Essential Ideas

Chemistry is the study of matter and the changes that matter goes through. This chapter hands you the tools that the rest of the book will run on. You will learn what matter is and how its particles are packed together. You will sort samples into solids, liquids and gases, and you will tell a physical change from a chemical one. Then you will start to measure. Every number in chemistry carries a unit, and that unit carries real meaning. You will meet the SI units, learn to report what your instruments actually tell you, and practice turning one unit into another. By the end of the chapter, your numbers will be worth trusting.

Section 1Matter and its states

What counts as matter

Matter is anything that has mass and takes up space, which covers nearly everything around you. Your chair is matter, and so is the air in the room and the water in your glass. Light and sound are not matter, because they carry energy without holding any mass or volume.

All matter is built from atoms, and an atom is the smallest piece of an element that still behaves like that element. Atoms bond together into molecules, which are small groups of joined atoms. A single water molecule is two hydrogen atoms bonded to one oxygen atom, which chemists write as H₂O.

Solid, liquid and gas

The three states you meet most often are solid, liquid and gas, and what separates them is how the particles are packed and how freely they travel.

In a solid, the particles sit close together and stay in fixed positions. They still vibrate in place but cannot trade spots, so a solid holds both its own shape and its own volume.

In a liquid, the particles still touch each other, but they slide past their neighbors freely. A liquid keeps its volume while taking the shape of its container, so pouring 50 mL of water into a wide dish still leaves you with 50 mL.

In a gas, the particles fly apart in every direction and spread out to fill whatever holds them. Most of a gas sample is actually empty space, which is why you can squeeze a gas into a smaller can while a solid barely budges.

Matter heated to extreme temperatures can reach a fourth state called plasma, in which electrons are torn away from their atoms. Stars shine with plasma, and so does the gas inside a glowing neon sign.

Heating or cooling moves a sample between these states. Ice melts into water, water boils into steam, and steam cools and condenses back into water again. Dry ice skips the liquid stage and turns straight into a gas, a change known as sublimation. Through every one of these changes the substance itself stays the same, and only the packing of the particles changes.

Pure substances and mixtures

A pure substance has exactly the same makeup all the way through the sample. An element is a pure substance built from only one kind of atom, such as copper or oxygen. A compound is a pure substance built from two or more elements bonded in a fixed ratio. Water always carries two hydrogen atoms for each oxygen atom, in every sample on Earth.

A mixture holds two or more substances that are not bonded together, so each one keeps its own identity. A homogeneous mixture looks completely uniform throughout, and salt water and clean air are both good examples. We also call a homogeneous mixture a solution. A heterogeneous mixture has visible parts you could pick out, like sand sitting in water or a bowl of trail mix.

Because nothing in a mixture is bonded, a mixture can always be separated by physical means. Boil a pot of salt water and the water escapes as steam, while all of the salt stays behind in the pot.

Section 2Physical and chemical properties

Two kinds of property

A physical property is a trait you can measure without turning the sample into something new. Color, smell, hardness, melting point, boiling point and density are all physical properties. You can melt an ice cube to find its melting point and still have ordinary water at the end.

A chemical property describes the way a sample turns into a new substance, so you have to let the change happen before you can observe one. Iron rusting in damp air is a chemical property of iron, and so is the way paper burns or the way an antacid tablet fizzes in acid.

Does the amount matter?

Some properties depend on how much of the sample you happen to have. Mass, volume and length are extensive properties, so two liters of milk carry twice the mass of one liter.

Other properties stay exactly the same at any amount, and density, color, melting point and temperature are these intensive properties. Cut a copper bar in half and each piece still has a density of 8.96 g/cm³. Intensive properties help you identify an unknown sample, because they do not care how big the sample is.

Physical change and chemical change

A physical change rearranges a sample without changing what the sample is made of. Melting, boiling, dissolving, cutting and bending are all physical changes, and the molecules are identical before and after.

A chemical change produces new substances, and we also call it a chemical reaction. Bonds break, atoms trade partners, and fresh bonds form in their place. Burning methane turns CH₄ and O₂ into CO₂ and H₂O, and those products carry their own new properties.

A handful of clues can hint that a chemical change is under way. You might notice bubbles of a brand new gas, a shift in color, a solid forming inside a clear liquid, or heat and light pouring out. Treat all of these as hints rather than proof, because boiling water bubbles hard while remaining a purely physical change. The real test is whether a new substance is present at the end.

Matter is conserved

The law of conservation of matter says that matter is never created or destroyed during a chemical change. The atoms are only rearranged, and every single atom is still present at the end. That means the total mass of the products equals the total mass of the reactants, and this one rule lets you solve for a mass you never measured.

Section 3Measurement and SI units

A number alone says nothing

Every measurement has two parts, a number and a unit, and neither part works alone. Asking a lab partner for five means nothing at all, because five grams, five liters and five kelvin are wildly different things. Write the unit down every single time.

Scientists in every country share one system of units, the SI units, and SI is short for the International System of Units. Seven base units hold the entire system up, and every other unit is built out of those seven.

Mass is the odd one out in that table, because the kilogram already carries a prefix inside its name. Grams are still what you will weigh samples in on most days in the lab.

Units built from other units

A derived unit is a combination of base units. Area is a length times a length, so area arrives in m², and volume is a length cubed, so its SI unit is m³. A cubic meter is enormous, roughly the size of a washing machine, so laboratories use the liter instead. One liter is exactly one cubic decimeter, and one milliliter is exactly one cubic centimeter.

Speed is measured in meters per second, written m/s, and density is mass over volume, often reported in g/cm³. Watch the pattern in those names, because the word "per" always signals division.

Mass and weight are two different quantities. Mass measures how much matter a sample contains, and it does not change when you travel. Weight is the pull of gravity on that mass, so a 1.00 kg block keeps its mass on the Moon while weighing about one sixth as much.

Prefixes do the heavy lifting

Real samples range from enormous to microscopic, so SI uses prefixes instead of long strings of zeros. Each prefix stands for a power of ten. Kilo means one thousand, so 1 km is 1000 m, and milli means one thousandth, so 1 mg is 0.001 g. Centi is one hundredth, micro is one millionth, and nano is one billionth.

A prefix can be glued onto whatever unit you like, so a nanosecond is a billionth of a second and a megajoule is a million joules. The rule behind the prefix never changes.

Section 4Uncertainty and significant figures

Every measurement carries doubt

No instrument is perfect, so a measurement is really a best value plus some uncertainty. The habit to build is straightforward. Read every digit you are certain of, then estimate one more digit past the finest mark on the scale. That final digit is the uncertain one, and you still write it down.

Picture a ruler with marks printed every centimeter. The edge of your sample sits past the 7 cm mark, about a third of the way toward 8, so you report 7.3 cm. The 7 is certain while the 3 is your estimate. A ruler marked every millimeter would let you report 7.34 cm instead.

A digital screen follows the same logic, so you trust the digits it displays while remembering that the last one still wobbles.

Accuracy is not precision

Accuracy describes how close a measurement lands to the true value, while precision describes how close repeated measurements land to each other. The two can easily come apart. A balance that always reads 2.00 g high is wonderfully precise and badly wrong. Readings scattered all around the true value can average out well while being anything but precise.

Counting significant figures

Significant figures are the digits in a measurement that carry real information about its size. Four rules cover nearly every case you will ever meet.

• Every nonzero digit counts, so 48.2 carries three significant figures.
• Zeros trapped between nonzero digits count, so 4.028 carries four.
• Leading zeros never count, because they only park the decimal point, so 0.0056 carries two.
• Trailing zeros count only when a decimal point is shown, so 1.200 carries four while 1200 stays unclear. Write 1.2 × 10³ whenever you mean two.

Counted objects and defined values are exact numbers, so they hold unlimited significant figures and never limit your answer. There are exactly 12 eggs in a dozen, and exactly 2.54 cm in an inch, because that is how the inch is defined.

Rounding a calculation

Your answer can never be more certain than the data behind it, and two rules keep you honest. For multiplication and division, keep as many significant figures as the measurement that carries the fewest. For addition and subtraction, keep as many decimal places as the measurement that carries the fewest.

Save the rounding for the very end, because rounding in the middle of a long chain lets small errors pile up.

Section 5Dimensional analysis

Multiply by one on purpose

Dimensional analysis is the habit of carrying units through a calculation and letting them steer the setup. Its main tool is a conversion factor, which is a fraction built out of an equality. Because 1 m = 100 cm, both of these fractions are equal to exactly one.

Multiplying by one never changes how much you have, and it only changes the label on the amount. So pick whichever version puts the unit you want on top and the unit you want gone on the bottom, and that unwanted unit then cancels.

The units are also your error check. If the surviving unit is wrong, then your setup was wrong, and no amount of careful arithmetic will rescue it. Flip the factor over and try the chain again.

Chain as many steps as you need

You often will not know the single factor that takes you straight to the target, and that is perfectly fine. String several factors together instead and let each unwanted unit cancel in turn. Write the whole chain first, confirm that only the target unit survives, and save the arithmetic for last.

Rates are conversion factors too

Any ratio that ties two quantities together can serve as a conversion factor. A dose of 0.750 mg per kilogram of body mass is a factor, and so is a syrup labeled 25.0 mg per milliliter. Those two factors let you walk straight from a body mass to a volume you can pour.

Section 6Density and temperature

Density packs mass into space

Density tells you how much mass is crammed into a given volume, and it is simply mass divided by volume.

Solids and liquids are usually reported in g/cm³, which is identical to g/mL, while gases are so light that we report them in g/L. Liquid water is very close to 1.00 g/cm³. It peaks near 4 °C at 1.000 g/cm³ and drops to 0.997 g/cm³ by 25 °C.

Density is an intensive property, so a gold ring and a gold brick share the same density of 19.3 g/cm³. That steadiness makes density a strong clue to the identity of an unknown sample.

Whether something floats depends on its density rather than its mass, so a sample floats in a liquid whenever its density is lower than the liquid's. Ice has a density of 0.917 g/cm³, so ice cubes bob on water, while a steel bolt sinks because steel sits near 7.8 g/cm³. An enormous ship still floats because its shape traps air, which lowers the average density of the whole hull.

Three temperature scales

Temperature tells you which direction heat will flow, because heat always moves from the hotter sample toward the cooler one on its own.

The Celsius scale sets 0 °C at the freezing point of water and 100 °C at its boiling point under 1 atm of pressure. That span is divided into 100 steps called degrees.

The Fahrenheit scale places those same two points at 32 °F and 212 °F. That gives 180 degrees across the identical span, so a Fahrenheit degree is smaller, and nine Fahrenheit degrees cover the same gap as five Celsius degrees.

The kelvin is the SI unit of temperature. Its steps are exactly the size of Celsius degrees, but the scale begins at absolute zero, the coldest temperature that can exist. Absolute zero is 0 K, which is −273.15 °C. Nothing can be colder, so kelvin never goes negative and it takes no degree sign at all. You say 298 kelvin rather than 298 degrees kelvin.

ReferenceWords worth knowing

Here are the terms this chapter leans on. Each one shows up again later in the book.

Absolute zero

The coldest temperature possible, 0 K or −273.15 °C.

Accuracy

How close a measurement is to the true value.

Chemical change

A change that breaks and forms bonds, producing new substances. Also called a reaction.

Chemical property

A trait you can only observe by letting the sample turn into something new.

Compound

A pure substance built from two or more elements bonded in a fixed ratio.

Conversion factor

A fraction made from an equality, so its value is exactly one.

Density

Mass divided by volume, usually in g/cm³ for solids and liquids.

Derived unit

A unit made by multiplying or dividing base units, such as m/s.

Dimensional analysis

Carrying units through a calculation and using them to check the setup.

Element

A pure substance made of only one kind of atom.

Extensive property

A property that changes with the amount of sample, such as mass.

Heterogeneous mixture

A mixture with parts you can see and separate, like sand in water.

Homogeneous mixture

A mixture that looks uniform throughout. Also called a solution.

Intensive property

A property that stays the same at any amount, such as density.

Law of conservation of matter

The rule that matter is neither created nor destroyed in a chemical change.

Physical change

A change in form or state that leaves the substance itself alone.

Physical property

A trait you can measure while the sample stays the same substance.

Precision

How close repeated measurements are to each other.

Significant figures

The digits of a measurement that carry real information about its size.

Uncertainty

The amount of doubt that rides along with every measurement.

Atoms, Molecules and Ions

Everything around you is built from a short list of parts. Protons, neutrons and electrons pack into atoms. Atoms join into molecules, or trade charge and become ions. From there they stack into every material you have ever touched. This chapter hands you the parts and the rules. You will count protons to name an element. You will mix isotopes to get an atomic mass. You will read the periodic table like a map. You will balance charge to write a formula. Then you will pick up the naming system chemists share. By the end, a label like Fe(NO₃)₃ will read like a sentence.

