粒子数，先把质量换算成摩尔数，再乘以阿伏伽德罗常数：

Number of particles = (m / M) × Nₐ

Mass → moles: divide by molar mass
Moles → particles: multiply by 6.02 × 10²³
Particles → moles: divide by 6.02 × 10²³
Moles → mass: multiply by molar mass

质量 → 摩尔：除以摩尔质量
摩尔 → 粒子数：乘以 6.02 × 10²³
粒子数 → 摩尔：除以 6.02 × 10²³
摩尔 → 质量：乘以摩尔质量

5. Worked Examples | 计算实例

Example 1: How many atoms are in 24.0 g of carbon-12? M(C) = 12.0 g mol⁻¹.

例1：24.0克碳-12中含有多少个原子？M(C) = 12.0 g mol⁻¹。

n = 24.0 / 12.0 = 2.00 mol

N = 2.00 × 6.02 × 10²³ = 1.20 × 10²⁴ atoms

Therefore, 24.0 g of carbon-12 contains 1.20 × 10²⁴ atoms.

因此，24.0克碳-12含有 1.20 × 10²⁴ 个原子。

Example 2: What is the mass of 0.250 mol of calcium nitrate, Ca(NO₃)₂?

例2：0.250摩尔硝酸钙 Ca(NO₃)₂ 的质量是多少？

M(Ca) = 40.08, M(N) = 14.01, M(O) = 16.00.

M(Ca(NO₃)₂) = 40.08 + 2 × (14.01 + 3 × 16.00) = 164.10 g mol⁻¹

m = n × M = 0.250 × 164.10 = 41.0 g

So 0.250 mol of calcium nitrate has a mass of 41.0 g.

因此，0.250摩尔硝酸钙的质量为41.0克。

Example 3: How many oxygen atoms are in 0.200 mol of CO₂?

例3：0.200摩尔CO₂中含有多少个氧原子？

Each CO₂ molecule contains 2 oxygen atoms.

每个CO₂分子含有2个氧原子。

N(O atoms) = 0.200 × 2 × 6.02 × 10²³ = 2.41 × 10²³

Be careful to multiply by the number of atoms per formula unit, not only the number of molecules.

注意：要乘以每个化学式单元中的原子数目，而不仅仅是分子数目。

6. Hydrated Compounds | 水合物

Many ionic compounds crystallise with a fixed number of water molecules, called water of crystallisation. For example, hydrated copper(II) sulfate has the formula CuSO₄·5H₂O.

许多离子化合物结晶时带有固定数目的水分子，称为结晶水。例如，五水硫酸铜的化学式为 CuSO₄·5H₂O。

When calculating molar mass, include the mass of water:

计算摩尔质量时，必须包含水的质量：

M(CuSO₄·5H₂O) = M(CuSO₄) + 5 × M(H₂O)

If 1 mole of CuSO₄·5H₂O is heated, it produces 1 mole of anhydrous CuSO₄ and 5 moles of H₂O.

1摩尔 CuSO₄·5H₂O 加热后，生成1摩尔无水CuSO₄和5摩尔H₂O。

The moles of water are always in a simple ratio to the moles of anhydrous salt, which allows us to determine the formula of a hydrate experimentally.

水的物质的量与无水盐的物质的量总是成简单整数比，这使我们能通过实验确定水合物的化学式。

7. Empirical and Molecular Formulas | 实验式与分子式

The empirical formula shows the simplest whole-number ratio of atoms of each element in a compound. The molecular formula shows the actual number of atoms of each element in one molecule.

实验式表示化合物中各元素原子数目的最简整数比。分子式表示一个分子中各元素原子的实际数目。

To find the empirical formula from percentage composition by mass:

根据质量百分比组成求实验式：

Assume 100 g of sample, so percentages become masses in grams.
Convert each mass to moles by dividing by atomic mass.
Divide all mole values by the smallest mole value.
If necessary, multiply to obtain whole numbers.

假设样品为100克，则百分比可直接视为质量（克）。
用各元素质量除以原子质量，得到物质的量（摩尔）。
将所有摩尔值除以其中最小值，得到比值。
若有必要，乘以整数使其变为整数比。

The molecular formula is a whole-number multiple of the empirical formula. For example, ethane has empirical formula CH₃ and molecular formula C₂H₆, so the multiplier is 2.

分子式是实验式的整数倍。例如，乙烷的实验式是CH₃，分子式是C₂H₆，倍数为2。

8. Gases and Molar Volume | 气体与摩尔体积

At standard temperature and pressure (STP: 273 K and 100 kPa), one mole of any ideal gas occupies a volume of 22.7 dm³.

