Chemistry 1A: General Chemistry

Modern Fundamental Laws

• Law of Conservation of Mass: mass is neither created nor destroyed in a chemical reaction.
• Law of Definite Proportion: a given compound always contains the same proportion of elements by mass.
• Law of Multiple Proportions: when two elements form multiple compounds, the mass ratios can be reduced to small whole numbers.

Atoms

• Protons: Positively charged subatomic particles found in the nucleus of an atom. They determine the atomic number and identity of an element.
• Neutrons: Chargeless subatomic particles also found in the nucleus. They contribute to the atomic mass but do not affect the charge of the atom.
• Electrons: Negatively charged subatomic particles that orbit the nucleus in electron shells. They are involved in chemical bonding and reactions.

Atomic Symbols

• Elemental Symbol: A one- or two-letter abbreviation for an element (e.g., H for hydrogen, O for oxygen).
• Atomic Number: The number of protons in the nucleus of an atom, which defines the element.
• Mass Number: The total number of protons and neutrons in the nucleus of an atom.
• Nuclear Symbol: A notation that includes the mass number and atomic number. (e.g. $^{12}_{6}C$ for carbon-12)
  $\text{Nuclear Symbol} = \begin{pmatrix} \text{Mass Number} \\ \text{Atomic Number} \end{pmatrix} \text{Element Symbol}$

Types of Atoms

• Elementary Substances: Pure substances that consist of only one type of atom. They can be classified as:
  • Monoatomic: meaning they consist of single atoms. (e.g., He, Ne)
  • Diatomic molecules (e.g., O₂, N₂) or polyatomic molecules (e.g., S₈).
• Isotopes: Atoms of the same element that have different numbers of neutrons, resulting in different mass numbers. Isotopes can be stable or radioactive.

Periodic Table

• Elements are organized by increasing atomic number.
• Each element in a cell contains its elemental symbol, atomic number, and the atomic mass. Atomic mass is the weighted average of the masses of an element's isotopes.
• Rows are called periods and columns are called groups.
• Broadly, metals are on the left and center, while nonmetals are on the upper right.

Molecules and Chemical Bonds

• Atoms can combine by sharing electrons in a covalent bond to form a molecule.

Ways to Represent Molecules

• Chemical formula: shows how many of each atom are present, but not connectivity. $C_6H_{12}O_6$
• Structural formula: shows atoms and bonds.
• Space-filling model: shows relative atom size and occupied space.
• Ball-and-stick model: shows bonds clearly with 3D spheres.

Ions

• Ions: atoms or molecules with a net charge due to an imbalance of protons and electrons.
  • Simple ions: single atoms that have gained or lost electrons. Examples include $Na^+$, $Cl^-$, and $O^{2-}$.
  • Polyatomic ions: groups of atoms that together have a net charge. Examples include $NH_4^+$ (ammonium) and $SO_4^{2-}$ (sulfate).
• Cations: positively charged ions with more protons than electrons. ($H^+$, $Na^+$, $NH_4^+$)
• Anions: negatively charged ions with more electrons than protons. ($OH^-$, $F^-$, $NH_2^-$)
• Ionic Compounds: compounds formed from cations and anions held together by ionic bonds. The electrons in ionic bonds are not shared but transferred from one atom to another, resulting in the formation of ions.

Determining Charge

• Group 1 elements usually form +1 ions.
• Group 2 elements usually form +2 ions.
• Group 13 elements usually form +3 ions.
• Group 14 elements can often be $\pm 4$.
• Group 15 elements usually form -3 ions.
• Group 16 elements usually form -2 ions.
• Group 17 elements usually form -1 ions.
• Group 18 elements are very difficult to ionize and are usually found as neutral atoms.

Covalent and Ionic Bonds

• Elements on the left side of the periodic table are generally more stable as cations. Elements on the right side are generally more stable as anions.
• Observations:
  • Metal + nonmetal usually gives an ionic compound.
  • Nonmetal + nonmetal usually gives a covalent compound.

Naming Chemical Compounds

Type I Binary Ionic Compounds

• These contain one type of cation and one type of anion.
• The cation has only one possible charge.

Naming Rules

• Name the cation first and the anion second.
• The cation keeps its element name.
• The anion uses the beginning of the element name plus -ide.

Examples

• NaCl: sodium chloride
• HBr: hydrogen bromide
• $Be_3N_2$: beryllium nitride

Acids

• Ionic compounds that begin with H are often treated separately as acids.
• For binary acids:
  • Use hydro-
  • Use the anion root
  • End with -ic acid

Examples

• HBr: hydrobromic acid.
• HCl: hydrochloric acid.

Type II Binary Ionic Compounds

• These involve metals, often transition metals, that can have more than one charge.

Naming Rules

• Name the cation first.
• Follow with a Roman numeral in parentheses showing the cation charge.
• Name the anion second using -ide.