Section 1Atomic theory and the subatomic particles

A very old idea, finally tested

People argued about matter for a long time. Could you cut a lump of gold in half forever? Or do you reach a smallest piece that will not split? For centuries nobody could test the question. Careful weighing finally settled it.

In 1808 John Dalton pulled the evidence into one theory. An atom is the smallest piece of an element that still behaves like that element. All atoms of one element act the same way in a reaction. Atoms of different elements differ. Atoms combine in fixed, whole-number ratios to make compounds. And a chemical reaction only rearranges atoms. It never creates them and never destroys them.

Two mass laws backed him up. The law of definite proportions says one compound always holds the same mass ratio of elements. Pure water is 11.19% hydrogen by mass, whether it came from a river or a flask. The law of multiple proportions compares two compounds made from the same pair of elements. In carbon monoxide, 1.00 g of carbon is joined to 1.33 g of oxygen. In carbon dioxide, 1.00 g of carbon is joined to 2.66 g of oxygen. The ratio of 2.66 to 1.33 is a clean 2 to 1. Tidy whole numbers like that point to whole particles.

Three particles, and where they sit

Dalton thought the atom had no parts. He was wrong, and the next hundred years showed why. J. J. Thomson ran a current through a thin gas and watched a beam bend toward the positive plate. He had found the electron, a very light particle with a negative charge. Ernest Rutherford then fired tiny positive bullets at thin gold foil. Most flew straight through. A few bounced almost straight back. That only happens if the positive charge is crammed into a tiny core. He named the core the nucleus. The positive particle inside it is the proton. In 1932 James Chadwick found the neutron, a nuclear particle with no charge at all.

Masses here are quoted in the atomic mass unit (u). One unit is exactly 1/12 the mass of a carbon-12 atom, or 1.6605 × 10^-24 g. A proton weighs 1.0073 u. A neutron weighs 1.0087 u. An electron weighs only 0.00055 u, which is about 1/1836 of a proton. So the nucleus carries almost all of the mass.

Size runs the other way. A whole atom is roughly 10^-10 m across. Its nucleus is roughly 10^-15 m across. The atom is about 100,000 times wider than its own core. Blow the nucleus up to the size of a marble and the atom would stretch across a sports field. Almost all of an atom is empty space with a few electrons moving through it.

Counting protons names the element

The atomic number (Z) is the number of protons in the nucleus. It is the atom's identity badge. Six protons means carbon, every single time. The mass number (A) is the protons plus the neutrons added together. Chemists write both on the symbol, with A above and Z below. A neutral atom holds as many electrons as protons. Change the electron count and the atom carries a charge instead.

Section 2Isotopes and atomic mass

Same element, different weight

Dalton said every atom of an element has the same mass. That part needed a repair. Atoms of one element can hold different numbers of neutrons. Those versions are isotopes. They share an atomic number and differ in mass number. Hydrogen shows it most clearly. Almost every hydrogen atom is one proton with no neutron at all. Roughly one in 8,700 carries a neutron, and that form is called deuterium. A third form, tritium, carries two neutrons and breaks down over time.

Isotopes of one element react the same way. Chemistry is run by electrons, and isotopes have matching electron counts. Mass does shift a few physical traits. Heavy water, D₂O, freezes at 3.8 °C instead of 0 °C. It is also about 11% denser than ordinary water.

A nuclide is one specific build of a nucleus, such as carbon-13 or uranium-235. Naming a nuclide means fixing both numbers, not just the element. Some nuclides are stable and last forever. Others come apart and release radiation.

Weighing a mixture

A mass spectrometer sorts atoms by mass. It knocks an electron off each atom, speeds the ions up, then bends their path with a magnet. Heavy ions bend less than light ones. A detector counts how many land at each mass. Out comes the mass of every isotope and its percent abundance, the share of atoms that are that isotope.

The number printed under each symbol on the periodic table is the average atomic mass. It is a weighted average, not a plain one. Multiply each isotope mass by its share, then add the results:

Use the fraction, so divide each percent by 100 first. The answer always leans toward the common isotope. Chlorine averages 35.45 u, far nearer 35 than 37, because the lighter form is about three times as common.

Mass number is a count, atomic mass is a mass

Keep these two apart. A mass number is a whole number and nothing else. It counts protons plus neutrons. An atomic mass is a measured mass in u, and it usually arrives with decimals. Carbon-13 has mass number 13 and an atomic mass of 13.00335 u. Only carbon-12 has both numbers equal, and that is on purpose.

Section 3The periodic table

A map built out of behavior

By the 1860s chemists knew dozens of elements and their masses. Dmitri Mendeleev lined them up in order of mass and spotted a repeat. Properties kept coming back around. So he broke the line into rows and stacked the look-alikes into columns. He even left holes for elements nobody had found, and guessed what they would be like. Gallium turned up in 1875 and matched his guess closely. That is when the table stopped looking like a filing trick and started looking like a law.

Today the elements are ordered by atomic number instead of mass. That fixed a few seats. Tellurium weighs more than iodine, yet tellurium comes first, because it has fewer protons. Order by protons and every look-alike lands in the right column.

Rows, columns and the long drawers

A row is a period, numbered 1 through 7. A column is a group, numbered 1 through 18. Elements in one group behave alike because they hold the same number of outer electrons. Four groups have names worth knowing. Group 1 is the alkali metals, soft metals that react hard with water. Group 2 is the alkaline earth metals. Group 17 is the halogens, which grab electrons eagerly. Group 18 is the noble gases, which mostly refuse to react with anything.

Groups 1, 2 and 13 through 18 are the main group elements. Groups 3 through 12 hold the transition metals, the familiar working metals like iron, copper and nickel. Two long rows are pulled out and printed underneath, which keeps the page from stretching too wide. They are the lanthanides, elements 57 through 71, and the actinides, elements 89 through 103. Slide them back into period 6 and period 7 and the table becomes one sheet 32 columns across.

Metals, nonmetals and the stair step

A jagged step runs down the right side of the table. Metals sit to the left of it. They shine, they bend rather than shatter, and they carry heat and current well. Nonmetals sit to the right. They are dull, brittle when solid, and poor conductors. Sitting right on the step are the metalloids, which act partly like each side. Boron, silicon, germanium, arsenic, antimony and tellurium are the usual six. Silicon's halfway conducting is exactly what computer chips need. Books draw the step in slightly different places, so polonium and astatine get sorted either way.

Hydrogen is printed above group 1, but it is a nonmetal and does its own thing. Of the 118 known elements, the first 94 turn up in nature, a few only in traces. The rest are built one atom at a time in accelerators and vanish in moments. At room temperature most elements are solid. Eleven are gases. Only two are liquid, mercury and bromine.

Section 4Ions and ionic compounds

Move an electron and you make a charge

An atom with equal protons and electrons has no net charge. Pull an electron away and the protons now outvote the electrons, so the particle is positive. Push an extra electron on and the particle is negative. A charged atom, or a charged group of atoms, is an ion. A positive ion is a cation. A negative ion is an anion. Losing one electron gives 1+. Losing two gives 2+. Gaining one gives 1−. Note that protons never move in ordinary chemistry. Only electrons do.

The group number predicts the charge for most main group elements. Group 1 forms 1+ ions. Group 2 forms 2+. Aluminum, in group 13, forms 3+. Group 15 often takes on three electrons for 3−. Group 16 takes two for 2−. Group 17 takes one for 1−. Noble gases usually form no ion at all. Every one of those ions ends up with the same outer electron count as a noble gas, which is the pattern behind the rule.

Transition metals are less tidy. Many of them form more than one ion. Iron forms Fe²⁺ and Fe³⁺. Copper forms Cu⁺ and Cu²⁺. A handful are fixed and easy. Zinc is always Zn²⁺ and silver is always Ag⁺.

Ions built from several atoms

A polyatomic ion is a bonded group of atoms carrying one charge for the whole group. Sulfate is SO₄^2-. Its sulfur and four oxygens travel together as a single unit, and the 2− belongs to the unit, not to any one atom. Ammonium, NH₄⁺, is the common positive one. Nitrate is NO₃^- and nitrite is NO₂^-. In pairs like that, the -ate form always holds one more oxygen than the -ite form.

Balancing the charge writes the formula

An ionic compound pairs cations with anions, and it is usually a metal with a nonmetal. There is no single molecule to point at. Instead the ions stack into a repeating three-dimensional grid called a lattice, which runs on for as long as the crystal does. So the formula gives the smallest whole-number ratio that cancels the charge. That ratio is called a formula unit.

The rule is one line. Total positive charge plus total negative charge must come to zero. Find the smallest counts that pull that off. One useful trick is the lowest common multiple of the two charge sizes.

Section 5Molecular compounds and formulas

Sharing instead of handing over

Two nonmetals do not pass electrons back and forth. Neither one gives up easily, so they share instead. A shared pair of electrons is a covalent bond. A neutral group of atoms held together by covalent bonds is a molecule. A molecular compound is built out of molecules, so it comes in countable units. One water molecule is two hydrogens and one oxygen, and the next one is the same.

That is a real difference from an ionic compound. Table salt has no smallest salt particle, only a ratio that repeats through the crystal. Sugar does have one. You can point at a single sucrose molecule and say that is the whole thing. Molecular compounds also tend to melt at lower temperatures and dissolve without splitting into ions.

Some elements travel as molecules of themselves. Seven are diatomic under normal conditions: H₂, N₂, O₂, F₂, Cl₂, Br₂ and I₂. Phosphorus goes around as P₄ and sulfur as S₈. A lone oxygen atom is rare and very reactive, which is why the air holds O₂ and not O.

Three ways to write the same compound

A molecular formula counts every atom in one molecule. Glucose is C₆H₁₂O₆, so six carbons, twelve hydrogens and six oxygens. An empirical formula gives the smallest whole-number ratio of those atoms. Divide every subscript in glucose by 6 and you get CH₂O. Something is lost on the way. Formaldehyde really is CH₂O, and acetic acid is C₂H₄O₂. All three share one ratio, and all three are completely different substances.

A structural formula shows which atom is bonded to which. Lines stand in for bonds. That is often the detail that decides behavior. Ethanol and dimethyl ether are both C₂H₆O. Ethanol is the alcohol in drinks. Dimethyl ether is a gas used as a propellant. Compounds that share a molecular formula and differ in structure are isomers.

Models push further still. A ball-and-stick model keeps the bonds visible so you can see the angles between them. A space-filling model drops the sticks and swells the balls, so you see the room the electrons actually occupy. Each view hides something, which is why chemists keep all of them.

Section 6Naming compounds

Two questions sort every formula

Chemists needed one naming system so that a formula reads the same in any country. The system splits by compound type, so start by deciding which type you have. Ask two questions in order. Does the formula start with hydrogen and act as an acid in water? Then name it as an acid. If not, does it start with a metal or with ammonium? If yes it is ionic, and if no it is molecular. Two questions, three roads.

Ionic names

Name the cation, then the anion, and never use prefixes. The metal keeps its ordinary name. A single-atom anion drops its ending and picks up -ide. So NaCl is sodium chloride and MgO is magnesium oxide. A polyatomic ion keeps whatever name it already has, so Na₂SO₄ is sodium sulfate.

If the metal can form more than one charge, you have to say which one. Put the charge in Roman numerals in parentheses right after the metal name. FeCl₂ is iron(II) chloride and FeCl₃ is iron(III) chloride. You read that charge off the formula by balancing, never by guessing. Metals with only one common charge, such as sodium or zinc, get no numeral at all. One more case: a hydrate traps water inside its crystal, and a prefix counts it. CuSO₄·5H₂O is copper(II) sulfate pentahydrate.

Molecular names

Here you name the subscripts out loud with prefixes. Mono is 1, di is 2, tri is 3, tetra is 4, penta is 5, hexa is 6, hepta is 7, octa is 8, nona is 9 and deca is 10. The first element keeps its name and skips mono. The second element takes the -ide ending and always gets its prefix. CO is carbon monoxide and CO₂ is carbon dioxide. SF₆ is sulfur hexafluoride. Drop the final vowel of a prefix before oxide, which is why you say monoxide and pentoxide.

Acid names

Acids come in two families. A binary acid is hydrogen plus one other element. Use hydro-, then the root of that element, then -ic acid. HCl in water is hydrochloric acid. An oxyacid is hydrogen plus a polyatomic ion that holds oxygen. Here the ending of the anion steers the name. An -ate anion gives -ic acid. An -ite anion gives -ous acid. So HNO₃ is nitric acid and HNO₂ is nitrous acid. H₂SO₄ is sulfuric acid and H₂SO₃ is sulfurous acid.

