Atoms, molecules and stoichiometry

Atoms 原子 are far too light to weigh in grams, so we compare every mass to one standard. The standard is the unified atomic mass unit 统一原子质量单位 (symbol u), defined as exactly one twelfth of the mass of one carbon-12 atom.

Using this unit, we state masses as simple numbers:

• the relative atomic mass 相对原子质量 $A_r$ of an element is the average mass of its atoms compared with $\tfrac{1}{12}$ of a carbon-12 atom. It is an average over all the isotopes 同位素, weighted by how common each one is.
• the relative isotopic mass 相对同位素质量 is the mass of one atom of a single isotope, compared with $\tfrac{1}{12}$ of a carbon-12 atom.
• the relative molecular mass 相对分子质量 $M_r$ of a molecule 分子 is the sum of the relative atomic masses of all its atoms.
• the relative formula mass 相对式量 is the same idea for a substance that is not made of molecules (such as an ionic compound). Add up the relative atomic masses shown in the formula.

Relative atomic mass is a weighted average: chlorine's two isotopes (75% ³⁵Cl, 25% ³⁷Cl) average to Aᵣ = 35.5

To find $A_r$ from isotope data, multiply each isotope mass by its percentage, add these up, and divide by 100.

Worked example. Chlorine is $75\%$ ${}^{35}\text{Cl}$ and $25\%$ ${}^{37}\text{Cl}$. Find its relative atomic mass.

Worked example. A mass spectrometer shows copper is $69.2\%$ ${}^{63}\text{Cu}$ and $30.8\%$ ${}^{65}\text{Cu}$. Find its relative atomic mass.

A modern electronic balance measures mass — the basis of mole calculations.

Chemists count particles in groups called moles, just as we count eggs in dozens.

One mole 摩尔 (symbol mol) is the amount of substance that contains the same number of particles as there are atoms in exactly 12 g of carbon-12. That number is the Avogadro constant 阿伏伽德罗常量:

So one mole of anything contains $6.02 \times 10^{23}$ particles. The particles may be atoms, molecules or ions 离子 — always say which.

The mass of one mole in grams equals the relative mass ($A_r$ or $M_r$). This is the molar mass 摩尔质量, with units $\text{g mol}^{-1}$. The key equation is:

where $n$ is the amount in moles, $m$ is the mass in grams, and $M$ is the molar mass.

Worked example. How many moles are in $8.0\ \text{g}$ of methane, $\text{CH}_4$? ($A_r$: C $= 12$, H $= 1$.)

The molar mass is $M = 12 + 4(1) = 16\ \text{g mol}^{-1}$, so

The mole is the hub of every amount calculation: convert to mass ($n=m/M$), particles ($\times N_A$), gas volume ($n=V/24$) or solution ($n=cV$)

A compound 化合物 is a substance made of two or more elements chemically joined.

Charges and formulas of ionic compounds

In an ionic compound 离子化合物 the total positive charge balances the total negative charge, so the compound is neutral overall.

You can predict the charge of many ions from the element's position in the Periodic Table:

Hydrogen forms $\text{H}^+$, and the Group 18 noble gases do not normally form ions.

Some ions you must know by name and formula:

For a metal that can have more than one charge, a Roman numeral shows the oxidation number 氧化数. For example, iron(II) is $\text{Fe}^{2+}$ and iron(III) is $\text{Fe}^{3+}$. To write a formula, balance the charges: iron(III) oxide is $\text{Fe}_2\text{O}_3$, because two $\text{Fe}^{3+}$ balance three $\text{O}^{2-}$.

Equations and state symbols

A chemical equation must be balanced 配平 — the same number of each kind of atom on both sides. Add state symbols 状态符号 to show the state of each species: (s) solid, (l) liquid, (g) gas, and (aq) aqueous 水溶液 (dissolved in water).

An ionic equation 离子方程式 shows only the ions and molecules that actually change. The ions that do not change are spectator ions 旁观离子, and you leave them out. For example, the reaction that forms silver chloride is:

Empirical and molecular formulas

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

For example, ethane has empirical formula $\text{CH}_3$ but molecular formula $\text{C}_2\text{H}_6$.

Hydrated and anhydrous solids

Some solids hold water inside their crystals. This water is the water of crystallisation 结晶水. A solid that contains it is hydrated 水合的; the same solid with the water removed is anhydrous 无水的.

For example, hydrated copper(II) sulfate is $\text{CuSO}_4 \cdot 5\text{H}_2\text{O}$. Heating it drives off the water to leave anhydrous $\text{CuSO}_4$.

Calculating empirical and molecular formulas

To find the empirical formula from masses (or percentages by mass):

1. divide each element's mass by its $A_r$ to get the moles.
2. divide all the mole values by the smallest one.
3. round to the nearest whole numbers — that ratio is the empirical formula.