Examples

• $FeCl_3$: iron (III) chloride
• $Cu_2S$: copper (I) sulfide
• $Co_3N_4$: cobalt (IV) nitride

Type III Binary Compounds

• These are covalently bonded molecules.
• They contain two nonmetal elements.

Naming Rules

• Name the first element first using its full element name.
• Name the second element like an anion, ending in -ide.
• Use numerical prefixes such as mono-, di-, tri-, and tetra- to show the number of atoms present. Mono- is used only on the second element, never the first.

Examples

• $CO$: carbon monoxide
• $N_2O_3$: dinitrogen trioxide
• $N_2O_4$: dinitrogen tetroxide

Chemical Formula vs. Empirical Formula

• Empirical formula: the simplest whole-number ratio of elements in a compound.
• Chemical formula: the formula that gives the actual number of each type of atom in the molecule.
• Different compounds can have the same empirical formula but different chemical formulas.

Stoichiometry

SI System

International System of Units.

Quantity	Meaning	SI Unit
Mass	measure of how a material resists change in motion	kilogram (kg)
Length	one-dimensional distance between two points	meter (m)
Time	duration of an event	second (s)
Temperature	thermal energy scale of a system	kelvin (K)
Amount of substance	quantity of particles / chemical entities	mole (mol)

Temperature Scales: Kelvin (K), Celsius ($^\circ C$), and Fahrenheit ($^\circ F$).

SI Unit Prefixes

Prefix	Symbol	Factor
atto	a	$10^{-18}$
femto	f	$10^{-15}$
pico	p	$10^{-12}$
nano	n	$10^{-9}$
micro	$\mu$	$10^{-6}$
milli	m	$10^{-3}$
centi	c	$10^{-2}$
deci	d	$10^{-1}$
deka	da	$10^1$
hecto	h	$10^2$
kilo	k	$10^3$
mega	M	$10^6$
giga	G	$10^9$
tera	T	$10^{12}$
peta	P	$10^{15}$
exa	E	$10^{18}$

Atomic Mass

• Mass number is an integer for a specific isotope.
• Atomic mass is the weighted average mass of all naturally occurring isotopes. In atomic mass units (amu).
• In chemical reactions, use the atomic mass from the periodic table rather than a single isotope's mass number.

The Mole and Avogadro's Number

Avogadro's number: $1\ \text{mol} = 6.022 \times 10^{23}\ \text{particles}$

Molar Mass and Mass Percent

The periodic table's atomic mass also gives the molar mass of a natural element in g/mol.

$$12.01\ \text{g natural C} = 1\ \text{mol natural C} = 6.022 \times 10^{23}\ \text{C atoms}$$

Molar mass is the mass of 1 mole of a substance. For an element, use the atomic mass from the periodic table and change the unit from amu/atom to g/mol. Example: C has atomic mass about 12.01 amu/atom, so its molar mass is 12.01 g/mol.

Chemical Reactions

• Reactants: starting materials
• Products: substances formed
• Reaction arrow: shows the direction from reactants to products

Percent Yield

• A limiting reactant is the reactant that gets used up first in a chemical reaction.
• Theoretical yield is the maximum amount of product that can be formed if the limiting reactant is used up completely.
• Percent yield compares actual product obtained to theoretical product expected.

$$\%\text{ yield} = \frac{\text{actual yield}}{\text{theoretical yield}} \times 100\%$$

Solution

• A solution is a homogeneous mixture of solute and solvent in the same phase.
• Solvent: the substance causing the dissolving.
• Solute: the substance being dissolved.
• Dissolve: moving the solute into the same phase as the solvent.

Electrolyte Solution

Dissociation: When ionic compounds dissolve in solution, they separate into ions (cations and anions).

• Compounds that dissociate in solution are electrolytes.
• Covalent molecules that dissolve but do not dissociate are nonelectrolytes.

Electrolyte Strength and Conductivity

• Strong electrolytes dissociate completely.
• Weak electrolytes dissociate only partially.
• Stronger electrolytes conduct electricity better because they produce more ions in solution.

Acids and Bases

Acids and bases are special types of electrolytes.

• Acids: ionic compounds containing $H^+$ ions.

• Bases: ionic compounds containing $OH^-$ ions or help make $OH^-$ in solution.

• Strong acids/bases dissociate essentially 100% in solution.

• Weak acids/bases dissociate only partially.

