Formula Sheet
Every key formula of the course on one page, grouped by chapter, with a link back to the section that develops it.
Chapter 1 · Some Basic Concepts of Chemistry
Density§ 1.1 →
$\rho=\dfrac{m}{V}$, $1\ \mathrm{g\,cm^{-3}}=1\ \mathrm{g\,mL^{-1}}=10^{3}\ \mathrm{kg\,m^{-3}}$
Temperature scales§ 1.1 →
$T(\mathrm{K})=t(^{\circ}\mathrm{C})+273.15$, $t(^{\circ}\mathrm{F})=\tfrac{9}{5}\,t(^{\circ}\mathrm{C})+32$ (kelvin temperatures are never negative)
Volume units and SI prefixes§ 1.1 →
$1\ \mathrm{L}=1\ \mathrm{dm^{3}}=10^{3}\ \mathrm{cm^{3}}=10^{-3}\ \mathrm{m^{3}}$, $1\ \mathrm{mL}=1\ \mathrm{cm^{3}}$; p $=10^{-12}$, n $=10^{-9}$, µ $=10^{-6}$, m $=10^{-3}$, c $=10^{-2}$, k $=10^{3}$, M $=10^{6}$
Significant figures in calculations§ 1.1 →
$+,-$: keep the fewest decimal places; $\times,\div$: keep the fewest significant figures; exact numbers never limit. Rounding a final exactly-$5$: keep an even preceding digit, raise an odd one ($7.45\to7.4$, $7.55\to7.6$).
Mass and weight§ 1.1 →
$W=mg$ (in N); mass $m$ (in kg) does not depend on $g$
Unified atomic mass unit§ 1.2 →
$1\ \mathrm{u}=\tfrac{1}{12}\,m\!\left(\mathrm{{}^{12}C}\right)=1.66054\times10^{-24}\ \mathrm{g}$
Average atomic mass§ 1.2 →
$\bar{A}=\sum_{i}f_{i}m_{i}$ ($f_{i}$ = fractional abundance, $\sum f_{i}=1$)
The mole§ 1.2 →
$n=\dfrac{m}{M}=\dfrac{N}{N_{A}}$, $N_{A}=6.022\times10^{23}\ \mathrm{mol^{-1}}$; mass of one particle $=\dfrac{M}{N_{A}}$
Reacting gas volumes§ 1.2 →
At the same $T$ and $p$: volume ratio $=$ ratio of molecules $=$ ratio of coefficients
Molecular / formula mass§ 1.2 →
Sum of atomic masses in one molecule (or formula unit), e.g. $M_{r}(\mathrm{H_{2}O})=2(1.008)+16.00=18.02$
Percentage composition and formulae§ 1.3 →
mass % $=\dfrac{(\text{atoms in formula})\times A}{M}\times100$; molecular formula $=(\text{empirical})_{n}$, $n=\dfrac{M}{\text{empirical formula mass}}$
Limiting reagent§ 1.3 →
For $a\mathrm{A}+b\mathrm{B}\rightarrow\ldots$: the smaller of $\dfrac{n_{\mathrm{A}}}{a}$, $\dfrac{n_{\mathrm{B}}}{b}$ is limiting
Percentage yield§ 1.3 →
$\%\ \text{yield}=\dfrac{\text{actual yield}}{\text{theoretical yield}}\times100$
Concentration terms (1 = solvent, 2 = solute)§ 1.3 →
$x_{2}=\dfrac{n_{2}}{n_{1}+n_{2}}$, $M=\dfrac{n_{2}}{V_{\text{soln}}\,(\mathrm{L})}$, $m=\dfrac{n_{2}}{w_{1}\,(\mathrm{kg})}$, mass % $=\dfrac{w_{2}}{w_{1}+w_{2}}\times100$
Dilution§ 1.3 →
$M_{1}V_{1}=M_{2}V_{2}$ (1 = before, 2 = after); $M_{\text{mix}}=\dfrac{M_{1}V_{1}+M_{2}V_{2}}{V_{1}+V_{2}}$
