The mole · 摩尔
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| molecules/ˈmɒlɪkjuːlz/ | 分子 | fèn zǐ |
| mole/məʊl/ | 摩尔 | mó ěr |
| amount of substance/əˈmaʊnt ɒv ˈsʌbstəns/ | 物质的量 | wù zhì dì liàng |
| kelvin/ˈkelvɪn/ | 开尔文 | kāi'ěrwén |
| Avogadro constant/ˌævəˈɡædrəʊ ˈkɒnstənt/ | 阿伏伽德罗常量 | ā fú gā dé luó cháng liàng |
| atoms/ˈætəmz/ | 原子 | yuán zi |
| monatomic/ˌmɒnəˈtɒmɪk/ | 单原子 | dān yuán zi |
| molar mass/ˈməʊlə mæs/ | 摩尔质量 | mó ěr zhì liàng |
A number nobody can picture
- Spread $6.02 \times 10^{23}$ grains of sand over the United Kingdom and the country would be buried ten metres deep. That is the number of water molecules in a tablespoon.
- Chemistry and physics both need to count particles, and no experiment can count them one at a time. So we count them in packets, and the packet holds that number.
- The packet is called the mole 摩尔, and it is an SI base unit alongside the metre, the kilogram and the second.
- This lesson is what the mole is, how to convert between moles, particles and mass, and the wording the examiner accepts.
谁也想象不出的一个数
- 把 $6.02 \times 10^{23}$ 粒沙铺在英国上,整个国家会被埋在十米之下。那就是一汤匙水里水分子的数量。
- 化学和物理都需要数粒子,而没有任何实验能一个一个地数。所以我们成包地数,而一包正好装那么多。
- 这一包叫做摩尔(mole),它和米、千克、秒并列,是一个 SI 基本单位。
- 这一课讲摩尔是什么、怎样在摩尔、粒子数和质量之间换算,以及考官接受的措辞。
Amount of substance and the mole
- Amount of substance 物质的量 is an SI base quantity, and its base unit is the mole (mol), one of the seven alongside the kilogram, metre, second, ampere and kelvin 开尔文.
- One mole of any substance contains a number of particles equal to the Avogadro constant 阿伏伽德罗常量:
- The examiner's one-mark definition: the Avogadro constant is the number of atoms or molecules in one mole of a substance. Adding "in 0.012 kg of carbon-12" is accepted but not required.
A fixed count, whatever the substance
物质的量与摩尔
- 物质的量(amount of substance)是 SI 基本量,它的基本单位是摩尔(mol),与千克、米、秒、安培和开尔文(kelvin)并列为七个基本单位之一。
- 任何物质的一摩尔都包含数量等于阿伏伽德罗常量(Avogadro constant)的粒子:
- 考官的一分定义:*阿伏伽德罗常量是一摩尔物质中原子或分子的数目。*补上"在 0.012 kg 碳-12 中"会被接受,但不是必需的。
不论什么物质,都是同一个固定的数目
One mole of a substance contains: · 一摩尔某物质包含：
A mole always contains the Avogadro number, $N_{\text{A}} = 6.02 \times 10^{23}$, of particles. · 一摩尔始终包含阿伏伽德罗数 $N_{\text{A}} = 6.02 \times 10^{23}$ 的粒子。
The mole is one of the SI base units. · 摩尔是国际单位制 (SI) 基本单位之一。
Yes — amount of substance (mole) is a base quantity, alongside the kilogram, metre, second, ampere and kelvin. · 是的——物质的量（摩尔）与千克、米、秒、安培和开尔文并列，属于基本物理量。
Say what you are counting
- A "particle" means whatever the substance is made of: atoms 原子 for a monatomic 单原子 element such as helium, molecules 分子 for $\text{O}_2$ or $\text{H}_2\text{O}$.
- One mole of oxygen gas contains $6.02 \times 10^{23}$ molecules, and therefore twice as many atoms. A question that says "molecules" and an answer that gives atoms is out by a factor of two.
- For $n$ moles the number of particles is:
A laboratory quantity on one side, an atomic count on the other
说清你在数什么
- "粒子"指的是这种物质由什么构成:像氦这样的单原子(monatomic)元素数原子(atoms),$\text{O}_2$ 或 $\text{H}_2\text{O}$ 数分子(molecules)。
- 一摩尔氧气含 $6.02 \times 10^{23}$ 个分子,因而含两倍数目的原子。题目说"分子"而答案给出原子,就差了两倍。
- 对 $n$ 摩尔,粒子数是:
一边是实验室的量,一边是原子的数目
Mole particle count lab · 摩尔粒子数实验
particles = n x Avogadro constant · 粒子数 = n × 阿伏伽德罗常数
Change amount of substance and see particle number scale directly. · 改变物质的量并观察粒子数是否按比例缩放。
A sample contains $3.01 \times 10^{23}$ molecules. How many moles is this? · 某样品含有 $3.01 \times 10^{23}$ 个分子。这相当于多少摩尔？
$n = \dfrac{N}{N_{\text{A}}} = \dfrac{3.01 \times 10^{23}}{6.02 \times 10^{23}} = 0.50\ \text{mol}$.