Examples:

• Strong acids: $\mathrm{HCl}$, $\mathrm{HBr}$, $\mathrm{HI}$, $\mathrm{HNO_3}$, $\mathrm{H_2SO_4}$, $\mathrm{HClO_4}$, $\mathrm{HClO_3}$
• Strong bases: $\mathrm{LiOH}$, $\mathrm{NaOH}$, $\mathrm{KOH}$, $\mathrm{RbOH}$, $\mathrm{CsOH}$, $\mathrm{Ca(OH)_2}$, $\mathrm{Sr(OH)_2}$, $\mathrm{Ba(OH)_2}$
• Weak acids: $\mathrm{HF}$, $\mathrm{CH_3COOH}$
• Weak bases: $\mathrm{NH_3}$, $\mathrm{NaOCl}$

Concentration

Molarity:

$$M = \frac{mol}{L} = \frac{\text{moles of solute}}{\text{liters of solution}}$$

Square-bracket notation like [NaCl] means the molar concentration of that solute.

Dilution

Dilution: lowering the concentration of a solution by adding solvent.
During dilution, the amount of solute stays constant while the total solution volume increases.

$$M_1V_1 = M_2V_2$$

where $M_1$ = initial concentration, $V_1$ = initial volume used, $M_2$ = final concentration, $V_2$ = final volume

Precipitation

Precipitation: the formation of a solid compound from ions in aqueous solution.
A precipitate forms when two soluble ionic compounds react to produce an insoluble ionic compound.

Example:

$$\mathrm{Pb(NO_3)_2(aq) + 2KOH(aq) \rightarrow Pb(OH)_2(s) + 2KNO_3(aq)}$$

Solubility Rules

Anion	Soluble	Slightly Soluble	Insoluble
$\mathrm{NO_3^-}$ nitrate	All	---	---
$\mathrm{CH_3COO^-}$ acetate	Most	---	$\mathrm{Be(CH_3COO)_2}$
$\mathrm{F^-}$ fluoride	Group 1, $\mathrm{AgF}$, $\mathrm{BeF_2}$	$\mathrm{SrF_2}$, $\mathrm{BaF_2}$, $\mathrm{PbF_2}$	$\mathrm{MgF_2}$, $\mathrm{CaF_2}$
$\mathrm{Cl^-}$ chloride	Most	$\mathrm{PbCl_2}$	$\mathrm{AgCl}$, $\mathrm{Hg_2Cl_2}$
$\mathrm{Br^-}$ bromide	Most	$\mathrm{PbBr_2}$, $\mathrm{HgBr_2}$	$\mathrm{AgBr}$, $\mathrm{Hg_2Br_2}$
$\mathrm{SO_4^{2-}}$ sulfate	Most	$\mathrm{CaSO_4}$, $\mathrm{Ag_2SO_4}$	$\mathrm{BaSO_4}$, $\mathrm{SrSO_4}$, $\mathrm{PbSO_4}$, $\mathrm{Hg_2SO_4}$
$\mathrm{S^{2-}}$ sulfide	Group 1 and 2, $\mathrm{(NH_4)_2S}$	---	Most
$\mathrm{CO_3^{2-}}$ carbonate	Group 1, $\mathrm{(NH_4)_2CO_3}$	---	Most
$\mathrm{SO_3^{2-}}$ sulfite	Group 1, $\mathrm{(NH_4)_2SO_3}$	---	Most
$\mathrm{PO_4^{3-}}$ phosphate	Group 1, $\mathrm{(NH_4)_3PO_4}$	---	Most
$\mathrm{OH^-}$ hydroxide	Group 1, $\mathrm{Ba(OH)_2}$, $\mathrm{NH_4OH}$	$\mathrm{Sr(OH)_2}$, $\mathrm{Ca(OH)_2}$	Most

Solubility categories:

• Soluble: dissolves more than about 10 g/L.
• Slightly soluble: dissolves about 0.1 to 10 g/L.
• Insoluble: dissolves less than about 0.1 g/L.

Net Ionic Equations

1. Write the molecular equation.
2. Write the complete ionic equation by dissociating all soluble strong electrolytes.
3. Cancel spectator ions, which appear unchanged on both sides.
4. The remaining reaction is the net ionic equation.

Example:

$$\mathrm{Pb(NO_3)_2(aq) + 2KOH(aq) \rightarrow Pb(OH)_2(s) + 2KNO_3(aq)}$$

Complete ionic equation:

$$\mathrm{Pb^{2+}(aq) + 2NO_3^-(aq) + 2K^+(aq) + 2OH^-(aq) \rightarrow Pb(OH)_2(s) + 2K^+(aq) + 2NO_3^-(aq)}$$

Net ionic equation:

$$\mathrm{Pb^{2+}(aq) + 2OH^-(aq) \rightarrow Pb(OH)_2(s)}$$

Selective Precipitation

• Selective precipitation uses solubility rules to separate different ions from a mixture.
• A reagent is added that precipitates one ion while leaving others dissolved.
• The solid can be filtered out before adding another reagent.

Example: Separate $\mathrm{Ag^+}$, $\mathrm{Ba^{2+}}$, and $\mathrm{Fe^{3+}}$ from nitrate salts.