Molarity → molality§ 1.3 →
$m=\dfrac{1000\,M}{1000\,d-M\,M_{2}}$ ($d$ in $\mathrm{g\,mL^{-1}}$, $M_{2}$ = molar mass of solute)
Chapter 2 · Structure of Atom
Fundamental particles§ 2.1 →
$\mathrm{e^{-}}$: $-1.602\times10^{-19}\ \mathrm{C}$, $9.109\times10^{-31}\ \mathrm{kg}$; $\mathrm{p}$: $+1.602\times10^{-19}\ \mathrm{C}$, $1.6726\times10^{-27}\ \mathrm{kg}$ ($1.00727\ \mathrm{u}$); $\mathrm{n}$: $0$, $1.6749\times10^{-27}\ \mathrm{kg}$ ($1.00867\ \mathrm{u}$); $m_{p}/m_{e}=1836$
Electron: $e/m_{e}$, $e$ and $m_{e}$§ 2.1 →
$\dfrac{e}{m_{e}}=1.758820\times10^{11}\ \mathrm{C\,kg^{-1}}$ (Thomson); $q=ne$, $e=1.602176\times10^{-19}\ \mathrm{C}$ (Millikan); $m_{e}=\dfrac{e}{e/m_{e}}=9.109\times10^{-31}\ \mathrm{kg}$
Balanced crossed fields§ 2.1 →
No deflection when $eE=evB$, so $v=\dfrac{E}{B}$
Counting particles in ${}^{A}_{Z}\mathrm{X}^{q}$§ 2.1 →
protons $=Z$, neutrons $=A-Z$, electrons $=Z-q$
Size of nucleus and atom§ 2.1 →
$r_{\text{nucleus}}\sim10^{-15}\ \mathrm{m}$, $r_{\text{atom}}\sim10^{-10}\ \mathrm{m}$; $\dfrac{V_{\text{nucleus}}}{V_{\text{atom}}}\sim10^{-15}$
Wavelength, frequency, wavenumber§ 2.2 →
$c=\nu\lambda$, $\bar{\nu}=\dfrac{1}{\lambda}=\dfrac{\nu}{c}$, $c=3.0\times10^{8}\ \mathrm{m\,s^{-1}}$
Planck: energy of a photon§ 2.2 →
$E=h\nu=\dfrac{hc}{\lambda}=hc\bar{\nu}$, $h=6.626\times10^{-34}\ \mathrm{J\,s}$; per mole multiply by $N_{A}$
Photoelectric equation§ 2.2 →
$h\nu=h\nu_{0}+\tfrac12m_{e}v^{2}$ (for $\nu\ge\nu_{0}$); $W_{0}=h\nu_{0}=\dfrac{hc}{\lambda_{0}}$; $1\ \mathrm{eV}=1.602\times10^{-19}\ \mathrm{J}$
Rydberg formula (hydrogen)§ 2.2 →
$\bar{\nu}=R_{H}\left(\dfrac{1}{n_{1}^{2}}-\dfrac{1}{n_{2}^{2}}\right)$, $R_{H}=109\,677\ \mathrm{cm^{-1}}$; $n_{1}=1$ Lyman (UV), $2$ Balmer (visible), $3$ Paschen, $4$ Brackett, $5$ Pfund (IR); for $\mathrm{He^{+}}$, $\mathrm{Li^{2+}}$ multiply by $Z^{2}$
Bohr orbits (H and hydrogen-like ions)§ 2.2 →
$E_{n}=-2.18\times10^{-18}\,\dfrac{Z^{2}}{n^{2}}\ \mathrm{J}$, $r_{n}=52.9\,\dfrac{n^{2}}{Z}\ \mathrm{pm}$, $m_{e}vr=\dfrac{nh}{2\pi}$
Transitions and lines§ 2.2 →
$\Delta E=h\nu=2.18\times10^{-18}Z^{2}\left(\dfrac{1}{n_{f}^{2}}-\dfrac{1}{n_{i}^{2}}\right)\ \mathrm{J}$; lines from level $n$ to ground: $\dfrac{n(n-1)}{2}$
de Broglie relation§ 2.3 →
$\lambda=\dfrac{h}{mv}=\dfrac{h}{p}=\dfrac{h}{\sqrt{2mK}}$
Uncertainty principle§ 2.3 →
$\Delta x\,\Delta p_{x}\ge\dfrac{h}{4\pi}$, $\Delta x\,\Delta v_{x}\ge\dfrac{h}{4\pi m}$
Quantum numbers§ 2.3 →
$n=1,2,3,\ldots$; $l=0,1,\ldots,n-1$ (s, p, d, f); $m_{l}=-l,\ldots,+l$; $m_{s}=\pm\tfrac12$