How many atoms are there in one mole of oxygen gas, O2? · 一摩尔氧气 (O2) 中有多少个原子？
One mole is 6.02 x 10^23 molecules, and each O2 molecule holds two atoms. Saying which particle you are counting is worth a mark on its own. · 一摩尔包含 6.02 x 10^23 个分子，且每个 O2 分子含有两个原子。明确说明你在计算哪种粒子本身就能得分。
Molar mass
- The molar mass 摩尔质量 $M_{\text{m}}$ is the mass of one mole of the substance, in $\text{kg/mol}$ or $\text{g/mol}$.
- The mass of $n$ moles is $M = n M_{\text{m}}$, so the number of moles in a sample of mass $M$ is $n = M / M_{\text{m}}$.
- The mass of a single particle follows by dividing the mass of a mole by the number in a mole:
摩尔质量
- 摩尔质量(molar mass)$M_{\text{m}}$ 是该物质一摩尔的质量,单位是 $\text{kg/mol}$ 或 $\text{g/mol}$。
- $n$ 摩尔的质量是 $M = n M_{\text{m}}$,所以质量为 $M$ 的样品的摩尔数是 $n = M / M_{\text{m}}$。
- 单个粒子的质量由一摩尔的质量除以一摩尔中的个数得到:
The molar mass is the mass of: · 摩尔质量是指：
Molar mass is the mass per mole; dividing it by $N_{\text{A}}$ gives the mass of one particle. · 摩尔质量是每摩尔的质量；将其除以 $N_{\text{A}}$ 可得单个粒子的质量。
The mass of a single particle equals the molar mass divided by the ____ constant. · 单个粒子的质量等于摩尔质量除以 ____ 常数。
$m_0 = \dfrac{M_{\text{m}}}{N_{\text{A}}}$ — molar mass shared over the $N_{\text{A}}$ particles in a mole. · $m_0 = \dfrac{M_{\text{m}}}{N_{\text{A}}}$ —— 将摩尔质量分摊到摩尔中的 $N_{\text{A}}$ 个粒子上。
Worked example: oxygen, both ways
- Oxygen has a molar mass of $32\ \text{g/mol}$. Find the mass of one oxygen molecule, and the number of molecules in $8.0\ \text{g}$ of oxygen.
- Mass of one molecule: work in kilograms. $m_0 = \dfrac{M_{\text{m}}}{N_{\text{A}}} = \dfrac{0.032}{6.02 \times 10^{23}} = 5.3 \times 10^{-26}\ \text{kg}$.
- Number of moles: $n = \dfrac{8.0}{32} = 0.25\ \text{mol}$. Then $N = n N_{\text{A}} = 0.25 \times 6.02 \times 10^{23} = 1.5 \times 10^{23}$ molecules.
- Keep molar masses in $\text{kg/mol}$ whenever the answer is wanted in kilograms. The stray factor of 1000 is the commonest slip in the whole topic.
例题:氧气,两个方向
- 氧的摩尔质量是 $32\ \text{g/mol}$。求一个氧分子的质量,以及 $8.0\ \text{g}$ 氧气中的分子数。
- 一个分子的质量:用千克计算。$m_0 = \dfrac{M_{\text{m}}}{N_{\text{A}}} = \dfrac{0.032}{6.02 \times 10^{23}} = 5.3 \times 10^{-26}\ \text{kg}$。
- 摩尔数:$n = \dfrac{8.0}{32} = 0.25\ \text{mol}$。于是 $N = n N_{\text{A}} = 0.25 \times 6.02 \times 10^{23} = 1.5 \times 10^{23}$ 个分子。
- 只要答案要以千克给出,摩尔质量就用 $\text{kg/mol}$。那个多出来的 1000 倍是整个单元里最常见的失误。
The three conversions
- Moles to particles: multiply by $N_{\text{A}}$. Particles to moles: divide by $N_{\text{A}}$.
- Moles to mass: multiply by $M_{\text{m}}$. Mass to moles: divide by $M_{\text{m}}$.
- Mass to particles: go through moles. There is no single step, and inventing one is where errors come from.
- Every question in this subtopic is one or two of these six conversions in a row. Write down which one you need before reaching for the calculator.