1. Add $\mathrm{NaCl}$ to precipitate $\mathrm{AgCl(s)}$.
2. Add $\mathrm{Na_2SO_4}$ to precipitate $\mathrm{BaSO_4(s)}$.
3. Add $\mathrm{Na_3PO_4}$ to precipitate $\mathrm{FePO_4(s)}$.

Acid-Base Reactions

• Acid: a molecule that donates $\mathrm{H^+}$.
• Base: a molecule that accepts $\mathrm{H^+}$.
• Acid-base reactions often form water.

$$\mathrm{H^+ + OH^- \rightarrow H_2O}$$

A neutralization reaction occurs when an acid and base react to form water and an ionic compound.

Titrations

• Titration: analysis of a solution where one solution is used to determine the amount or concentration of another.
• Titrant: the solution with known concentration used to analyze the unknown solution.
• Equivalence point: the point where enough titrant has been added to react exactly with the unknown substance.

Redox Reactions

Reduction-oxidation reactions, or redox reactions, involve electron transfer between species.

• Oxidation: losing electrons; oxidation state becomes more positive or less negative.

• Reduction: gaining electrons; oxidation state becomes less positive or more negative.

• Oxidizing agent: causes another species to be oxidized; it gets reduced.

• Reducing agent: causes another species to be reduced; it gets oxidized.

Memory device:

• LEO: Lose Electrons = Oxidation
• GER: Gain Electrons = Reduction

Oxidation States

Oxidation states describe the effective charge assigned to each element in a compound, ion, or reaction.

Rules:

• An element in its elemental form has oxidation state 0. Examples: $\mathrm{Na(s)}$, $\mathrm{H_2(g)}$, $\mathrm{O_2(g)}$.
• Hydrogen is usually +1, except in $\mathrm{H_2(g)}$ and metal hydrides.
• Oxygen is usually -2, except in $\mathrm{O_2(g)}$ and peroxides such as $\mathrm{H_2O_2}$.
• In binary compounds with multiple nonmetals, the more electronegative element gets the negative oxidation state.
• The sum of all oxidation states equals the overall charge of the compound or ion.

Gases

Pressure

$$P = \frac{F}{A}$$

Unit	Meaning
atm	atmosphere
torr	unit named after Torricelli
mmHg	millimeters of mercury
psi	pounds per square inch
Pa	pascal

$1\ \mathrm{atm} = 760\ \mathrm{torr} = 760\ \mathrm{mmHg}$

Ideal Gas Law

An ideal gas is a model gas whose particles:

• Have no attractive or repulsive interactions with each other.
• Collide elastically, meaning collisions do not change total energy.
• Have particle volume that is negligible compared with the container volume.

$$PV = nRT$$

Gas Density:

$$d = \frac{m}{V} = \frac{PM}{RT}$$

Because $R$ is constant, two gas states can be related:

$$R = \frac{P_1V_1}{n_1T_1} = \frac{P_2V_2}{n_2T_2}$$

Variable	Meaning	Common Unit
$P$	pressure	atm
$V$	volume	L
$n$	moles of gas	mol
$R$	gas constant	$\mathrm{0.0821\ \frac{atm \cdot L}{mol \cdot K}}$
$T$	temperature	K

When using energy units:

$$R = 8.3145\ \mathrm{\frac{J}{mol \cdot K}}$$

Standard Temperature and Pressure (STP)

• Standard temperature: 273 K or 0°C
• Standard pressure: 1 atm

At STP: $1\ \mathrm{mol\ gas} = 22.4\ \mathrm{L}$

Partial Pressure

Law of Partial Pressures: For a mixture of gases, the total pressure is the sum of the partial pressures of each gas.

$$P_{\text{total}} = P_A + P_B + P_C + \cdots$$

Mole Fraction

Mole fraction describes the portion of a mixture made up by one component.

$$\chi_A = \frac{\text{moles of A}}{\text{total moles}}$$

Partial pressure can be found using mole fraction: $P_A = \chi_A P_{\text{total}}$

Kinetic Molecular Theory

Average kinetic energy:

$$KE_{\text{avg}} = \frac{3}{2}RT$$

Kinetic energy depends on temperature, not the identity of the gas.

Common speed equations:

$$u_{mp} = \sqrt{\frac{2RT}{M}}, \ u_{avg} = \sqrt{\frac{8RT}{\pi M}}, \ u_{rms} = \sqrt{\frac{3RT}{M}}$$

Where:

• $u_{mp}$ = most probable speed
• $u_{avg}$ = average speed
• $u_{rms}$ = root-mean-square speed
• $M$ = molar mass in kg/mol when using $R = 8.3145\ \mathrm{\frac{J}{mol \cdot K}}$

Diffusion and Effusion

Gases and liquids both belong to a class of substances called fluids, which can flow and do not have a fixed shape.