ReferenceWords worth knowing

Every term below showed up in bold somewhere above. Come back here whenever a word stops making sense.

alkali metals

Group 1 metals. Soft, very reactive, and they form 1+ ions.

anion

An ion with a negative charge, made by gaining one or more electrons.

atomic mass unit (u)

A mass unit equal to 1/12 the mass of a carbon-12 atom, or 1.6605 × 10^-24 g.

atomic number (Z)

The number of protons in a nucleus. It decides which element the atom is.

average atomic mass

The isotope masses of an element weighted by their abundances and added. This is the value printed on the table.

cation

An ion with a positive charge, made by losing one or more electrons.

covalent bond

A pair of electrons shared between two atoms.

empirical formula

The smallest whole-number ratio of atoms in a compound, such as CH₂O.

formula unit

The smallest whole-number ratio of ions that makes an ionic compound neutral.

group

A column of the periodic table, numbered 1 to 18. Members behave alike.

ion

An atom or bonded group of atoms that carries a net electric charge.

ionic compound

A compound of cations and anions held in a lattice, usually a metal with a nonmetal.

isotopes

Atoms of one element with different neutron counts, so the same Z and different A.

mass number (A)

Protons plus neutrons in one nucleus. Always a whole number.

metalloid

An element on the step between metals and nonmetals that behaves partly like each.

molecular formula

The actual count of each kind of atom in one molecule, such as C₆H₁₂O₆.

oxyacid

An acid of hydrogen plus a polyatomic ion containing oxygen, such as HNO₃.

percent abundance

The share of atoms in a natural sample that are one particular isotope.

period

A row of the periodic table, numbered 1 to 7.

polyatomic ion

A bonded group of atoms carrying one charge for the whole group, such as SO₄^2-.

Composition of Substances and Solutions

Atoms are far too small to count one at a time. A single grain of salt holds more of them than you could tally in a lifetime. So chemists count by weighing instead. This chapter hands you the tool that makes that trade work. It is called the mole, and it ties the mass on a balance to the number of particles you cannot see. From there you will find the mass of any formula, split a compound into the share of mass each element owns, and work backward from lab data to a formula. At the end you will mix and water down solutions to an exact strength.

Section 1The mole and Avogadro's number

Chemistry needs a counting word

You already use counting words every day. A dozen means 12. A ream of paper means 500. Each word stands for a fixed count, whatever you point it at. A dozen eggs and a dozen bricks hold the same number of things. Their masses are nothing alike.

Chemistry needs a counting word too, but a huge one. A teaspoon of water holds roughly 1.7 × 10²³ molecules. Counting them one per second would take longer than the age of the universe. The mole is the counting word that makes such numbers workable. One mole is 6.022 × 10²³ of whatever you are counting. The unit is written mol.

That count has a name of its own. Avogadro's number, written N_A, is 6.022 × 10²³ particles per mole. The exact value is now fixed by definition. One mole is 6.02214076 × 10²³ particles, no more and no less. Four digits is plenty for the work in this book.

Why such a strange count

The number looks random. It was chosen with great care. It was set so that one mole of carbon-12 atoms weighs almost exactly 12 grams. That link is the whole point of the mole. It means the mass of one mole in grams matches the mass of one particle in atomic mass units. Carbon has an atomic mass of 12.011 amu, so one mole of carbon atoms weighs 12.011 g. You weigh grams on a balance. You get a count for free.

The word particle is doing real work in that definition. A mole counts whatever you aim it at, so you have to say what. A mole of neon is 6.022 × 10²³ atoms. A mole of water is that many molecules. An ionic solid like NaCl has no molecules at all, so we count formula units instead. One formula unit of NaCl is one Na⁺ paired with one Cl^-.

Crossing the bridge

Moles sit in the middle of every problem in this chapter. Mass sits on one side. Particle count sits on the other. To go from moles to particles, multiply by 6.022 × 10²³ per mole. To come back, divide by it. Write the unit on every number and let the units cancel. If what is left over is not the unit you wanted, your setup is wrong, not your calculator.

Section 2Molar mass

The mass of one mole

Molar mass is the mass of one mole of a substance. Its unit is grams per mole, written g/mol. For a single element you can read it straight off the periodic table. Take the atomic mass and attach g/mol to it. Carbon is 12.011 g/mol. Oxygen is 15.999 g/mol. Iron is 55.845 g/mol.

For a compound you add up the atoms in the formula. Each subscript tells you how many of that atom to count. Multiply each atomic mass by its subscript, then add. The total is the formula mass in amu. It is also the molar mass in g/mol. The number is the same and only the unit changes.

Water is H₂O. Two hydrogens at 1.008 plus one oxygen at 15.999 gives 18.015. So water is 18.02 g/mol. Through this book we round molar masses to two decimal places. That is already finer than any balance you will use.

Parentheses and raised dots

Some formulas carry a group inside parentheses. The subscript outside multiplies everything within the group. Ca(NO₃)₂ holds one Ca, two N and six O. Miss that and your answer lands far from the truth. The same rule holds for a long formula like Al₂(SO₄)₃, which holds 2 Al, 3 S and 12 O for a total of 342.13 g/mol.

Other formulas carry a raised dot, as in CuSO₄·5H₂O. A solid like this is a hydrate. Water molecules sit inside the crystal in a fixed ratio. The dot does not mean multiply. It means that much water is locked in. Add the water in as five whole units of H₂O. The pentahydrate comes out at 249.68 g/mol. Drive the water off with heat and the dry salt is only 159.61 g/mol. Forgetting the water throws a weighing off by more than a third.

One factor, two faces

Molar mass is a conversion factor you can flip. Written as g/mol it turns moles into grams. Flipped to mol/g it turns grams into moles. Choose the face that cancels the unit you are holding. That choice is the only decision in most of these problems.

Section 3Percent composition

Who owns the mass

A pure compound has a fixed recipe. Every sample of pure glucose carries the same share of carbon by mass, whether it came from a grape or a beaker. Percent composition reports that share for each element in the formula. It answers a very practical question. In 100 g of this substance, how many grams are carbon?

The method is one line long. Take exactly one mole of the compound. For each element, divide the mass that element contributes by the molar mass of the whole thing. Multiply by 100 to turn the fraction into a percent. The result for one element is its mass percent.

mass percent of X = (mass of X in one mole ÷ molar mass) × 100%

Notice that the compound's identity is doing all the work. You never need to know how big your sample is. A grain and a truckload give the same percents. That is why percent composition is printed on labels and quoted in catalogs.

A number you can reuse

Percent composition is more than a tidy fact. Once you have it, you can jump straight from the mass of a sample to the mass of one element inside it. Turn the percent into a decimal and multiply. A fertilizer bag marked 35.00% nitrogen says just that. A 50.0 kg bag holds 0.3500 × 50.0 kg = 17.5 kg of nitrogen. No moles needed.

The shares always add to 100%, give or take a little rounding. That hands you a free check. If your percents sum to 94% or 108%, something is wrong. Go back and recount the atoms in your molar mass. A missed oxygen or a dropped parenthesis is the usual culprit.

Section 4Empirical and molecular formulas

Two kinds of formula

A molecular formula tells you exactly how many atoms sit in one molecule. An empirical formula gives the smallest whole-number ratio of those same atoms. Glucose is C₆H₁₂O₆. Divide all three subscripts by 6 and you get CH₂O. That is the empirical formula of glucose.

Very different compounds can share one empirical formula. Formaldehyde really is CH₂O. Acetic acid is C₂H₄O₂, which also reduces to CH₂O. All three have the very same percent composition. So percent data by itself can never hand you a molecular formula. You need one more measurement, and that measurement is the molar mass.

Ionic compounds are a special case. A crystal has no molecules, so its formula is already the simplest ratio. NaCl and Fe₂O₃ are empirical formulas.

From lab data to a ratio

Analysis gives you masses or percents. It never hands you subscripts. The fix is to turn mass into moles, because subscripts are mole ratios. Four steps do it every time.

1. Start with grams. If you were given percents, pretend you hold exactly 100.00 g of the compound. Then each percent is already a mass in grams.
2. Divide each mass by that element's molar mass. Now every number is in moles.
3. Divide every mole value by the smallest of them. The smallest becomes 1.000 and the rest fall close to whole numbers.
4. If a value sits near .50, .33 or .25, multiply all of them by 2, 3 or 4 to clear it.

Judgment lives in step 4. A value of 1.99 is 2, since lab data is never perfect. A value of 1.50 is not 2. Rounding it would change the compound. Multiply instead.

Scaling up to the molecule

The empirical formula has a mass of its own, called the empirical formula mass. Divide the real molar mass by it. The answer is a whole number n, the number of empirical units per molecule. Multiply every subscript by n and you have the molecular formula.

n = molar mass ÷ empirical formula mass

If n comes out as 1, the two formulas are the same. Water and ammonia are both like that. If n lands on 2.98 or 3.03, treat it as 3 and move on.

Section 5Molarity

Strength is not amount

A solution is a mixture so evenly blended that it acts like one substance. The part that dissolves is the solute. The part it dissolves into is the solvent. When the solvent is water we call it an aqueous solution and tag the formula with (aq).

Picture two glasses of the same salt water. One holds a sip and one holds a liter. The amount of salt in them is very different. The strength is identical. Concentration is the word for that strength. It is the amount of solute per amount of solution, and it does not care how much you poured.

Molarity is the concentration unit chemistry leans on hardest. It is moles of solute per liter of solution. The symbol is a capital M. A bottle marked 2.0 M is read out loud as two molar.

molarity (M) = moles of solute ÷ liters of solution

Of solution, not of solvent

Read the bottom of that fraction again. It says liters of solution. It does not say liters of solvent. The difference is real work in the lab. You do not pour 1.00 L of water onto your solid. You drop the solid into a volumetric flask, swirl it into part of the water, then top up to the etched mark on the neck. The solute takes up room of its own. Filling to the mark is the only way to know your final volume.

Molarity has one soft spot. Liquids swell when warmed, so a volume shifts a little with temperature. A 0.100 M solution on a hot bench is a hair weaker than the same flask in a cold room. For ordinary lab work the drift is too small to matter.

Three ways the equation gets used

Molarity, moles and volume form a small triangle. Know any two and you can get the third. Moles = M × V. Volume = moles ÷ M. Convert the volume to liters before anything else. 250.0 mL is 0.2500 L, and 25.0 mL is 0.0250 L.

Section 6Dilution

Add water, keep the solute

Labs rarely weigh out a solid for every weak solution they need. They mix a strong one and water it down. The strong one is the stock solution. Watering it down is dilution.

One idea makes every dilution problem easy. Adding water adds no solute and removes none. The moles before equal the moles after. Only the volume grew, so the concentration fell.

Moles are M × V, so that sentence becomes an equation. Subscript 1 is before and 2 is after.

M₁V₁ = M₂V₂

Both sides of that line are moles of solute. The same volume unit appears on each side, so you may leave both volumes in milliliters. The only rule is to use one unit throughout.

Working out the pour

Usually you know the stock concentration, the one you want and the final volume. The unknown is how much stock to measure out. Rearrange for V₁.

V₁ = M₂V₂ ÷ M₁

Measure out that much stock, then add solvent up to V₂. With a strong acid the order matters. Acid goes into water, never water into acid, because mixing releases a lot of heat.

The dilution factor

The dilution factor is V₂ ÷ V₁. It also equals M₁ ÷ M₂, a handy second check. Take 10.0 mL up to 100.0 mL and the factor is 10. The concentration drops to one tenth. That is a 1:10 dilution.

Factors multiply when you chain steps. Two 1:10 steps in a row give a factor of 100. This is called a serial dilution. It is how a lab reaches very low concentrations without weighing out a speck of solid.

ReferenceWords worth knowing

These are the terms this chapter leans on. Each one shows up in the worked examples above, so read them with those numbers in mind.

Aqueous solution

A solution whose solvent is water. Written with the tag (aq) after a formula.

Avogadro's number

The number of particles in one mole, N_A = 6.022 × 10²³ per mole. Fixed by definition at 6.02214076 × 10²³.

Concentration

The amount of solute per amount of solution. A measure of strength, not of quantity.

Dilution

Adding solvent to a solution to lower its concentration. The moles of solute stay the same.

Dilution factor

V₂/V₁, equal to M₁/M₂. A factor of 10 means a 1:10 dilution.

Empirical formula

The smallest whole-number ratio of atoms in a compound. CH₂O is the empirical formula of glucose.

Formula mass

The sum of the atomic masses in a formula, in amu. Numerically equal to the molar mass in g/mol.

Formula unit

The smallest neutral group an ionic compound's formula describes, such as one Na⁺ with one Cl^-.

Hydrate

A solid that holds water inside its crystal in a fixed ratio, shown with a raised dot, as in CuSO₄·5H₂O.

Mass percent

The share of a compound's mass owned by one element, as a percent. Also called percent by mass.

Molar mass

The mass of one mole of a substance, in grams per mole. The bridge between grams and moles.

Molarity

Moles of solute per liter of solution, symbol M. The most used concentration unit in this book.