三组换算
- 摩尔到粒子数:乘以 $N_{\text{A}}$。粒子数到摩尔:除以 $N_{\text{A}}$。
- 摩尔到质量:乘以 $M_{\text{m}}$。质量到摩尔:除以 $M_{\text{m}}$。
- 质量到粒子数:要经过摩尔。没有一步到位的做法,而自己发明一个正是错误的来源。
- 这个子专题的每道题都是这六种换算中的一两步。先写下你需要哪一步,再去按计算器。
Oxygen has a molar mass of 32 g/mol. What is the mass of one oxygen molecule, in units of 10^-26 kg? · 氧气的摩尔质量为 32 g/mol。一个氧分子的质量是多少（单位为 10^-26 kg）？
Convert to kilograms first: 0.032 / (6.02 x 10^23) = 5.3 x 10^-26 kg. Dividing 32 by the Avogadro constant instead gives an answer 1000 times too large. · 先换算为千克：0.032 / (6.02 x 10^23) = 5.3 x 10^-26 kg。如果直接用 32 除以阿伏伽德罗常数，答案会大 1000 倍。
How many moles are there in 8.0 g of oxygen, molar mass 32 g/mol? · 8.0 g 氧气（摩尔质量 32 g/mol）中含有多少摩尔？
n = M / Mm = 8.0 / 32 = 0.25 mol, which is 1.5 x 10^23 molecules. Both masses are in grams here, so no conversion is needed. · n = M / Mm = 8.0 / 32 = 0.25 mol，即 1.5 x 10^23 个分子。此处两个质量单位均为克，无需换算。
How the mole links to the gas constants
- The mole is what connects the per mole and per molecule versions of the gas equations you meet next.
- The Boltzmann constant is the gas constant per molecule: $k = R / N_{\text{A}}$, equivalently $R = N_{\text{A}} k$.
- When a question asks for "the relationship between $N_{\text{A}}$, $R$ and $k$", that one line is the whole answer.
摩尔怎样连接气体常量
- 摩尔正是连接你接下来会遇到的气体方程的每摩尔版本和每分子版本的东西。
- 玻尔兹曼常量是每个分子的气体常量:$k = R / N_{\text{A}}$,等价地 $R = N_{\text{A}} k$。
- 当题目问"$N_{\text{A}}$、$R$ 和 $k$ 之间的关系"时,那一行就是全部答案。
Match each conversion to what you do. · 将每种转换与其对应的操作匹配。
Every question in this subtopic is one or two of these in a row. Decide which before reaching for the calculator. · 本子主题中的每个问题都是上述操作的连续一次或两次组合。在拿起计算器之前请先确定具体步骤。
Marks that slip away
- State what you are counting. One mole of $\text{O}_2$ is $6.02 \times 10^{23}$ molecules and twice that many atoms.
- Convert grams to kilograms before dividing by $N_{\text{A}}$, or the particle mass is out by 1000.
- The mole is a base unit, and amount of substance a base quantity. It is not derived from anything.
- Mass to particles has no shortcut: go mass, then moles, then particles.
容易丢掉的分
- 说清你在数什么。一摩尔 $\text{O}_2$ 是 $6.02 \times 10^{23}$ 个分子,以及两倍数目的原子。
- 在除以 $N_{\text{A}}$ 之前把克换成千克,否则粒子质量差 1000 倍。
- 摩尔是基本单位,物质的量是基本量。它不是从别的东西推导出来的。
- 质量到粒子数没有捷径:质量、摩尔、粒子数,依次走。
Write the relationship between the Boltzmann constant k, the molar gas constant R and the Avogadro constant NA, in the form k = ... · 写出玻尔兹曼常数 k、摩尔气体常数 R 和阿伏伽德罗常数 NA 之间的关系式，形式为 k = ...
k is the gas constant per molecule, as R is per mole. That single line is the whole answer when the relationship is asked for. · k 是每个分子的常数，正如 R 是每摩尔的常数。当要求写出关系时，这一行表达式就是完整的答案。
You've got it
- amount of substance is an SI base quantity with base unit the mole; one mole holds $N_{\text{A}} = 6.02 \times 10^{23}$ particles
- $N = n N_{\text{A}}$, and a "particle" is an atom or a molecule depending on the substance, so say which
- molar mass is the mass of one mole: $M = n M_{\text{m}}$ and one particle has mass $M_{\text{m}} / N_{\text{A}}$
- $k = R / N_{\text{A}}$: the Boltzmann constant is the gas constant per molecule
你掌握了
- 物质的量是 SI 基本量,基本单位是摩尔;一摩尔含 $N_{\text{A}} = 6.02 \times 10^{23}$ 个粒子
- $N = n N_{\text{A}}$,而"粒子"是原子还是分子取决于物质,所以要说清是哪一种
- 摩尔质量是一摩尔的质量:$M = n M_{\text{m}}$,单个粒子的质量是 $M_{\text{m}} / N_{\text{A}}$
- $k = R / N_{\text{A}}$:玻尔兹曼常量是每个分子的气体常量