To get the molecular formula, you also need $M_r$. Find how many times the empirical formula mass fits into $M_r$, then multiply the formula by that number.

Worked example. A compound is $40.0\%$ carbon, $6.7\%$ hydrogen and $53.3\%$ oxygen by mass. Find its empirical formula. ($A_r$: C $= 12$, H $= 1$, O $= 16$.)

Take $100\ \text{g}$ and divide each mass by its $A_r$ to get moles: C $= 40.0/12 = 3.33$, H $= 6.7/1 = 6.7$, O $= 53.3/16 = 3.33$. Dividing through by the smallest ($3.33$) gives a ratio C : H : O $= 1 : 2 : 1$, so the empirical formula is $\text{CH}_2\text{O}$.

The empirical-formula recipe: divide by Ar, divide by the smallest, read off the ratio

A titration finds reacting volumes precisely.

Reacting masses and percentage yield

The numbers in front of each species in a balanced equation give the mole ratio — this is the stoichiometry 化学计量. To find a reacting mass: change the known mass to moles, use the mole ratio to find the moles you want, then change back to mass.

Worked example. What mass of magnesium oxide forms when $4.8\ \text{g}$ of magnesium burns completely? $2\text{Mg} + \text{O}_2 \rightarrow 2\text{MgO}$. ($A_r$: Mg $= 24$, O $= 16$.)

Moles of Mg $= 4.8/24 = 0.20\ \text{mol}$. The ratio Mg : MgO is $1 : 1$, so $0.20\ \text{mol}$ of MgO forms. Its molar mass is $24 + 16 = 40\ \text{g mol}^{-1}$, so

The mole bridge: mass to moles, ratio, then back to mass

In real reactions you usually get less product than the maximum. The percentage yield 产率 compares the amount you actually made with the most you could make:

Limiting and excess reagent

When two reactants are mixed, one usually runs out first. The limiting reagent 限量试剂 is the one that runs out — it decides how much product forms. The other is the excess reagent 过量试剂, because there is more than enough of it. Always base the calculation on the limiting reagent.

To find it: work out the moles of each reactant, divide each by its number in the equation, and the smallest result is the limiting reagent.

The limiting reagent runs out first and decides how much product forms; the leftover reactant is in excess

Volumes of gases

At the same temperature and pressure, equal volumes of any gases contain equal numbers of molecules. At room temperature and pressure (r.t.p.), one mole of any gas takes up $24.0\ \text{dm}^3$, so:

Equal volumes of gases at the same temperature and pressure hold equal numbers of molecules, whatever the gas

This is used when burning hydrocarbons 碳氢化合物 (compounds of only carbon and hydrogen), where you compare gas volumes.

Volumes and concentrations of solutions

The concentration 浓度 of a solution is the amount of solute 溶质 in each cubic decimetre of solution 溶液, measured in $\text{mol dm}^{-3}$:

where $c$ is the concentration and $V$ is the volume in $\text{dm}^3$. Remember that $1000\ \text{cm}^3 = 1\ \text{dm}^3$.

Worked example. In a titration, $25.0\ \text{cm}^3$ of sodium hydroxide solution is exactly neutralised by $20.0\ \text{cm}^3$ of $0.100\ \text{mol dm}^{-3}$ hydrochloric acid: $\text{NaOH} + \text{HCl} \rightarrow \text{NaCl} + \text{H}_2\text{O}$. Find the concentration of the sodium hydroxide.

Moles of HCl $= cV = 0.100 \times \dfrac{20.0}{1000} = 2.00 \times 10^{-3}\ \text{mol}$. The ratio is $1 : 1$, so there are $2.00 \times 10^{-3}\ \text{mol}$ of NaOH in $25.0\ \text{cm}^3$:

This is the basis of a titration 滴定, where you find an unknown concentration by reacting it with a solution whose concentration you already know.

In a titration a burette adds a solution of known concentration to the unknown in the conical flask, until the indicator changes

Significant figures

Give your answer to a sensible number of significant figures 有效数字 — usually match the data in the question. Do not write more digits than the data supports, and do not round so early that you lose accuracy.

• Convert volumes to $\text{dm}^3$ before using concentration ($25.0\ \text{cm}^3 = 0.0250\ \text{dm}^3$); the missing $\div 1000$ is the most common titration error.
• Use the balancing numbers as the mole ratio between species — never the $M_r$ values.
• Empirical formula: divide each mass/percentage by $A_r$, then by the smallest, then scale to whole numbers; use $M_r$ to reach the molecular formula.
• For gases at r.t.p. use $\text{volume} = \text{moles} \times 24\ \text{dm}^3$; quote the equation you use every time.
• Give the answer to the same significant figures as the data (usually 3) and always include units.