Mole

The counting unit for particles. One mole is 6.022 × 10²³ atoms, molecules or formula units.

Molecular formula

The actual number of atoms of each element in one molecule, such as C₆H₁₂O₆.

Percent composition

The full set of mass percents for a compound. The shares add to 100%.

Solute

The substance that dissolves in a solution. It is usually the smaller part by mass.

Solution

A mixture blended so evenly that it behaves as a single substance.

Solvent

The substance that does the dissolving. Water is the most common one.

Stock solution

A concentrated solution kept on hand and diluted when a weaker one is needed.

Volumetric flask

A flask with one etched line on its neck, used to make a solution up to an exact volume.

Stoichiometry of Chemical Reactions

A chemical equation is a recipe written in formulas. It names what goes in, what comes out, and in what ratio. This chapter turns that recipe into numbers you can trust. First you will balance an equation so that no atom appears or vanishes. Then you will sort reactions into a few families, so a new one looks familiar. You will track oxidation numbers to see which atoms gave electrons away. Last you will use the balanced equation as a bridge. Grams become moles, moles of one substance become moles of another, and moles become grams again. By the end you can predict how much product a real run should give.

Section 1Writing and balancing equations

What a chemical equation says

A chemical equation is a short written record of a chemical change. The starting substances, called the reactants, sit on the left side. The substances that form, called the products, sit on the right side. An arrow joins the two sides, and that arrow means turns into. Here is methane burning in air.

The letters in parentheses record the physical state of each substance. Use (s) for a solid, (l) for a liquid and (g) for a gas. Use (aq) whenever the substance is dissolved in water. That last label is short for aqueous, which simply means dissolved in water.

Two kinds of number show up in an equation, and they do completely different jobs. The large number in front of a formula is a coefficient, and it counts whole units of that substance. The small lowered number inside a formula is a subscript, and it counts atoms inside a single unit. You may change a coefficient whenever balancing requires it. You may never change a subscript, because a different subscript names a different substance. Carbon dioxide, CO₂, behaves nothing like carbon monoxide, CO.

Why the two sides must match

Atoms are never created or destroyed during a chemical change, and that statement is the law of conservation of matter. Bonds break and new bonds form, but the very same atoms turn up again in the products. So every element must appear in equal numbers on both sides of the arrow. An equation that passes that test is balanced.

The reliable way to get there is an atom inventory. List every element down the page, count it on the left, then count it on the right. Change one coefficient and count again. Two habits make the process a great deal faster. Begin with an element that appears in only one formula on each side. Save the element that appears in the most formulas for last, which is usually oxygen. Finish by reducing everything to the smallest whole numbers that still work.

Section 2Classifying reactions

One reaction, three equations

A huge share of chemistry happens in water. A reaction in water can be written three ways, and each way shows something the others hide. The molecular equation shows every compound as a whole formula. The complete ionic equation splits each dissolved salt, strong acid and strong base into its free ions. The net ionic equation keeps only the parts that change. Ions that start out free and end up free took no part at all, and those are the spectator ions you cross off.

Precipitation: a solid falls out

Pour two clear solutions together and a cloud of solid can suddenly appear. That solid is a precipitate, and the change is a precipitation reaction. To predict one in advance you need the solubility rules, which are short statements about which ionic solids dissolve well in water and which hardly dissolve at all.

Using the table takes two simple moves. Swap the partners in the two compounds to see the possible new pairings, then test each new pair against the rules. Any pair the table calls insoluble is your precipitate.

Acids and bases trade a proton

An acid gives away a hydrogen ion, H⁺. A base accepts one. When an acid meets a base in water the two cancel out, and chemists call that an acid-base reaction, or neutralization. The products are water and a dissolved salt. A strong acid comes apart fully in water. The common ones are HCl, HBr, HI, HNO₃, H₂SO₄ and HClO₄. A weak acid only comes apart part way, like the acetic acid in vinegar.

Strip out the Na⁺ and Cl^- spectators and a single short line remains. Every strong acid and strong base pairing boils down to exactly the same change.

Redox moves electrons

In the third family, electrons actually move from one atom to another. Chemists call that a redox reaction, which is short for oxidation and reduction. Burning fuel is redox, and so is iron rusting or a battery running your phone. Combustion is the rapid reaction of a fuel with oxygen. In a single displacement reaction, one element shoves another element out of its compound. Zinc metal does exactly that to the hydrogen in hydrochloric acid.

Some reactions sit in two families at once, and the next section gives you a firm test for the redox family.

Section 3Oxidation numbers and redox

A bookkeeping charge

An oxidation number is the charge an atom would carry if every bond around it were completely ionic. You pretend each shared pair belongs entirely to the greedier atom, and then you count. The result is usually not the real charge on that atom at all. It is still worth the trouble, because any change in the number proves that electrons moved.

The rules, in order

Work down this list and stop as soon as a rule settles an atom, because the higher rules always win.

1. An atom in a free element is 0. In O₂, in Na metal and in S₈, every atom is 0.
2. A single atom ion takes its own charge. Na⁺ is +1 and S^2- is -2.
3. Fluorine in a compound is always -1.
4. Group 1 metals are +1. Group 2 metals are +2.
5. Hydrogen is +1 with a nonmetal and -1 with a metal, as in NaH.
6. Oxygen is -2, except in a peroxide like H₂O₂, where it is -1.
7. Other halogens are -1 unless bonded to oxygen or to fluorine.
8. The numbers sum to 0 in a neutral compound. In an ion they sum to the charge.

Rule 8 is the one that does the real work, because it lets you solve for the single atom you do not know yet.

Which atom was oxidized?

Oxidation is a rise in oxidation number, which means the atom gave electrons away. Reduction is a drop in oxidation number, which means the atom took electrons in. Neither one can happen alone, since the electrons always have to go somewhere. The substance that accepts the electrons is the oxidizing agent, and the substance that hands them over is the reducing agent. Notice the twist in those names, because the oxidizing agent is itself the species that gets reduced.

You can split the whole change into two half-reactions, one for the loss and one for the gain. Zinc metal dropped into copper sulfate solution shows both halves clearly.

Section 4Mole to mole stoichiometry

The coefficients are a ratio

A balanced equation is far more than a tidy record of a change. Its coefficients give you a ratio you can actually count with, so read them as moles. In the reaction that makes ammonia, one mole of N₂ reacts with three moles of H₂ and yields two moles of NH₃.

Any pair of those numbers is a mole ratio. From this single equation you can pull 1 mol N₂ per 3 mol H₂, or 2 mol NH₃ per 1 mol N₂, and several others. A mole ratio is simply a conversion factor, so write it as a fraction and let the units cancel. The study of these reacting amounts is called stoichiometry.

The road map

Here is the snag. A laboratory balance weighs in grams, but the equation speaks only in moles. So almost every amount problem walks the same four stop road. Start with grams of the substance you know, then divide by its molar mass to reach moles. Multiply by the mole ratio to cross over into moles of the substance you want. Finally multiply by that second molar mass to reach grams of your answer.

Notice that the mole ratio is the only step that ever crosses the arrow, since the two outer steps never leave their own substance. If a problem hands you moles instead of grams, you simply skip a step and begin further along the road.

Section 5Limiting reactant

When one ingredient runs out

Think about building sandwiches, where each one takes two slices of bread and one slice of cheese. You have 10 slices of bread and only 3 slices of cheese. The bread could cover five sandwiches, but the cheese stops you at three. Cheese is the ingredient that sets the limit, and four slices of bread are left on the counter.

Reactions behave in exactly the same way. The reactant that runs out first is the limiting reactant, and it alone decides how much product can form. Any reactant still sitting there afterward is present in excess, and that leftover material is the excess reactant.

How to find the limiter

Do not compare masses. Grams are a trap here, because a mole of one substance can weigh fifteen times as much as a mole of another. Compare what each reactant could make instead, by converting each starting amount into moles of the same product. The smallest answer is the real one, and the reactant that produced it is limiting.

Once you know the limiter, the rest is simply the road map from the previous section, using only the limiting amount. To find what remains, work out how much of the excess reactant the reaction actually consumed, then subtract that from what you started with.

Look again at the masses in that example, because there was far more oxygen on the balance, yet oxygen still ran out first. The ratio inside the equation, and not the weight on the balance, is what really matters.

Section 6Theoretical, actual and percent yield

Three different yields

Stoichiometry tells you the most product you could possibly get, but real flasks rarely deliver it. Chemists keep three separate numbers apart so they can report how a run actually went.

Theoretical yield is the amount the limiting reactant allows, calculated from the road map, so it is a prediction. Actual yield is what you really collect, dry and weigh, so it is a measurement. Actual yield is almost always the smaller of the two. Some product clings to the glassware. Some reactant never finds a partner. A side reaction can steal material, or the reaction may reach equilibrium before it finishes. Percent yield puts the two numbers side by side.

Both yields must be expressed in the same unit before you divide. Grams over grams works, and moles over moles works, but you cannot mix the two.

Working the formula backward

The same relationship answers two other useful questions. If you know the usual percent yield, you can predict the real haul before you even begin. If you need a set amount of product, you can work out how much starting material to weigh.

One warning before you go. A percent yield above 100% is not a triumph, because it means the measurement itself is wrong. The product is probably still damp with solvent, or some unreacted solid came along with it. Dry the sample thoroughly and then weigh it again.

ReferenceWords worth knowing

These are the terms this chapter leans on. Come back to the list whenever an equation stops making sense.

acid-base reaction

A reaction in which an acid transfers H⁺ to a base, usually giving water and a dissolved salt. Also called neutralization.

actual yield

The mass or moles of product you really collect and weigh at the end of a run.

balanced equation

A chemical equation whose coefficients give every element the same atom count on both sides of the arrow.

coefficient

The whole number written in front of a formula. It counts units of that substance and may be changed to balance an equation.

combustion

The rapid reaction of a fuel with oxygen, releasing heat and light. Hydrocarbon combustion gives CO₂ and H₂O.

excess reactant

A reactant still present after the reaction stops, because the limiting reactant ran out first.

limiting reactant

The reactant that is used up first. It sets the maximum amount of product that can form.

mole ratio

A conversion factor built from two coefficients of a balanced equation, such as 2 mol NH₃ per 1 mol N₂.

net ionic equation

An equation in which spectator ions are removed, leaving only the species that change.

oxidation

An increase in oxidation number, which means the atom lost electrons.

oxidation number

The charge an atom would have if every bond in its compound were fully ionic. A bookkeeping device, not a measured charge.

oxidizing agent

The substance that accepts electrons in a redox reaction. It is the species that gets reduced.

percent yield

Actual yield divided by theoretical yield, multiplied by 100%. A measure of how well a run performed.

precipitate

An insoluble solid that forms and separates when two solutions are mixed.

reducing agent

The substance that donates electrons in a redox reaction. It is the species that gets oxidized.

reduction

A decrease in oxidation number, which means the atom gained electrons.

solubility rules

A short set of statements that predict whether a given ionic compound dissolves well in water.

spectator ion

An ion that is present in solution before and after the reaction and takes no part in the change.

stoichiometry

The use of a balanced equation to relate amounts of reactants and products, in moles, mass or volume.

theoretical yield

The maximum product the limiting reactant allows, calculated from the balanced equation.

Thermochemistry

Every chemical change moves energy. Burn methane and the room gets warm. Open a cold pack and your hand gets cold. Thermochemistry is the part of chemistry that tracks that energy. In this chapter you learn to say how much heat a reaction gives up or takes in. You learn to measure it with a cup of water and a thermometer. Then you learn three ways to predict it on paper. One adds up known steps. One looks up each compound in a table. One counts the bonds that break and form. By the end, a number like −890.5 kJ will tell you a whole story.

Section 1Energy, heat and work

Two kinds of energy

Energy is the ability to do work or to warm something up. It shows up in two broad kinds. Kinetic energy is the energy of motion. A rolling ball has it. So does a single molecule darting across a flask. You find it with KE = ½mv², so speed counts twice over. Potential energy is stored energy. It depends on where a thing sits, or on what it is made of. A book on a high shelf holds it. So do the bonds in a tank of gasoline.

Heat is not a third kind of energy that a thing owns. Thermal energy is the energy of random particle motion inside a sample. Temperature reports how fast those particles move, on average. Heat is thermal energy on the move, from a warm place to a cooler one. A hot pan does not hold heat. It holds thermal energy, and it gives off heat.

Draw a box around the problem

Before you can count energy, you must say what you are counting. The system is the part you care about. The surroundings are everything else in the universe. For a reaction in a beaker, the chemicals are the system. The glass, the air and your hand are the surroundings. A closed system can trade energy but not matter. An isolated system trades neither one.

Energy is never destroyed. The law of conservation of energy says it only moves or changes form. Whatever the system gains, the surroundings gave up. Written for a chemical system, that idea is the first law of thermodynamics.