Counting orbitals and electrons§ 2.3 →
subshell: $2l+1$ orbitals, $2(2l+1)$ e⁻; shell: $n^{2}$ orbitals, $2n^{2}$ e⁻
Nodes§ 2.3 →
radial $=n-l-1$, angular $=l$, total $=n-1$
Order of filling (Aufbau, $(n+l)$ rule)§ 2.4 →
$1s\lt2s\lt2p\lt3s\lt3p\lt4s\lt3d\lt4p\lt5s\lt4d\lt5p\lt6s\lt4f\lt5d\lt6p\lt7s$; lower $(n+l)$ first, ties: lower $n$ first
Pauli and Hund§ 2.4 →
≤ 2 electrons per orbital, spins opposite (↑↓); degenerate orbitals are filled singly with parallel spins before pairing
Exceptions: Cr and Cu§ 2.4 →
$\mathrm{Cr}=[\mathrm{Ar}]\,3d^{5}4s^{1}$, $\mathrm{Cu}=[\mathrm{Ar}]\,3d^{10}4s^{1}$ (half-filled / filled $d$ extra stable)
Exchange pairs§ 2.4 →
$k$ parallel spins give $\dfrac{k(k-1)}{2}$ exchange pairs ($d^{5}$: $10$, $d^{4}$: $6$)
Configurations of ions§ 2.4 →
Cations: remove from the highest $n$ first ($4s$ before $3d$): $\mathrm{Fe^{2+}}=[\mathrm{Ar}]\,3d^{6}$, $\mathrm{Fe^{3+}}=[\mathrm{Ar}]\,3d^{5}$; anions: add by Aufbau ($\mathrm{Cl^{-}}=[\mathrm{Ne}]\,3s^{2}3p^{6}$)
Chapter 3 · Classification of Elements and Periodicity in Properties
Döbereiner's triad§ 3.1 →
$A_{\text{middle}}\approx\tfrac12(A_{\text{first}}+A_{\text{third}})$, e.g. $\tfrac12(6.94+39.10)=23.02\approx A(\mathrm{Na})$
Periodic laws§ 3.1 →
Mendeleev (1869): properties = periodic function of atomic mass. Modern: properties = periodic function of atomic number $Z$.
Long form of the table§ 3.1 →
7 periods of $2,8,8,18,18,32,32$ elements (total $118$); 18 groups; lanthanoids $Z=58$–$71$, actinoids $Z=90$–$103$
IUPAC names for $Z\gt100$§ 3.1 →
0 nil, 1 un, 2 bi, 3 tri, 4 quad, 5 pent, 6 hex, 7 sept, 8 oct, 9 enn; add -ium; symbol = first letters. bi/tri + ium → bium/trium; enn + nil → ennil. E.g. $114$ ununquadium (Uuq), $120$ unbinilium (Ubn)
Period from configuration§ 3.2 →
period = $n$ of the outermost shell; period lengths $2,8,8,18,18,32,32$ ($=2\times$ orbitals filled)
General configurations of the blocks§ 3.2 →
s: $ns^{1-2}$; p: $ns^{2}np^{1-6}$; d: $(n-1)d^{1-10}ns^{0-2}$; f: $(n-2)f^{1-14}(n-1)d^{0-1}ns^{2}$
Group from configuration§ 3.2 →
s-block: group = $ns$ electrons; p-block: group = $10+(ns+np)$ electrons; d-block: group = $(n-1)d+ns$ electrons
Metallic character§ 3.2 →
decreases left → right across a period; increases down a group. Metalloids: B, Si, Ge, As, Sb, Te
Summary of periodic trends§ 3.3 →
Across a period (→): radius ↓, $\Delta_{i}H$ ↑, $\Delta_{eg}H$ more negative, electronegativity ↑, metallic character ↓. Down a group (↓): the reverse.