• Diffusion: particles spread from one area to another, usually from high concentration to low concentration.
• Effusion: particles pass through a tiny opening into another space.

Graham's Law of Effusion/Diffusion:

$$\text{rate} = \frac{1}{\sqrt{M}}$$ $$\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$$

Molecular Collisions

Fluid molecules are constantly moving and colliding with each other and with container walls.

$$\text{collision rate} = \frac{\text{number of collisions}}{\text{time}}$$ $$Z = \frac{N}{V} 4 d^2 \sqrt{\frac{\pi RT}{M}}$$

where $Z$ is the collision rate per unit volume, $N$ is the number of particles, $V$ is the volume, $d$ is the particle diameter, and $M$ is the molar mass.

Quantum Mechanics

Light and Electromagnetic Radiation

Light behaves as both a wave and a particle. This is called wave-particle duality.

• Wavelength ($\lambda$): distance between repeating points in a wave.
• Frequency ($\nu$): number of wave cycles passing a point per unit time.
• Speed: distance traveled per unit time.

$$\text{speed} = \lambda \nu$$

Speed of light: $c = 3.0 \times 10^8\ \mathrm{m/s}$

Light Energy

$$E = h\nu = \frac{hc}{\lambda}$$

where $h = 6.626 \times 10^{-34}\ \mathrm{J \cdot s}$ (Planck's constant)

Higher frequency means higher energy. Shorter wavelength also means higher energy.

Electromagnetic Spectrum

Sort by wavelength (from shortest to longest): Gamma rays, X-rays, ultraviolet, visible light, infrared, microwaves, and radio waves (FM, AM, Shortwave).
In visible light: violet, blue, green, yellow, orange, and red.

• Microwaves are lower-energy electromagnetic radiation.
• Blackbody radiation occurs when hot objects emit light due to their temperature. As a metal is heated, it can glow red, orange, yellow, and eventually white because hotter objects emit more intense and higher-energy visible light.

Photoelectric Effect

Light energy is emitted in packets called photons.
The photoelectric effect occurs when light hits a metal surface and ejects electrons.

• Electrons eject only if the light has high enough frequency.
• Increasing intensity alone does not eject electrons if the frequency is too low.
• The minimum energy needed to eject an electron is the threshold energy.

$$E_0 = h\nu_0$$

If the incoming light has energy greater than the threshold, the excess energy becomes kinetic energy of the ejected electron.

$$KE_{e^-} = h\nu - h\nu_0$$

Einstein's mass-energy relationship:

$$E = mc^2$$

As a system loses energy, it also loses a corresponding amount of mass.

De Broglie Wavelength

Particles can also behave like waves.

$$\lambda = \frac{h}{mv}$$

where $v$ is the velocity of the particle.

Diffraction

Diffraction is the scattering or bending of waves when they interact with particles, openings, or edges.

• Light scatters when it collides with particles.
• Different wavelengths scatter differently because they have different energies.
• Diffraction patterns can be used to determine structures of solids.
• Different atom arrangements create different diffraction patterns.

Bohr Model of the Hydrogen Atom

Hydrogen Line Spectrum

Hydrogen does not emit a continuous spectrum. It emits only specific lines because only certain electron energies are allowed.

• Electron energy in hydrogen is quantized.
• Light is emitted or absorbed when an electron moves between allowed energy levels.
• The photon energy equals the energy difference between levels.

$$\Delta E = E_{\text{final}} - E_{\text{initial}}$$

• Emit light: electron moves to a lower energy level ($n$ decreases).
• Absorb light: electron moves to a higher energy level ($n$ increases).

The electron in a hydrogen atom moves around the nucleus only in allowed circular orbits.

Allowed energy levels:

$$E = -2.178 \times 10^{-18}\ \mathrm{J} \left(\frac{Z^2}{n^2}\right)$$

where $Z$ = nuclear charge and $n$ = principal quantum number

For hydrogen, $Z = 1$.

Quantum Atomic Model

The quantum model treats the electron as a standing wave bound to the nucleus.

• A standing wave must fit whole-number combinations of half-wavelengths.
• Only certain standing waves are allowed.
• This naturally produces quantized electron energies.

Schrodinger Equation

$$\hat{H}\psi = E\psi$$

Where $\hat{H}$ is the Hamiltonian operator, $\psi$ is the wave function, and $E$ is the energy.

The Schrodinger equation has many possible solutions. Each solution describes an allowed electron state.

The equation introduces the principal quantum number $n$, which describes the size and energy of the region where an electron can be found.

Quantum Numbers

Each electron in an atom is described by four quantum numbers.

Quantum Number	Name	Meaning	Possible Values
$n$	principal quantum number	size and energy of orbital	$1, 2, 3, ...$
$l$	angular momentum quantum number	orbital shape	$0$ to $n - 1$
$m_l$	magnetic quantum number	orbital orientation	$-l$ to $+l$
$m_s$	spin quantum number	electron spin	$+\frac{1}{2}$ or $-\frac{1}{2}$

Electron spin describes the orientation of one electron and helps explain how multiple electrons can occupy orbitals.