Here ΔE is the change in the system's internal energy, the total of all its kinetic and potential energy. The letter q stands for heat. The letter w stands for work, which in chemistry is usually a gas pushing on its container or getting squeezed.

Getting the signs right

Both q and w carry a sign, and the sign always describes the system. Think of a bank account. Money coming in is positive. Money going out is negative.

• q is positive when heat flows into the system. That change is endothermic.
• q is negative when heat flows out of the system. That change is exothermic.
• w is positive when the surroundings do work on the system, such as squeezing a gas.
• w is negative when the system does work on the surroundings, such as a gas expanding.

The SI unit of energy is the joule, J. One joule is one kg·m²/s², which is tiny. A calorie is larger, and it equals 4.184 J by definition. The Calorie on a food label is really a kilocalorie, so one of them is 4184 J.

Section 2Heat capacity and calorimetry

Why water fights back

Put the same amount of heat into a steel pan and into a pot of water. The pan gets hot fast. The water barely budges. That difference has a name. Heat capacity, C, is the heat needed to raise a whole object by one degree. Its unit is J/°C. It depends on how much stuff you have, so a big pot has a larger value than a small one.

Specific heat capacity, c, is the heat needed to raise one gram by one degree. Its unit is J/g·°C. This one is a property of the substance itself, so you can look it up. Molar heat capacity is the same idea per mole instead of per gram.

Water has a specific heat of 4.184 J/g·°C. That is high for a liquid. Hydrogen bonds soak up energy before the molecules speed up much. The result shapes daily life. Lakes warm and cool slowly. Your blood holds your body near 37 °C. Car engines use water to carry heat away.

The equation that does the work

Heat, mass, specific heat and temperature change all tie together in one line.

Write ΔT as final minus initial every time. Then the sign takes care of itself. A sample that warms up has a positive ΔT and a positive q, so it gained heat. A sample that cools has a negative ΔT and a negative q. One more bonus: a change of one degree Celsius is the same size as a change of one kelvin. So you may use either unit for ΔT.

Measuring heat in a cup

Calorimetry is the practice of measuring heat by watching a temperature change. The device is a calorimeter. The simplest one is a foam cup of water with a lid and a thermometer. Foam is a poor conductor, so very little heat escapes. The cup sits open to the room, so pressure stays constant.

A bomb calorimeter is the heavy-duty version. A sealed steel vessel holds the sample and pure oxygen. The whole vessel sits in a water bath. Because the volume cannot change, it measures heat at constant volume instead.

Either way the logic is the same. If no heat leaks out, the heat one part loses is the heat the other part gains.

Section 3Enthalpy and thermochemical equations

A quantity built for the lab bench

Most reactions you run sit in an open container. The pressure stays near 1 atm the whole time. If a gas forms, it has to push the air back, and that costs energy as work. So ΔE is awkward to measure in an open cup, because part of the energy slipped away as work you never saw. Chemists get around this with a quantity called enthalpy, H.

Enthalpy is defined as H = E + PV. You will never need to find H itself, and that is fine. What matters is the change. At constant pressure, the enthalpy change is exactly the heat you measure.

That is a gift. The thermometer reading from an open cup gives you ΔH with no extra work term to chase. The sign rules match the ones you already know. A negative ΔH means heat left the system, so the reaction is exothermic. A positive ΔH means heat went in, so the reaction is endothermic. Enthalpy is an extensive property, which means the amount of substance changes the size of the answer.

Thermochemical equations

A thermochemical equation is a balanced equation with two extra pieces of care. Every substance carries its physical state. And a value of ΔH is written beside it for the equation exactly as balanced.

Read that as a package deal. Burn one mole of methane with two moles of oxygen, and make liquid water, and 890.5 kJ of heat leaves. States are not decoration. If the water comes off as vapor instead, the same reaction gives only 802.5 kJ. The missing 88 kJ stayed in the vapor, because turning 2 mol of liquid water into vapor costs about 44.0 kJ per mole.

A value marked ΔH° is a standard enthalpy change. The little circle means every substance was in its standard state: a pressure of 1 bar for gases, a concentration of 1 M for solutions, and a stated temperature that is almost always 25 °C.

Scaling and reversing

Two rules let one equation answer many questions. Multiply the whole equation by a number, and ΔH gets multiplied by the same number. Reverse the equation, and ΔH keeps its size but flips its sign. Watch all three forms below say the same thing.

The last line is worth a pause. Splitting water back into its elements takes in exactly the heat that forming it gave off. Nature keeps a tidy ledger.

Section 4Hess's law

Enthalpy does not care how you got there

Imagine hiking to a lookout 300 m above the car park. You could go straight up the steep face. You could take a long switchback trail instead. Either way you climbed 300 m. Altitude depends only on where you start and where you stop. Enthalpy behaves the same way, which makes it a state function. Its change depends on the starting state and the final state, never on the route between them.

That single fact is Hess's law. If a reaction can be written as the sum of two or more steps, then the reaction's ΔH is the sum of the steps' ΔH values. The steps do not even have to be real. They only have to add up on paper.

Hess's law is how chemists get numbers they cannot measure. Take the reaction of carbon with oxygen to make carbon monoxide. You cannot measure it cleanly, because some of the CO always burns on to CO₂. So you measure two reactions you can control, then combine them.

Three moves and nothing else

Every Hess's law problem uses the same three moves on the steps you were given.

• Use a step as written, and use its ΔH as given.
• Reverse a step, and change the sign of its ΔH.
• Multiply a step by any factor, and multiply its ΔH by that same factor.

Then stack the adjusted steps and add. Anything that appears on the left in one step and on the right in another will cancel. What is left should be the target equation. If it is not, go back and adjust a factor.

A good habit is to work from the target, not from the steps. Find a substance that appears in the target. Hunt for it in the given steps. Ask two questions. Is it on the correct side? Is the coefficient correct? Flip or scale until both answers are yes, then move to the next substance.

Section 5Standard enthalpies of formation

One table, thousands of reactions

Hess's law is powerful, but hunting for the right steps gets old. There is a shortcut that works for almost any reaction. Pick one special step and tabulate it for every compound. That step is formation from the elements.

The standard enthalpy of formation, written ΔH_f°, is the enthalpy change when one mole of a substance forms from its elements, with everything in its standard state. The standard state of an element is its most stable form at 1 bar and the stated temperature, usually 25 °C. For carbon that is graphite, not diamond. For oxygen it is O₂(g), not ozone. For bromine it is a liquid.

Two details follow right away. First, the equation must make exactly one mole of the product, so fractional coefficients are normal. Second, an element already in its standard state has nothing to form, so its ΔH_f° is zero.

Products minus reactants

Once every compound has a number, any reaction enthalpy is one subtraction.

Hess's law is why this works. Think of a two leg route. First take the reactants apart into their elements, which is the reverse of forming them, so you subtract their values. Then build the products from those elements, which adds their values. The elements act like sea level, a shared zero that both sides are measured from.

Three cautions save most of the lost points. Multiply each value by the coefficient from the balanced equation. Match the state in the table to the state in your equation, since H₂O(l) and H₂O(g) differ by 44.0 kJ/mol. And subtract the whole reactant sum, not just the first term.

Section 6Bond energies

Breaking costs, making pays

Formation enthalpies tell you about whole compounds. Bond energies let you look inside them. A bond dissociation energy, D, is the energy needed to break one mole of a given bond in the gas phase, pulling the pieces apart into free atoms. Breaking a bond always takes energy in, so D is always positive. Forming that same bond gives back the same amount, so it always counts as negative.

There is a wrinkle. The strength of a bond shifts a little depending on what sits nearby. The first C–H bond in methane takes about 439 kJ/mol to break. The four of them average out to about 415 kJ/mol. Tables list that kind of average bond energy, pooled across many molecules. So answers from bond energies are good estimates, not exact values. Expect to land within a few percent.

Bond order matters a lot. More shared pairs pull the atoms closer and hold them tighter. A C–C single bond is 347 kJ/mol. The double bond is 614, and the triple is 839. Notice the double bond is not twice the single, and the triple is not three times it. Each extra pair adds less than the one before.

Counting a whole reaction

Picture a reaction taking the long way around. First tear every reactant apart into loose atoms. That climb costs the sum of all the bonds you broke. Then let those atoms snap together into the products. That drop pays back the sum of all the bonds you made. The difference between the climb and the drop is the reaction enthalpy.

This is Hess's law again, with free atoms as the stopping point. If the new bonds are stronger than the old ones, the drop beats the climb and ΔH comes out negative. That is the real reason fuels release heat. Carbon dioxide and water hold unusually strong bonds, stronger than the ones in the fuel and in O₂.

One shortcut helps on paper. Bonds that appear unchanged on both sides cancel out. Still, counting every bond is safer while you are learning, because a missed bond is easy to spot in a full list.

ReferenceWords worth knowing

These are the words this chapter leans on. Come back to the list whenever a problem uses one you are not sure of.

Bomb calorimeter

A sealed steel vessel used to burn a sample in oxygen at constant volume, sitting in a measured water bath.

Bond dissociation energy (D)

The energy needed to break one mole of a given bond in the gas phase. Always a positive number.

Calorimetry

Measuring heat by tracking the temperature change of a known mass.

Endothermic

A change that takes heat in from the surroundings, so q and ΔH are positive.

Enthalpy (H)

A thermodynamic quantity defined as H = E + PV. Its change is what chemists measure at constant pressure.

Enthalpy change (ΔH)

The enthalpy of the products minus the enthalpy of the reactants. At constant pressure it equals the heat exchanged.

Exothermic

A change that gives heat out to the surroundings, so q and ΔH are negative.

First law of thermodynamics

Energy is conserved. For a system, ΔE = q + w.

Heat (q)

Thermal energy moving because of a temperature difference. It flows from warmer to cooler.

Heat capacity (C)

The heat needed to raise a whole object by one degree. Measured in J/°C, and it depends on size.

Hess's law

If a reaction is the sum of several steps, its ΔH is the sum of those steps' ΔH values.

Internal energy (E)

The total kinetic and potential energy stored inside a system.

Joule (J)

The SI unit of energy, equal to 1 kg·m²/s². One calorie is 4.184 J.

Specific heat capacity (c)

The heat needed to raise one gram of a substance by one degree, in J/g·°C. A property of the substance.

Standard enthalpy of formation (ΔH_f°)

The enthalpy change for making one mole of a substance from its elements, all in standard states.

Standard state

The reference condition for tabulated values: 1 bar for gases, 1 M for solutions, and the element's most stable form, at a stated temperature.

State function

A property whose change depends only on the starting and final states, never on the path taken.

Surroundings

Everything outside the system that can trade energy or matter with it.

System

The part of the universe you choose to study, such as the contents of a beaker.

Work (w)

Energy transferred by a force acting through a distance. In chemistry it is usually a gas expanding or being squeezed.

Electronic Structure and Periodic Properties

Light carries energy in small packets. An atom can take in or give off those packets only in fixed amounts. That one fact explains the red glow of a neon sign and the orange of a sodium streetlight. It also explains the shape of the periodic table. In this chapter you follow the electron. You will see where it can sit, how much energy each seat holds, and what happens when it moves. Then you will use that map to write the electron layout of any element. By the end the table will look less like a chart and more like a record of how electrons stack up.

Section 1Light and quantized energy

Light is a wave you can measure

Light is a moving pair of electric and magnetic fields. In empty space it always travels at one speed. That speed is c = 2.998 × 10⁸ m/s. Every wave has a wavelength, written λ. Wavelength is the distance from one crest to the next crest. Every wave also has a frequency, written ν. Frequency counts how many crests pass a fixed point each second. Its unit is the hertz, which is just s^-1.

The speed of light ties those two numbers together.

Because c never changes, λ and ν must trade off. A short wave has to come fast. A long wave has to come slowly. Blue light has a shorter wavelength than red light. So blue light must have the higher frequency. The amplitude is the height of the wave. Amplitude sets how bright the light looks. It does not set the color.

Visible light is a thin slice of the whole electromagnetic spectrum. Red light sits near 700 nm. Violet light sits near 400 nm. Radio waves can be meters long. X-rays are smaller than an atom. All of them are the same kind of wave at different sizes.

Energy comes in packets

For a long time light looked purely like a wave. Then two experiments broke that picture. Hot objects do not glow the way wave theory predicts. Light shining on a metal knocks electrons loose only above a threshold color. Max Planck and Albert Einstein fixed both problems with one idea. Light energy comes in countable pieces.

One piece of light energy is a photon. The energy of a photon depends only on frequency.

Here h is Planck's constant, 6.626 × 10^-34 J·s. The number is tiny, so one photon carries very little energy. Still, the rule is strict. A photon of 400 nm light carries almost twice the energy of a 700 nm photon.