在标准状况（STP：273 K，100 kPa）下，1摩尔任何理想气体所占体积为22.7 dm³。

V = n × 22.7 dm³ mol⁻¹ (at STP)

For example, 0.500 mol of O₂ gas at STP occupies:

例如，标准状况下0.500摩尔O₂气体所占体积为：

V = 0.500 × 22.7 = 11.35 dm³

Using the ideal gas equation PV = nRT is also common, where R = 8.31 J K⁻¹ mol⁻¹ and pressure is in kPa, volume in dm³, temperature in K.

使用理想气体状态方程 PV = nRT 也很常见，其中 R = 8.31 J K⁻¹ mol⁻¹，压力单位为kPa，体积单位为dm³，温度单位为K。

9. Concentrations and Moles | 浓度与摩尔

In solution chemistry, concentration is defined as the amount of solute per unit volume of solution.

在溶液化学中，浓度定义为单位体积溶液中所含溶质的物质的量。

c = n / V

where c is concentration in mol dm⁻³ (often written M), n is amount in mol, and V is volume in dm³.

其中 c 是浓度（mol dm⁻³，常写作M），n 是物质的量（mol），V 是体积（dm³）。

To find the number of particles dissolved, first calculate moles from c and V, then multiply by Nₐ. Pay attention to the number of ions in ionic compounds; for example, 0.10 mol dm⁻³ Na₂SO₄ gives 0.20 mol dm⁻³ Na⁺ and 0.10 mol dm⁻³ SO₄²⁻.

要求溶解粒子的数目，先从浓度和体积算出摩尔数，再乘以Nₐ。注意离子化合物中离子的数目，例如 0.10 mol dm⁻³ Na₂SO₄ 产生 0.20 mol dm⁻³ Na⁺ 和 0.10 mol dm⁻³ SO₄²⁻。

10. Common Pitfalls | 常见错误

Using relative atomic mass without units as molar mass — always write g mol⁻¹.
Forgetting to multiply by subscripts when calculating molar mass, e.g. in H₂SO₄, oxygen is 4 × 16.00.
Confusing molecules with atoms: 1 mol of O₂ contains 2 mol of O atoms.
Using 22.4 dm³ instead of 22.7 dm³ for STP in IB exams. Check which conditions are given.
Mixing up dm³ and cm³: 1 dm³ = 1000 cm³.

把相对原子质量直接当作摩尔质量而不写单位——必须写 g mol⁻¹。
计算摩尔质量时忘记乘以角标，例如H₂SO₄中的氧原子应为 4 × 16.00。
混淆分子与原子：1摩尔O₂含有2摩尔氧原子。
在IB考试中错误使用22.4 dm³代替22.7 dm³作为标准状况下的摩尔体积。注意题目给出的条件。
混淆dm³与cm³：1 dm³ = 1000 cm³。

11. Exam Tips | 考试技巧

Always lay out your working in three clear steps: write the formula, substitute numbers with units, then calculate. This allows you to pick up method marks even if the final answer is wrong.

作答时务必分三步清晰书写：写出公式，代入带单位的数值，再计算结果。这样即使最终答案有误，也能获得步骤分。

Before starting a calculation, underline the given quantities and the target quantity. Decide which equation connects them: n = m/M, c = n/V, or PV = nRT.

开始计算前，先标出已知量和待求量，判断用哪个公式联系它们：n = m/M、c = n/V 或 PV = nRT。

If the question asks for atoms, do not stop at molecules. Multiply by the number of atoms per molecule. If it asks for ions, consider the ionic formula carefully.

如果题目要求原子数，不要只算到分子数。要乘以每个分子中的原子数。如果要求离子数，需仔细考虑离子式。

12. Summary | 总结

The mole is a bridge between the mass we measure in the laboratory and the particles we cannot see. By using molar mass and the Avogadro constant, we can count atoms, molecules, and ions from a simple weighing.

摩尔是连接实验室中可称量的质量与不可见的粒子之间的桥梁。利用摩尔质量和阿伏伽德罗常数，我们可以通过一次简单的称量来数出原子、分子和离子。

n = m / M = N / Nₐ

Mastering these conversions is fundamental to stoichiometry, gas calculations, and solution chemistry in IB Chemistry.

掌握这些换算是IB化学中化学计量学、气体计算和溶液化学的基础。

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The Mole and Counting Particles by Mass | 摩尔与粒子质量计数