Examples:

• For $n = 1$: $l = 0$, so $m_l = 0$. There is 1 orbital.
• For $n = 2$: $l = 0$ and $l = 1$. There are 4 orbitals total: one 2s orbital and three 2p orbitals.

Orbitals

An orbital is a region in space where an electron is likely to be found.

Subshell letters:

$l$	Subshell	Number of Orbitals	Electron Capacity
0	s	1	2
1	p	3	6
2	d	5	10
3	f	7	14

Orbital shapes:

• s orbitals are spherical.
• p orbitals have two lobes.
• d orbitals are usually four-lobed.
• f orbitals are more complex, often shown with multiple lobes.

Polyelectronic Atoms

Atoms with more than one electron follow three major rules.

1. Aufbau principle: electrons occupy lower-energy orbitals before higher-energy orbitals.
2. Pauli exclusion principle: no two electrons in an atom can have the same set of four quantum numbers.
3. Hund's rule: when orbitals have the same energy, electrons occupy them singly before pairing.

Electron Configuration

An electron configuration describes how electrons are arranged in orbitals.

General filling order:

$$1s,\ 2s,\ 2p,\ 3s,\ 3p,\ 4s,\ 3d,\ 4p,\ 5s,\ 4d,\ 5p,\ 6s,\ 4f,\ 5d,\ 6p,\ 7s,\ 5f,\ 6d,\ 7p$$

Example: carbon has 6 electrons.

$$\mathrm{C}: 1s^2\ 2s^2\ 2p^2$$

Configuration examples:

Atom	Electron Configuration
Fe	$\mathrm{[Ar]\ 4s^2\ 3d^6}$
U	$\mathrm{[Rn]\ 7s^2\ 5f^3\ 6d^1}$
Cr	$\mathrm{[Ar]\ 4s^1\ 3d^5}$
Cu	$\mathrm{[Ar]\ 4s^1\ 3d^{10}}$

Chromium and copper are common exceptions because half-filled and fully filled d subshells are especially stable.

Special Electron Configurations

Some atoms have observed electron configurations that differ from the simple Aufbau prediction.

• Noble gases have especially stable electron configurations.
• A subshell is more stable when it is half-filled or fully filled.
• Electrons can shift between nearby subshells if doing so creates a more stable half-filled or filled subshell.
• Breaking one or more electron-configuration rules gives an excited-state electron configuration.

Examples:

Atom	Expected Pattern	Observed Stable Configuration
Cr	would place two electrons in 4s before filling 3d	$\mathrm{[Ar]\ 4s^1\ 3d^5}$
Cu	would place two electrons in 4s before filling 3d	$\mathrm{[Ar]\ 4s^1\ 3d^{10}}$

Periodic Trends

Electron configurations help predict physical properties from an element's position on the periodic table.

• Ionization energy: energy r equired to remove an electron from a gaseous atom or ion. This is always positive.
• Electron affinity: energy change when an electron is added to a neutral gaseous atom.
• Atomic radius: size of an atom from the nucleus to the outermost electrons.

General trends:

Property	Trend
Ionization energy	increases up and to the right
Electron affinity	generally becomes more negative to the right, especially for nonmetals
Atomic radius	increases down and to the left

These trends depend on effective nuclear charge.

• More protons increase attraction between the nucleus and electrons.
• More inner electrons shield outer electrons from the nucleus.
• Stronger effective nuclear charge pulls valence electrons closer and usually makes atoms smaller.

Chemical Bonding

Chemical bonds are forces that hold atoms together in molecules and compounds.

• Molecules: two or more atoms bonded together, usually all nonmetals.
• Ionic compounds: metals and nonmetals held together by attraction between ions.
• Covalent compounds: nonmetals sharing electrons through orbital overlap.

Ionic and Covalent Bonds

Ionic bonds form when electrons are transferred between a metal and a nonmetal.

• The metal loses electrons and becomes a cation.
• The nonmetal gains electrons and becomes an anion.
• The ions attract through electrostatic forces.

Coulomb's law describes between charged particles:

$$V = \frac{Q_1Q_2}{4\pi\epsilon_0r} = (2.31 \times 10^{-19}\ \mathrm{J \cdot nm})\left(\frac{Q_1Q_2}{r}\right)$$

Covalent bonds form when two nonmetals share electrons. Similar atomic orbitals overlap, and a pair of electrons is shared between atoms.

Electronegativity

Electronegativity measures how strongly an atom attracts electrons in a bond.