Energy that comes in fixed pieces is quantized. Think about coins. You can pay 25 cents or 50 cents. You cannot pay 25.3 cents, because no coin that size exists. Light works the same way. Brighter light means more photons, not bigger ones.

Section 2The Bohr model and line spectra

A glowing gas gives only certain colors

Put hydrogen gas in a glass tube and run current through it. The tube glows pink. Send that glow through a prism and the pink splits apart. You do not get a smooth rainbow. You get four sharp colored lines on a dark field. That pattern is a line spectrum.

Hydrogen's four visible lines sit at 656 nm, 486 nm, 434 nm and 410 nm. Every hydrogen sample in the universe gives the same four. Sodium gives a different set. Helium gives another. A line spectrum is a fingerprint for an element. That is how we know what stars are made of.

A smooth rainbow would mean an electron can hold any energy at all. Sharp lines mean something else. The electron can only hold certain fixed energies. When it drops from a higher one to a lower one, the leftover energy leaves as a single photon. One jump gives one color.

Bohr's ladder of allowed energies

In 1913 Niels Bohr built a model that got hydrogen right. He let the electron circle the nucleus only in certain orbits. Each orbit carries a fixed energy set by a whole number n, called the principal quantum number.

The energies are negative because the electron is trapped. Zero energy means a free electron, far from the nucleus. The lowest rung, n = 1, is the ground state. Any higher rung is an excited state. Notice how fast the rungs crowd together. The gap from n = 1 to n = 2 is huge. The gap from n = 5 to n = 6 is small.

To find the photon, subtract the two energies.

A drop gives a negative ΔE. The atom lost that energy, so a photon of that size left. A climb gives a positive ΔE, which means the atom took a photon in. The four visible lines all end on n = 2. Jumps that end on n = 1 release far more energy, so they land in the ultraviolet.

Section 3Quantum numbers and orbital shapes

From orbits to clouds

Bohr's neat circles cannot be right. An electron is small and fast, and the Heisenberg uncertainty principle says we cannot pin down both its place and its motion. So we stop asking where the electron is. We ask where it is likely to be. Erwin Schrödinger wrote an equation whose answers give exactly that. Each answer is an orbital.

An orbital is a region around the nucleus where one electron is likely to be found. Picture a swarm of gnats around a porch light. No single gnat has a path you can trace. Still, the swarm has a clear shape and a clear center. The darker part of the cloud means higher probability density.

Four numbers name every orbital

Each orbital is labeled by three quantum numbers. A fourth one describes the electron sitting in it.

• The principal quantum number n is 1, 2, 3 and up. It sets size and most of the energy. All orbitals with the same n form a shell.
• The angular momentum quantum number l runs from 0 up to n − 1. It sets shape. We write l = 0 as s, 1 as p, 2 as d and 3 as f. Orbitals that share n and l form a subshell.
• The magnetic quantum number m_l runs from −l through 0 to +l. It sets direction in space. That gives 2l + 1 orbitals in a subshell.
• The spin quantum number m_s is +½ or −½. It is the electron's own two-way property, drawn as an up or down arrow.

So s subshells hold 1 orbital, p hold 3, d hold 5 and f hold 7. Each orbital takes at most 2 electrons. A full shell therefore holds 2n² electrons.

What the shapes look like

An s orbital is a ball centered on the nucleus. A p orbital is two lobes on opposite sides, like a long balloon pinched in the middle. The three p orbitals point along x, y and z. A d orbital usually has four lobes in a clover pattern, and the fifth d shape has two lobes with a ring around the middle. The seven f shapes are more involved still.

Most orbitals have places where the electron is never found. Those places are nodes. The count is fixed. Angular nodes equal l, so a p orbital has one flat nodal plane through the nucleus. Radial nodes equal n − l − 1, and those are thin shells of empty space. Add them and any orbital has n − 1 nodes in all. More nodes go with more energy.

Section 4Electron configurations

Fill the cheapest seats first

An atom at rest holds its electrons in the lowest energy arrangement it can. The Aufbau principle says to add electrons one at a time, always to the lowest open subshell. The written result is the electron configuration. Oxygen has 8 electrons, so oxygen is 1s² 2s² 2p⁴. The superscripts must add up to the atomic number.

The filling order is not simply 1s, 2s, 3s. Once several electrons crowd an atom, each one feels the pull of the nucleus plus the push of the others. That reshuffles the subshell energies. The useful order is this.

Look at 4s and 3d. The 4s subshell fills first, even though its n is larger. The diagonal chart below builds that order for you. Follow each gold arrow from its tail to its head, then start the next one.

Shorthand, core and valence

Long configurations get clumsy, so we use the last noble gas as a stand-in. Calcium is 1s² 2s² 2p⁶ 3s² 3p⁶ 4s², which we shorten to [Ar] 4s². The electrons hidden in the bracket are the core electrons. The ones written out are the valence electrons, and those are the ones that do chemistry. Elements in the same group share a valence pattern, which is exactly why they behave alike.

The exceptions worth knowing

In the first row of transition metals, 4s and 3d sit very close in energy. Two elements break the simple pattern because of that.

Moving one s electron into the d set costs almost nothing here. Spreading the electrons out lowers repulsion, so the swapped version wins. Chromium ends with six unpaired electrons, which is more than any other element in its row.

Ions follow one more rule. When an atom loses electrons, the highest n goes first. Iron is [Ar] 4s² 3d⁶. Fe²⁺ is [Ar] 3d⁶, not [Ar] 4s² 3d⁴. Fe³⁺ is [Ar] 3d⁵. Anions just add electrons to the next open spot, so Cl^- is [Ne] 3s² 3p⁶, the same as argon.

Section 5Orbital diagrams and Hund's rule

Boxes and arrows

A configuration like 2p⁴ tells you the count. It does not tell you how those four electrons are spread over the three 2p orbitals. An orbital diagram shows that. Draw one box for each orbital and one arrow for each electron. An up arrow is spin +½ and a down arrow is spin −½. Two rules decide where the arrows go.

Pauli caps every box at two

The Pauli exclusion principle says no two electrons in one atom can share all four quantum numbers. An orbital already fixes n, l and m_l. Only spin is left, and spin has two settings. So a box holds at most two electrons, and they must point opposite ways. Those two are a paired set. This single rule is why s holds 2, p holds 6, d holds 10 and f holds 14.

Hund spreads them out first

Orbitals in the same subshell have equal energy. We call them degenerate. Hund's rule says to put one electron in each degenerate box, all spinning the same way, before doubling up anywhere. Electrons repel each other, so two in one box cost extra energy. Keeping them in separate boxes with parallel spins keeps them farther apart and lowers the total energy.

Nitrogen shows the rule plainly. Its 2p³ electrons go one per box, all three arrows up. Oxygen has one more. That fourth electron must pair somewhere, so oxygen ends with two unpaired electrons.

Unpaired electrons are measurable. An atom or ion with unpaired electrons is paramagnetic, and a magnet pulls on it. With every electron paired it is diamagnetic, and a magnet pushes it away very weakly. Zinc, [Ar] 4s² 3d¹⁰, is diamagnetic. Iron, with four unpaired 3d electrons, is strongly drawn to a magnet.

Section 6Periodic trends

One idea runs all four trends

Every trend in this section comes from a tug of war. The nucleus pulls the outer electrons in. The inner electrons get in the way and push back. The net pull that an outer electron actually feels is the effective nuclear charge, written Z_eff. The blocking by inner electrons is shielding.

Move right across a period and protons pile up fast. Shielding barely changes, because the new electrons go into the same shell. So Z_eff climbs and the outer electrons get held tighter. Move down a group and the valence electrons start a brand new shell. They sit farther out, with a thick layer of core electrons in between. The grip loosens.

Atomic radius

The atomic radius shrinks left to right and grows top to bottom. Across period 2 it falls from 167 pm at lithium to 42 pm at fluorine. Down group 1 it rises from 167 pm at lithium to 298 pm at cesium. Adding a shell always wins over adding protons.

Ions break the pattern in a useful way. A cation lost electrons, so the remaining ones feel a stronger pull per electron. A sodium atom is near 190 pm, while Na⁺ is about 102 pm. An anion gained electrons, which adds repulsion. Cl^- is about 181 pm, much wider than the chlorine atom it came from. In an isoelectronic set, every ion has the same electron count. Then the one with the most protons is smallest, so the order runs Na⁺ < Ne < F^-.

Ionization energy

The ionization energy is the energy needed to pull the loosest electron off a gaseous atom. It rises left to right and falls top to bottom. Sodium needs 496 kJ/mol, while argon in the same period needs 1521 kJ/mol. Cesium, far down group 1, needs only 376 kJ/mol.

Two small dips break the rise, and both are worth knowing. Boron (801 kJ/mol) is lower than beryllium (899 kJ/mol), because boron's electron leaves a higher-energy 2p orbital. Oxygen (1314 kJ/mol) is lower than nitrogen (1402 kJ/mol), because oxygen already has a pair crowding one 2p orbital. Those dips are the orbital diagrams showing up in measured data.

Removing a second electron always costs more. Sodium's first ionization energy is 496 kJ/mol. Its second is 4562 kJ/mol, roughly nine times as much. The second electron has to come out of the neon-like core. That huge jump is why sodium forms Na⁺ and never Na²⁺.

Electron affinity and electronegativity

The electron affinity is the energy change when a gaseous atom accepts an electron. A negative value means energy is released, so the atom wants the electron. Chlorine is the champion at −349 kJ/mol, with fluorine close behind at −328 kJ/mol. Beryllium and the noble gases have positive values, because their subshells are already full and a new electron has nowhere good to sit.

Finally, electronegativity measures how hard an atom pulls on electrons it is sharing inside a bond. The Pauling scale runs from 0.79 at cesium to 3.98 at fluorine. It rises toward the upper right, just like ionization energy. Oxygen is 3.44, nitrogen is 3.04, carbon is 2.55 and hydrogen is 2.20. You will use those four numbers constantly in the next chapter to decide whether a bond is polar.

ReferenceWords worth knowing

These are the terms this chapter leans on most. Come back to the list whenever a problem uses a word you want to pin down.

amplitude

The height of a wave. It controls brightness, not color or photon energy.

Aufbau principle

The rule that electrons fill the lowest open subshell first.

core electrons

The inner electrons hidden inside the noble gas bracket. They rarely take part in bonding.

degenerate orbitals

Orbitals in the same subshell, which all have equal energy.

effective nuclear charge

Written Z_eff. The net pull an outer electron feels after inner electrons shield it.

electron affinity

The energy change when a gaseous atom gains one electron. Negative means energy is released.

electron configuration

The list of subshells with electron counts, such as 1s² 2s² 2p⁴.

electronegativity

How strongly an atom pulls on electrons it shares in a bond. Fluorine leads at 3.98.

frequency

Written ν. The number of wave crests passing a point each second, in s^-1.

ground state

The lowest energy arrangement of an atom's electrons.

Hund's rule

Fill degenerate orbitals singly with parallel spins before any orbital gets a pair.

ionization energy

The energy needed to remove the loosest electron from a gaseous atom.

isoelectronic

Having the same number of electrons. Na⁺, Ne and F^- are an isoelectronic set.

node

A place inside an orbital where the electron is never found.

orbital

A region around a nucleus where one electron is likely to be. It holds two electrons at most.

paramagnetic

Having one or more unpaired electrons, so a magnet attracts the sample.

Pauli exclusion principle

No two electrons in an atom may share all four quantum numbers.

photon

One packet of light energy, with E = hν.

quantized

Allowed only in fixed amounts, like coins rather than a smooth stream.

wavelength

Written λ. The distance from one wave crest to the next.

Chemical Bonding and Molecular Geometry

Atoms almost never travel alone. The way they join sets the rules for everything that happens next. This chapter follows two ways of joining. In the first, one atom hands an electron over for good. In the second, two atoms share. Then we get to shape, because a bonded group of atoms is not a shapeless blob. It has angles you can predict, and you predict them by counting. By the end you can take a formula like SF₄, draw it, name its shape, and say whether its pull is lopsided or even.

Section 1Ionic bonding and lattice energy

One atom gives, the other takes

Picture sodium next to chlorine. Sodium has one electron in its outer shell, while chlorine is one electron short of a full shell. So sodium hands its electron over. Sodium is left with 11 protons and 10 electrons, so it becomes Na⁺. A positive ion like this is a cation. Chlorine gains the electron and becomes Cl⁻. A negative ion is an anion. Opposite charges pull on each other, and that pull is an ionic bond.

There are no tidy pairs here. Each Na⁺ tugs on every Cl⁻ nearby, and each Cl⁻ tugs back on every Na⁺. The ions stack into a repeating solid called a lattice. In table salt, every sodium ion has six chloride ions around it, and every chloride ion has six sodium ions around it. So NaCl is a ratio, not a tiny two-atom unit you could pull out with tweezers.

