Isotopic Abundance and Relative Atomic Mass Calculations | 同位素丰度与相对原子质量计算

📚 Isotopic Abundance and Relative Atomic Mass Calculations | 同位素丰度与相对原子质量计算

Atoms of the same element can have different numbers of neutrons. These variants are called isotopes, and they are the reason why most elements do not have a whole-number atomic mass. The relative atomic mass (Aᵣ) of an element is the weighted average mass of all its naturally occurring isotopes compared to one-twelfth of the mass of a carbon-12 atom.

同一种元素的原子，其内部中子数可以不同，这些原子互称为同位素。由于同位素的存在，大多数元素的原子质量并不是整数。元素的相对原子质量（Aᵣ）是以碳-12 原子质量的 1/12 为基准，该元素在自然界中所有同位素的加权平均质量。

1. Why Isotopes Matter | 为什么同位素很重要

Isotopes contain the same number of protons but different numbers of neutrons. For example, carbon-12 (¹²C) and carbon-14 (¹⁴C) both have 6 protons, but they contain 6 and 8 neutrons respectively.

同位素是质子数相同而中子数不同的同种元素的原子。例如，碳-12（¹²C）和碳-14（¹⁴C）都含有 6 个质子，但分别含有 6 个和 8 个中子。

Because isotopes of the same element have identical electron configurations, their chemical properties are almost the same. However, their masses differ, and this mass difference explains why the relative atomic masses on the periodic table are rarely whole numbers.

由于同一元素的同位素具有相同的电子排布，它们的化学性质几乎相同；但它们的质量不同，这正是元素周期表中相对原子质量很少为整数的原因。

• Atomic number (Z) = number of protons | 原子序数（Z）= 质子数
• Mass number (A) = number of protons + number of neutrons | 质量数（A）= 质子数 + 中子数

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2. Definition of Relative Atomic Mass | 相对原子质量的定义

Relative atomic mass is defined as the weighted mean mass of an atom of an element compared to one-twelfth of the mass of an atom of carbon-12. Since both masses are measured in the same units, the ratio is dimensionless; Aᵣ therefore has no units.

相对原子质量的定义是：某元素一个原子的平均质量与碳-12 原子质量的 1/12 之比。由于二者使用相同的质量单位，这一比值为无量纲量，因此 Aᵣ 没有单位。

Aᵣ = (mean mass of one atom of the element) ÷ (1/12 × mass of one ¹²C atom)

Aᵣ =（该元素一个原子的平均质量）÷（1/12 × 一个 ¹²C 原子的质量）

The carbon-12 scale was chosen because the mass of one atom of ¹²C is exactly 12 atomic mass units, giving a convenient reference for all other elements.

之所以选择碳-12 作为标准，是因為一个 ¹²C 原子的质量恰好为 12 个原子质量单位，为其他元素提供了方便统一的参照基准。

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3. Isotopic Abundance | 同位素丰度

Isotopic abundance is the percentage of atoms of a particular isotope in a natural sample of the element. In a natural sample of chlorine, about 75.77% of the atoms are ³⁵Cl and about 24.23% are ³⁷Cl.

同位素丰度是指某元素在自然界样品中，某一种同位素的原子数占该元素原子总数的百分数。在天然氯中，大约 75.77% 的原子是 ³⁵Cl，约 24.23% 是 ³⁷Cl。

Abundance values are usually either given as percentages or as fractions. A fraction of 0.7577 is the same as 75.77%.

丰度通常以百分数或小数分数形式给出。例如，0.7577 与 75.77% 是等价的。

These abundance values must be used as weightings when calculating Aᵣ. A simple arithmetic average would be incorrect because the two isotopes are not present in equal amounts.

在计算 Aᵣ 时，必须用丰度作为权重。简单算术平均值是错误的，因为两种同位素在自然界中的含量并不相等。

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4. Mass Spectrometry | 质谱法

Mass spectrometry is the experimental technique used to measure the exact masses of isotopes and their relative abundances. A gaseous sample is first ionised to form positive ions, then accelerated by an electric field and deflected by a magnetic field.

质谱法是用于测定同位素精确质量及其相对丰度的实验技术。气态样品首先被电离形成阳离子，再经电场加速，然后在磁场中发生偏转。

The amount of deflection depends on the mass-to-charge ratio (m/z): lighter ions are deflected more than heavier ions. The detector records the relative abundance of each ion as a peak in a mass spectrum.

偏转程度取决于质荷比（m/z）：较轻的离子偏转更大，较重的离子偏转更小。检测器将每种离子的相对丰度记录为质谱图中的一个峰。

In the mass spectrum of chlorine, two main peaks appear at m/z = 35 and m/z = 37, with peak heights in a ratio of approximately 3:1. This directly reflects the natural abundance of the two isotopes.

在氯的质谱图中，m/z = 35 和 m/z = 37 处出现两个主峰，其峰高比约为 3:1，直接反映两种同位素的天然丰度关系。

Isotope | 同位素	m/z | 质荷比	Relative abundance | 相对丰度
³⁵Cl | 氯-35	35	75.77%
³⁷Cl | 氯-37	37	24.23%

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5. The Weighted Average Formula | 加权平均公式

To calculate the relative atomic mass of an element with two isotopes, multiply the exact mass of each isotope by its fractional abundance, then add the products.

计算有两种同位素的元素的相对原子质量时，应将每个同位素的精确质量乘以其丰度分数，然后将各项相加。

Aᵣ = (m₁ × a₁ + m₂ × a₂) ÷ 100

Here, m₁ and m₂ are the exact masses of the isotopes, and a₁ and a₂ are their percentage abundances. If the abundances are given as fractions, the denominator 100 is simply omitted.

其中，m₁ 和 m₂ 是同位素的精确质量，a₁ 和 a₂ 是它们的百分丰度。若丰度以小数分数形式给出，则无需再除以 100。

For three or more isotopes, the same pattern continues:

当元素有三种或更多同位素时，计算方式依此类推：

Aᵣ = (m₁ × a₁ + m₂ × a₂ + m₃ × a₃ + …) ÷ 100

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6. Worked Example – Chlorine | 示例计算：氯

The two isotopes of chlorine have the following data: ³⁵Cl has a mass of 34.969 and an abundance of 75.77%; ³⁷Cl has a mass of 36.966 and an abundance of 24.23%. Calculate Aᵣ.

氯的两种同位素数据如下：³⁵Cl 的质量为 34.969，丰度为 75.77%；³⁷Cl 的质量为 36.966，丰度为 24.23%。计算氯的相对原子质量。

Step 1: Multiply each isotope mass by its percentage abundance.

第 1 步：将每个同位素的质量乘以对应的百分丰度。

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