The difference in electronegativity, $\Delta EN$, helps predict bond type:

$\Delta EN$	Bond Type	Example
$\Delta EN > 2.0$	ionic	$\mathrm{Na-Cl}$
$0.5 < \Delta EN < 2.0$	polar covalent	$\mathrm{H-F}$
$\Delta EN < 0.5$	nonpolar covalent	$\mathrm{C-C}$

In a polar covalent bond, electrons are shared unequally. The more electronegative atom has partial negative charge, $\delta^-$, and the less electronegative atom has partial positive charge, $\delta^+$.

Lewis Dot Structures

Valence electrons are the outermost electrons involved in bonding.

1. Count total valence electrons.
2. Add electrons for a negative charge or subtract electrons for a positive charge.
3. Choose a central atom, usually the least electronegative atom that is not H.
4. Connect surrounding atoms to the central atom with single bonds.
5. Complete octets on outer atoms.
6. Put remaining electrons on the central atom.
7. If the central atom lacks an octet, form double or triple bonds.
8. Check formal charges.

Formal Charge

Formal charge helps determine the best Lewis structure when multiple structures are possible.

$$\text{formal charge} = \text{valence electrons} - \text{nonbonding electrons} - \frac{1}{2}(\text{bonding electrons})$$

In most cases, the best Lewis structure has:

• The lowest combination of formal charges.
• Negative formal charge on the more electronegative atom.
• Formal charges closest to zero when possible.

Octet Rule and Exceptions

Octet rule: atoms in molecules are often most stable when they have 8 electrons around them, counting bonds and lone pairs.

Important exceptions:

• The octet rule applies most strictly to second-row elements.
• Hydrogen can only have 2 electrons.
• Group 13 elements, such as B and Al, can have fewer than 8 electrons.
• Elements in the third row or lower can have expanded octets with more than 8 electrons.

Example: $\mathrm{PO_4^{3-}}$. Phosphate has resonance structures and can be drawn with one $\mathrm{P=O}$ double bond and three $\mathrm{P-O^-}$ single bonds to reduce formal charge.

Resonance Structures

Resonance structures occur when multiple valid Lewis structures differ only in electron placement.

• Atom positions stay the same.
• Double bonds and formal charges can move.
• The real molecule is a resonance hybrid of the valid structures.

Example: $\mathrm{NO_3^-}$. Nitrate has three equivalent resonance structures. Each places the $\mathrm{N=O}$ double bond on a different oxygen.

Bond Order

Bond order measures the degree of bonding between atoms.

$$\text{bond order} = \frac{\text{number of bonds}}{\text{number of bond locations}}$$

Higher bond order means:

• Stronger bond.
• Higher bond energy.
• Shorter bond length.

Example: In nitrate, the average N-O bond order is:

$$\frac{4\ \text{total N-O bonds}}{3\ \text{N-O locations}} = \frac{4}{3}$$

Molecular Structure and VSEPR

Lewis structures are two-dimensional. To predict three-dimensional shape, use VSEPR: valence shell electron pair repulsion theory.

VSEPR idea:

• Electron groups around a central atom repel each other.
• The molecule adopts a geometry that keeps electron groups as far apart as possible.
• Lone pairs repel more strongly than bonding pairs.

Steps:

1. Draw the Lewis structure.
2. Determine the steric number of the central atom.
3. Use steric number to determine electron geometry.
4. Use the number of bonded atoms and lone pairs to determine molecular geometry.

$$\text{steric number} = \text{bonded atoms} + \text{lone pairs on central atom}$$

Multiple bonds count as one electron group for VSEPR.

VSEPR Geometries

Steric Number	Lone Pairs	Molecular Geometry	Approx. Bond Angle
2	0	linear	$180^\circ$
3	0	trigonal planar	$120^\circ$
3	1	bent	$<120^\circ$
4	0	tetrahedral	$109.5^\circ$
4	1	trigonal pyramidal	$<109.5^\circ$
4	2	bent	$<109.5^\circ$
5	0	trigonal bipyramidal	$90^\circ$, $120^\circ$
5	1	seesaw	varied
5	2	T-shaped	about $90^\circ$
5	3	linear	$180^\circ$
6	0	octahedral	$90^\circ$
6	1	square pyramidal	about $90^\circ$
6	2	square planar	$90^\circ$

Examples:

Molecule	Steric Number	Molecular Geometry
$\mathrm{CH_4}$	4	tetrahedral
$\mathrm{NH_3}$	4	trigonal pyramidal
$\mathrm{H_2O}$	4	bent
$\mathrm{CO_2}$	2	linear

Molecular Polarity

Molecular polarity depends on both bond polarity and molecular shape.

• If bond dipoles cancel, the molecule is nonpolar.
• If bond dipoles do not cancel, the molecule is polar.
• Lone pairs often make a molecule polar because they change the molecular shape.