Lattice energy measures the glue

Lattice energy (ΔH_lattice) is the energy it takes to pull one mole of the solid apart into free gas-phase ions. A large value means a tough solid. For NaCl it is about 769 kJ/mol, while for MgO it is about 3795 kJ/mol. So MgO takes nearly five times as much energy to break up. That shows up in the melting points. Salt melts at 801 °C, while MgO holds out past 2800 °C.

Two things set the size, and both come from Coulomb's law. Charge is the first, and the pull grows with the two charges multiplied together. Distance is the second, and the pull fades as the ion centers move apart.

Here Q₁ and Q₂ are the ion charges, and r₀ is the gap between the centers. Double both charges and the energy roughly quadruples, while swapping in a fatter ion makes the energy drop. MgO wins on both counts. Mg²⁺ and O²⁻ each carry twice the charge of the salt ions, and both are smaller, so r₀ is shorter.

Section 2Covalent bonds, electronegativity and polarity

Sharing instead of handing over

Two chlorine atoms cannot both win. Each one needs a single electron to fill its shell, and neither will give one up. So they share a pair instead. A shared pair of electrons holding two atoms together is a covalent bond. The pair sits in the space between the two nuclei, and both nuclei pull on it. That shared pull is the bond.

Sharing is rarely fair, because some atoms tug harder on a shared pair than others. Electronegativity is the number that scores that tug. Linus Pauling built the scale we still use, and he put fluorine at the top at 3.98, which makes it the greediest atom there is. Cesium sits near the bottom at 0.79. Electronegativity rises as you go right across a row and falls as you go down a column. Those two trends track the same cause, because nuclear charge grows to the right and the outer shell drifts farther from the nucleus as you go down.

The gap in the numbers tells you the bond type

Subtract the smaller value from the larger one. That difference, written ΔEN, sorts bonds along a smooth range. A tiny gap means the pair sits in the middle, which is a nonpolar covalent bond. A medium gap drags the pair toward the greedier atom, which is a polar covalent bond, and one end picks up a partial negative charge written δ−. The other end goes δ+. A huge gap means the electron is taken, not shared, and the bond counts as ionic.

Treat those cutoffs as road signs, not walls. H−F has ΔEN = 1.78 and still behaves as a very polar covalent bond. NaI has ΔEN = 1.73 yet is a true ionic solid. The better habit is to read ΔEN as a dial. Larger gap, more lopsided bond.

Putting a number on the lopsidedness

A polar bond has a dipole moment, written μ. It equals the size of the partial charge times the distance between the charge centers. The unit is the debye (D), where 1 D = 3.336 × 10⁻³⁰ C·m. Chemists often report the same idea as percent ionic character. To get it you compare the real dipole moment with the one you would see if a whole electron had moved across.

Section 3Lewis structures

Dots for electrons, lines for bonds

A Lewis structure is a map of where the outer electrons sit. Only valence electrons appear, because only they do the bonding. A shared pair is drawn as a line and called a bonding pair. A pair that belongs to one atom alone is drawn as two dots and called a lone pair. Two lines between the same two atoms make a double bond, and three lines make a triple bond. The number of shared pairs between two atoms is the bond order.

Most atoms stop when their valence shell holds eight electrons. That habit is the octet rule, and it comes straight from the s and p orbitals of the outer shell. Hydrogen is the standing exception, since its shell is full at two.

Five steps that always work

Build every structure the same way. First, add up the valence electrons from all the atoms. Add one more for each unit of negative charge, and take one away for each unit of positive charge. Second, sketch a skeleton with one line per bond. The atom with the lowest electronegativity usually goes in the middle, and hydrogen is never in the middle. Third, give every outer atom a full octet using lone pairs. Fourth, park any electrons that are left on the central atom. Fifth, if the central atom is still short of eight, pull a lone pair in from a neighbor to make a double or triple bond.

Keep a running count the whole time, because every line you draw uses two electrons and every pair of dots uses two. When your total matches the number from step one, the structure is complete. That count is the one thing that catches most mistakes.

Three ways the octet rule breaks

The rule is a strong habit, not a law. It fails in three tidy families.

Some molecules have an odd electron count, so no structure can pair everything up. Nitrogen monoxide, NO, has 5 + 6 = 11 valence electrons. One electron has to stay alone, and the nitrogen ends up with seven. A molecule with an unpaired electron is a free radical, and radicals like this tend to be reactive.

Some central atoms settle for less than eight, and beryllium and boron do this often. BF₃ has 3 + 3(7) = 24 valence electrons. Three B−F bonds use 6, and 18 fill the lone pairs on fluorine. Boron is left with only 6 electrons, and the molecule is stable anyway. That gap is why BF₃ grabs a lone pair from almost anything.

Atoms in period 3 and below can hold more than eight. This is an expanded valence shell, and it needs a central atom big enough to fit the extra neighbors. PCl₅ has 5 + 5(7) = 40 valence electrons, and phosphorus carries 10. SF₆ has 6 + 6(7) = 48, and sulfur carries 12. XeF₄ has 8 + 4(7) = 36. Four bonds use 8, the fluorines take 24 in lone pairs, and the last 4 sit on xenon as two lone pairs. Xenon ends up with 12 electrons. Second-row atoms such as C, N, O and F never do this, because they have no room for it.

Section 4Resonance and formal charge

When one drawing is not enough

Ozone, O₃, has 18 valence electrons. Follow the five steps and you get a bent molecule with one double bond and one single bond. Now ask which side gets the double bond. Nothing in the rules picks a side, and you can draw it either way. Measurement settles the argument, because both bonds in ozone are 127.8 pm long. A true O=O double bond is about 121 pm and a true O−O single bond is about 148 pm. The real bonds are identical and sit between the two.

So neither drawing is the molecule. The two drawings are called resonance forms, and the real molecule is a single blend of them called the resonance hybrid. The double-headed arrow between forms is important. It does not mean the molecule flips back and forth, but rather that one structure is not enough to hold the truth, so you average the set. In ozone each bond is one and a half bonds, which fits the measured length exactly.

Average the shared pairs to get the real bond order, so ozone, with 3 bonding pairs spread over 2 bond positions, comes out at 3/2 = 1.5. Carbonate, CO₃²⁻, has three resonance forms and 4 bonding pairs over 3 positions, so its bond order is 4/3 = 1.33. All three carbonate bonds measure about 129 pm, which again sits between a single bond near 143 pm and a double bond near 123 pm. Spreading electrons out this way also lowers the energy, which is why these ions are so stable.

Formal charge keeps the bookkeeping honest

Formal charge is the charge an atom would carry if every bonding pair were split exactly down the middle. It is a bookkeeping tool rather than a real measured charge, and it is still the fastest way to rank competing structures.

Three habits make the tool reliable. The formal charges must add up to the overall charge on the species. The best structure keeps those charges as close to zero as it can. When a negative charge has to land somewhere, it belongs on the more electronegative atom.

Section 5VSEPR and molecular shapes

Electron groups push each other apart

A Lewis structure is flat on the page. Real molecules are not. To get the shape, use VSEPR, which stands for valence shell electron pair repulsion. The whole idea fits in one line: electron groups around a central atom are all negative, so they spread out as far from each other as they can.

Count groups, not atoms. Each group is an electron domain, and one single bond counts as one domain. A double or triple bond is still one domain, because those pairs sit in the same region. A lone pair is one domain too. Add them up, and the number of domains fixes the electron-domain geometry for you.

Two domains spread to 180°, which is linear. Three settle at 120° in a flat triangle, which is trigonal planar. Four go to 109.5° and point at the corners of a tetrahedron. Five give a trigonal bipyramid, with three equatorial positions 120° apart and two axial positions at 90° to that belt. Six give an octahedron, with every neighbor at 90°.

Two shapes, one molecule

Here is the step that is easiest to skip. The domain geometry counts every domain, including lone pairs. The molecular geometry counts only the atoms, because lone pairs are invisible to a camera. So one molecule gets two names.

Water makes the point. Oxygen has 2 bonds and 2 lone pairs, so 4 domains, and the domain geometry is tetrahedral. The molecular geometry is bent, because you only see the two hydrogen atoms. Ammonia has 3 bonds and 1 lone pair, so also 4 domains. Its domain geometry is tetrahedral and its molecular shape is trigonal pyramidal.

Lone pairs also squeeze the angles. A lone pair is held by one nucleus instead of two, so it spreads wider and pushes harder on its neighbors. Methane has 4 bonds and no lone pairs, and its angle is the ideal 109.5°. Ammonia loses a little, down to 107°, and water loses more, down to 104.5°. Each lone pair shaves off about 2 to 3 degrees. For five domains, lone pairs always take equatorial spots, where there is more elbow room.

Section 6Molecular polarity

Arrows that can cancel

Every polar bond has a little arrow on it. The arrow points toward the greedier atom, and its length tracks ΔEN. That arrow is the bond dipole, and a whole molecule is polar only if those arrows fail to cancel. Arrows are vectors, so you add them head to tail, the same way you add forces. The leftover arrow is the molecular dipole moment.

Carbon dioxide shows the catch. Both C=O bonds are strongly polar, since ΔEN = 3.44 − 2.55 = 0.89. But CO₂ is linear, so the two arrows point in exactly opposite directions, which means they cancel and the measured dipole moment is 0 D. A molecule can be built entirely from polar bonds and still be nonpolar overall.

Water has the same two polar bonds and a bigger ΔEN of 1.24. It is bent at 104.5°, so the two arrows do not oppose each other, and they add to a strong net arrow pointing from the hydrogen side toward the oxygen. Water's dipole moment is 1.85 D, and that one number explains much of its behavior.

A two-question test

You can usually judge polarity without any math. Ask two questions. Are all the outer atoms the same? Are they placed symmetrically around the center, with no lone pairs on the central atom? If both answers are yes, the arrows cancel and the molecule is nonpolar. Any no, and it is polar.

That test sorts the common cases fast. CH₄, CCl₄, BF₃, CO₂, PCl₅ and SF₆ all pass, so all are nonpolar. Swap one atom and symmetry dies. CHCl₃ has a dipole moment of 1.04 D because one corner is now hydrogen, and a lone pair usually breaks symmetry just as well. SO₂ is bent, and its dipole moment is 1.63 D, while linear CO₂ sits at zero. XeF₂ and XeF₄ are the exception. Their lone pairs sit exactly opposite each other, so the bond arrows still cancel.

ReferenceWords worth knowing

Twenty terms carry this chapter. Read the list once now, then come back to it when you are drawing structures and naming shapes.

anion

An ion with a negative charge, made when an atom gains one or more electrons.

bond dipole

The arrow on one bond, pointing to the more electronegative atom. Its length tracks ΔEN.

bond order

The number of shared pairs between two atoms. Resonance can make it a fraction, such as 1.5 in ozone.

cation

An ion with a positive charge, made when an atom loses one or more electrons.

covalent bond

A bond made of one or more electron pairs shared between two atoms.

dipole moment (μ)

The measured size of a charge separation, in debye. 1 D = 3.336 × 10⁻³⁰ C·m.

electron domain

One group of valence electrons around a central atom. A single, double or triple bond counts as one, and so does a lone pair.

electronegativity

A number for how strongly an atom pulls on a shared pair. On the Pauling scale F is 3.98 and Cs is 0.79.

expanded valence shell

A central atom holding more than eight valence electrons, as phosphorus does in PCl₅. Only period 3 and heavier atoms can.

formal charge

Valence electrons minus lone pair electrons minus half the bonding electrons. A bookkeeping score used to rank structures.

ionic bond

The pull between oppositely charged ions after an electron has moved from one atom to another.

lattice energy (ΔH_lattice)

The energy needed to pull one mole of an ionic solid apart into gas-phase ions. About 769 kJ/mol for NaCl.

Lewis structure

A drawing of the valence electrons, with lines for shared pairs and dots for lone pairs.

lone pair

A valence pair that belongs to one atom and is not shared.

molecular geometry

The shape traced by the atoms alone, with lone pairs left out. Water is bent.

nonpolar covalent bond

A bond with ΔEN under about 0.4, where the shared pair sits near the middle.

octet rule

The habit of ending up with eight valence electrons. Hydrogen stops at two.

polar covalent bond

A bond with ΔEN of about 0.4 to 1.8, giving δ+ on one atom and δ− on the other.

resonance

Two or more valid Lewis structures for one species. The real molecule is their blend, not a switch between them.

VSEPR

Valence shell electron pair repulsion. The model that predicts shape by spreading electron domains as far apart as possible.

Advanced Theories of Covalent Bonding

Lewis dots tell you which atoms share pairs. They do not tell you why a shared pair holds, why a double bond refuses to twist, or why liquid oxygen sticks to a magnet. For that you need orbitals. In this chapter you will build two pictures of the same bond. The first keeps each atom's own orbitals and lets them overlap. The second mixes orbitals across the whole molecule. Each one pays off somewhere the other falls short. Together they explain shape, stiffness, bond strength and magnetism with the same small set of rules.

Section 1Valence bond theory and orbital overlap

A bond is a shared pair in a shared space

Valence bond theory keeps the atoms you already know. Each atom brings its own orbitals to the meeting. A bond forms when an orbital on one atom overlaps an orbital on the other. Two electrons then move into that shared space. Both nuclei pull on that pair at once. The double pull is the glue.

The key idea is orbital overlap. Overlap means two orbitals share some of the same space. More overlap piles more electron density between the nuclei. More density there makes a stronger bond. Overlap also has a direction, because orbitals point. A p orbital points along one axis. The bond it builds points along that axis too. That is why this theory predicts shapes so well.

One rule limits every bond to a pair. The two shared electrons must have opposite spins. Spin‑paired electrons in overlapping orbitals lower the energy. The same two electrons with matching spins raise it instead. So a full orbital has nothing to offer a new partner. That is why two He atoms meet and bounce apart.

Why atoms stop at one best distance

Slide two H atoms together and track the energy. Far apart they barely notice each other, so we call that energy zero. As they close in, each nucleus starts to pull on the other atom's electron. The energy falls. Lower energy means a steadier pair.

Keep pushing and the fall stops. Both nuclei carry positive charge, so they begin to shove. The energy climbs fast. Between the two trends sits a dip. The bottom of the dip is the bond length, the distance the atoms really keep. The depth of the dip is the bond energy, the price of pulling them apart again.

For H₂ that dip sits at 74 pm and runs 436 kJ/mol deep. A picometre is 10⁻¹² m, so 74 pm is a very short reach. Note that 436 kJ/mol is a mole of bonds, not one bond. The next example turns it into the cost of a single bond.

Section 2Hybrid orbitals

The shape problem that forces mixing

Carbon has one 2s orbital and three 2p orbitals in its valence shell. The 2s is round. The three 2p orbitals point along x, y and z, at 90° to each other. Yet CH₄ has four identical bonds at 109.5°. Plain s and p orbitals cannot give that. Something has to change.

The fix is hybridization. Orbitals on the same atom are waves, and waves on the same atom can add together. Mix one s with three p orbitals and you get four new hybrid orbitals. Each one has the same shape, the same energy, and one fat lobe pointing out. Four fat lobes spread as far apart as they can. That lands them at 109.5°, which is exactly what CH₄ shows.

Mixing never creates or destroys orbitals. Put four orbitals in and you take four out. The names just record the recipe. An sp³ set comes from one s plus three p. An sp² set comes from one s plus two p, leaving one p orbital untouched. An sp set comes from one s plus one p, leaving two p orbitals untouched. Those leftover p orbitals matter a lot in section 3.

Counting your way to the answer

You never have to guess a hybrid set. Count regions of electron density on the central atom. A region is one sigma bond or one lone pair. A double or triple bond still counts as one region, because the whole group points one way. The number of regions equals the number of hybrid orbitals you need.

Two regions give sp. Three give sp². Four give sp³. Five and six need more room, so the bookkeeping adds d orbitals: sp³d for five and sp³d² for six. The s character of a hybrid is the share of s in the mix. For sp it is 1 of 2, or 50%. For sp² it is 1 of 3, near 33%. For sp³ it is 1 of 4, or 25%. More s character pulls the pair in closer, so those bonds run shorter and stronger.

Section 3Sigma and pi bonds

Two ways for orbitals to meet

Overlap can happen in two shapes. Orbitals can meet end to end, along the line joining the nuclei. That line is the internuclear axis. End‑to‑end overlap builds a sigma bond, written σ. The electron density piles up right on the axis, between the two nuclei. Look down the axis and the bond looks like a plain round tube. Nothing changes as you spin it.

Orbitals can also meet side by side. Two p orbitals that stay parallel overlap above the axis and below it at the same time. That is a pi bond, written π. A pi bond puts no density on the axis at all. The axis sits inside a nodal plane, a flat sheet where the chance of finding the electron is zero. So a pi bond is really one bond in two slabs, one over and one under.

Sigma overlap is the better deal. Pointing two lobes straight at each other gets them as close as they can get. Side‑on lobes have to lean past one another, so they share less space. Less overlap means a weaker bond. Every single bond you have ever drawn is a sigma bond. Pi bonds only show up as extras, on top of a sigma bond that is already there.

Counting sigma and pi bonds

The counting rule is short. A single bond is 1 σ. A double bond is 1 σ + 1 π. A triple bond is 1 σ + 2 π. The second pi bond in a triple bond uses the other pair of p orbitals, at right angles to the first. The two pi bonds wrap the sigma bond from four sides.

Section 4Multiple bonds and rotation

Why a double bond will not twist

Take ethene, C₂H₄. Each carbon is sp², so each one keeps one unhybridized p orbital. Those two p orbitals stand up at right angles to the flat part of the molecule. When they line up parallel, they overlap side by side and make the pi bond. The whole molecule goes flat and stays flat.

Now try to twist one CH₂ group. The sigma bond does not care, because it is round about the axis. The pi bond cares a great deal. Twisting tips one p orbital out of line with the other. Their overlap shrinks. It falls roughly as the cosine of the twist angle. At 90° the two p orbitals sit at right angles and share nothing. The pi bond is simply gone.

That is restricted rotation. To spin a double bond you must first pay for breaking a pi bond. For ethene that bill is near 270 kJ/mol. Room temperature cannot cover it, so the molecule stays locked. A single bond, as in ethane, only has to slide past a small bump of about 12 kJ/mol. Ethane spins millions of times a second.

What the lock buys you

Restricted rotation is not a nuisance. It is the reason shape can carry information. Two compounds can share a formula and still differ, because the groups on a double bond are pinned in place. Cis means two chosen groups sit on the same side. Trans means they sit on opposite sides. Cis‑2‑butene and trans‑2‑butene are different substances, with different melting points and different boiling points.

Your eyes run on this. A molecule called retinal sits in the back of your eye with one cis double bond. Light hits it and that bond flips to trans. The shape change starts the nerve signal. Without restricted rotation the molecule would wobble all day and you would see nothing.

Bond data backs the picture. A C–C single bond is 154 pm long. A C=C double bond is 134 pm. A C≡C triple bond is 120 pm. More shared pairs pull the nuclei closer. The bonds get stronger too, but not in equal steps, as the next example shows.

Section 5Molecular orbital theory

Spread the electrons over the whole molecule

Valence bond theory has one blind spot. It says O₂ has a tidy double bond with every electron paired. Hang liquid oxygen between the poles of a magnet and it clings there. Only unpaired electrons do that. So O₂ must have some, and the shared‑pair picture misses them.

Molecular orbital theory fixes this by letting go of the idea that an electron belongs to one bond. Atomic orbitals are waves, so add them across the whole molecule. Adding two atomic orbitals in step builds up density between the nuclei. That gives a bonding molecular orbital, lower in energy than either atomic orbital. Adding them out of step cancels density between the nuclei. That gives an antibonding molecular orbital, higher in energy, with a node right between the atoms. The method is called the linear combination of atomic orbitals.

Two atomic orbitals always give two molecular orbitals, one down and one up. Electrons then fill those molecular orbitals by the same rules you already use. Lowest energy first. Two electrons per orbital, with opposite spins. Equal‑energy orbitals each take one electron before any of them takes a second. Orbitals at the same energy are called degenerate.

Period 1 and period 2 diatomics

For H₂ and He₂ only 1s orbitals matter. They give σ_1s and σ*_1s. The star marks antibonding. H₂ puts both electrons in σ_1s and holds together. He₂ has to fill the star as well, which undoes the gain, so He₂ does not exist.

Period 2 brings the 2s and 2p orbitals. The 2s pair gives σ_2s and σ*_2s. The six 2p orbitals give six molecular orbitals: σ_2p and σ*_2p from the pair pointing along the axis, plus two degenerate π_2p and two degenerate π*_2p from the two side‑on pairs.

Here is the twist that trips people up. The order of those levels is not the same across the row. For B₂, C₂ and N₂ the π_2p pair sits below σ_2p. For O₂, F₂ and Ne₂ the σ_2p sits below the π_2p pair. The reason is mixing. The 2s and 2p orbitals lie close in energy on the left of the row, so the sigma orbitals from 2s and 2p push on each other and σ_2p gets shoved up. Moving right, the nuclear charge grows and the 2s drops far below the 2p. The mixing fades and the normal order returns.

Section 6Bond order and magnetism

One number that ties it together

Bond order is the net number of bonding pairs holding two atoms together. Count the electrons in bonding molecular orbitals. Count the electrons in antibonding ones. Subtract, then halve.

bond order = (bonding electrons − antibonding electrons) ÷ 2

Antibonding electrons cancel bonding ones, so they carry a minus sign. A bond order of 0 means no bond at all. That is the whole story of He₂, Be₂ and Ne₂. They never form, because their stars fill as fast as their bonding orbitals do. Bond order can also land on a half, such as 2.5. Nothing is wrong. It just means one lone electron is pulling its weight in a bonding or antibonding orbital.

Bond order sets length and strength

Bond order is not just bookkeeping. It predicts two things you can measure. Raise the bond order and the two nuclei pull closer, so the bond length shrinks. Raise the bond order and the bond also gets harder to break, so the bond energy climbs. The row from N₂ to F₂ shows it clearly. N₂ has bond order 3, a length of 110 pm, and needs 945 kJ/mol. O₂ has bond order 2, a length of 121 pm, and needs 498 kJ/mol. F₂ has bond order 1, a length of 142 pm, and needs only 159 kJ/mol.

Reading the magnet

Magnetism is the other prize. A substance with one or more unpaired electrons is paramagnetic. Each unpaired electron acts like a tiny magnet, so the sample is drawn into a magnetic field. A substance with every electron paired is diamagnetic. Paired spins cancel, and the sample is pushed out of the field, very weakly. So the test is simple. Finish the diagram, then look for a lone arrow. One or more lone arrows means paramagnetic.

This is where molecular orbital theory earns its keep. Pour liquid oxygen between the poles of a strong magnet and it hangs there in a bridge. Liquid nitrogen in the same gap pours straight through. The diagrams say why. O₂ keeps two lone arrows in its π* level. N₂ has none.

ReferenceWords worth knowing

These are the words this chapter leans on. Each one is defined the way it is used here.

antibonding molecular orbital

A molecular orbital that is higher in energy than the atomic orbitals that built it. It has a node between the nuclei and weakens the bond. Its symbol carries a star, as in σ*_2p.

bond energy

The energy needed to break one mole of a given bond. It is the depth of the dip in the energy curve. For H₂ it is 436 kJ/mol.

bond length

The average distance between two bonded nuclei. It sits at the bottom of the energy dip. For H₂ it is 74 pm.

bond order

Half the difference between bonding and antibonding electrons. It gives the net number of bonds and can be a half number.

bonding molecular orbital

A molecular orbital lower in energy than the atomic orbitals that built it. It piles electron density between the nuclei and holds the atoms together.

degenerate orbitals

Two or more orbitals with exactly the same energy, such as the pair of π_2p orbitals. Electrons spread out singly across them before pairing up.

diamagnetic

Having every electron paired. A diamagnetic sample is pushed out of a magnetic field, and only very weakly.

hybrid orbital

A new orbital on one atom built by mixing that atom's own s and p orbitals. It has one large lobe and one small back lobe.

hybridization

The mixing of atomic orbitals on the same atom to make an equal number of hybrid orbitals that point where the bonds go.

linear combination of atomic orbitals

The method of adding and subtracting atomic orbital waves to build molecular orbitals. Adding in step gives a bonding orbital, out of step gives an antibonding one.

molecular orbital

An orbital that spreads over a whole molecule rather than one atom. Electrons in it belong to the molecule.

molecular orbital theory

The model that builds molecular orbitals from atomic orbitals and then fills them with all the electrons of the molecule.

node

A place where the chance of finding the electron is zero. A pi bond has a nodal plane through both nuclei.

orbital overlap

Two orbitals sharing some of the same space. More overlap puts more density between the nuclei and makes a stronger bond.

paramagnetic

Having one or more unpaired electrons. A paramagnetic sample is pulled into a magnetic field. Liquid O₂ does this.

pi bond

A bond from side‑on overlap of two parallel p orbitals. Its density sits above and below the internuclear axis, never on it.

region of electron density

One sigma bond or one lone pair on an atom. A double or triple bond still counts as a single region.

restricted rotation

The block on twisting about a double or triple bond. Twisting would break a pi bond, which costs far too much energy.

sigma bond

A bond from head‑on overlap along the internuclear axis, the straight line joining the two nuclei. Its density is highest on that axis and it spins freely.

valence bond theory

The model in which a bond is two overlapping atomic or hybrid orbitals sharing one pair of electrons with opposite spins.