Examples:

Molecule	Shape	Polarity
$\mathrm{CO_2}$	linear	nonpolar because equal C=O dipoles cancel
$\mathrm{COS}$	linear	polar because C=O and C=S dipoles are not equal
$\mathrm{H_2O}$	bent	polar because O-H dipoles do not cancel

Bonding Orbitals

Overlapping Orbitals

Valence atomic orbitals on neighboring atoms overlap to form bonds.

Types of orbital overlap:

• Sigma bond ($\sigma$): direct overlap along the bond axis.
• Pi bond ($\pi$): side-by-side overlap of orbitals above and below the bond axis.

General bond patterns:

Bond Type	Orbital Overlap
single bond	1 $\sigma$ bond
double bond	1 $\sigma$ bond and 1 $\pi$ bond
triple bond	1 $\sigma$ bond and 2 $\pi$ bonds

Hybridization

Hybridization is the mixing of atomic orbitals on the same atom to form hybrid orbitals.

• Orbitals with similar energy can mix.
• Hybrid orbitals give a better model for observed molecular shapes.
• Hybridization connects VSEPR geometry with orbital bonding.

Hybridization from Steric Number

Steric Number	Hybridization	Electron Geometry
2	$\mathrm{sp}$	linear
3	$\mathrm{sp^2}$	trigonal planar
4	$\mathrm{sp^3}$	tetrahedral
5	$\mathrm{dsp^3}$ or $\mathrm{sp^3d}$	trigonal bipyramidal
6	$\mathrm{d^2sp^3}$ or $\mathrm{sp^3d^2}$	octahedral

Examples:

Molecule / Atom	Steric Number	Hybridization
C in $\mathrm{CH_4}$	4	$\mathrm{sp^3}$
C in $\mathrm{CO_2}$	2	$\mathrm{sp}$
O in $\mathrm{CO_2}$	3	$\mathrm{sp^2}$
O in $\mathrm{H_2O}$	4	$\mathrm{sp^3}$

For elements in the third row or lower, d orbitals may be included in expanded-octet hybridization models.

Ethylene Bonds

Ethylene: $\mathrm{C_2H_4}$

Each carbon has three bonded atoms, so each carbon is approximately trigonal planar.

• Each carbon is $\mathrm{sp^2}$ hybridized.
• Each C-H bond forms by overlap of H 1s with C $\mathrm{sp^2}$.
• The C=C double bond contains:
  • one $\sigma$ bond from $\mathrm{sp^2}$-$\mathrm{sp^2}$ overlap
  • one $\pi$ bond from side-by-side overlap of unhybridized 2p orbitals

Acetylene Bonds

Acetylene: $\mathrm{C_2H_2}$

Each carbon has steric number 2, so each carbon is $\mathrm{sp}$ hybridized.

• The 2s orbital mixes with one 2p orbital to form two $\mathrm{sp}$ hybrid orbitals.
• Two unhybridized 2p orbitals remain on each carbon.
• The $C \equiv C$ triple bond contains:
  • one $\sigma$ bond
  • two $\pi$ bonds

Molecular Orbital Theory

Atomic orbitals can combine to form molecular orbitals.

Because electrons and orbitals behave like waves, orbital overlap can be:

• Constructive interference: orbitals of the same sign combine to form a lower-energy bonding orbital.
• Destructive interference: orbitals of opposite sign combine to form a higher-energy antibonding orbital.

Important rule:

$$\text{number of atomic orbitals} = \text{number of molecular orbitals}$$

Molecular orbitals are filled using the same rules as atomic orbitals:

1. Aufbau principle: fill lower-energy orbitals first.
2. Pauli exclusion principle: maximum two electrons per orbital with opposite spins.
3. Hund's rule: fill degenerate orbitals singly before pairing.

MO Diagrams

An MO diagram shows how atomic orbitals combine into bonding and antibonding molecular orbitals.

• Bonding orbitals are lower in energy.
• Antibonding orbitals are higher in energy and marked with an asterisk, such as $\sigma^*$.
• MO diagrams can determine bond order and molecular stability.

MO bond order:

$$\text{bond order} = \frac{\text{bonding electrons} - \text{antibonding electrons}}{2}$$

MO Diagrams with p Orbitals

When p orbitals combine, they can form both sigma and pi molecular orbitals.

• End-to-end p-orbital overlap forms $\sigma$ and $\sigma^*$ orbitals.
• Side-by-side p-orbital overlap forms $\pi$ and $\pi^*$ orbitals.
• Degenerate $\pi$ orbitals fill according to Hund's rule.

Magnetism from MO Diagrams

MO diagrams can predict magnetic behavior.

• Paramagnetic: has unpaired electrons.
• Diamagnetic: all electrons are paired.

Example: $\mathrm{O_2}$

• MO theory predicts two unpaired electrons in $\mathrm{O_2}$.
• Therefore, $\mathrm{O_2}$ is paramagnetic.
• Its bond order is 2, corresponding to a double bond.