Covalent radius§ 3.3 →
$r=\tfrac12\times$ bond length of $\mathrm{X{-}X}$, e.g. $\mathrm{Cl_{2}}$: $198/2=99\ \mathrm{pm}$
Isoelectronic series§ 3.3 →
radius falls as $Z$ rises: $\mathrm{N^{3-}\gt O^{2-}\gt F^{-}\gt Na^{+}\gt Mg^{2+}\gt Al^{3+}}$
Ionisation enthalpy anomalies§ 3.3 →
$\mathrm{Be\gt B}$, $\mathrm{Mg\gt Al}$ ($p$ electron removed); $\mathrm{N\gt O}$, $\mathrm{P\gt S}$ (half-filled $p^{3}$)
Electron gain enthalpy (group 17)§ 3.3 →
$\mathrm{Cl}\ (-349)\lt\mathrm{F}\ (-328)\lt\mathrm{Br}\ (-325)\lt\mathrm{I}\ (-295)\ \mathrm{kJ\,mol^{-1}}$
Nature of oxides (period 3)§ 3.3 →
$\mathrm{Na_{2}O}$, $\mathrm{MgO}$ basic → $\mathrm{Al_{2}O_{3}}$ amphoteric → $\mathrm{SiO_{2}}$ … $\mathrm{Cl_{2}O_{7}}$ acidic
Chapter 4 · Chemical Bonding and Molecular Structure
Formal charge§ 4.1 →
$\text{FC}=V-L-\tfrac12B$ ($V$ valence e⁻ of free atom, $L$ lone-pair e⁻, $B$ bonding e⁻); $\sum\text{FC}=$ charge of species
Electrons in a Lewis structure§ 4.1 →
total $=\sum V$ $+$ (negative charge) $-$ (positive charge); e.g. $\mathrm{CO_{3}^{2-}}$: $4+18+2=24$
Ionic bond energetics (NaCl)§ 4.1 →
$\Delta_{i}H(\mathrm{Na})+\Delta_{eg}H(\mathrm{Cl})=+147.1$; lattice $788\ \mathrm{kJ\,mol^{-1}}$ makes the solid stable
Bond order and resonance§ 4.1 →
order ↑ ⇒ length ↓, enthalpy ↑; resonance average $=n_{\mathrm{bonds}}/n_{\mathrm{positions}}$ ($\mathrm{CO_{3}^{2-}}$, $\mathrm{NO_{3}^{-}}$: $4/3$; $\mathrm{O_{3}}$: $3/2$)
Dipole moment§ 4.1 →
$\mu=Q\times r$; $\ 1\ \mathrm{D}=3.33564\times10^{-30}\ \mathrm{C\,m}$
Fajans' rules§ 4.1 →
more covalent character: small cation, large anion, high charges, $d^{10}$ (pseudo-noble-gas) cation — e.g. $\mathrm{NaCl\lt MgCl_{2}\lt AlCl_{3}}$, $\mathrm{LiF\lt LiI}$, $\mathrm{NaCl\lt CuCl}$
Electron pairs on the central atom§ 4.2 →
pairs $=\tfrac12(V+M-c+a)$; $M$ = monovalent atoms (H, halogens), O/S count $0$; lone pairs = pairs − bonded atoms
Repulsion order§ 4.2 →
lp–lp $\gt$ lp–bp $\gt$ bp–bp; lone pairs equatorial in a trigonal bipyramid, trans in an octahedron
Shapes with no lone pairs§ 4.2 →
2 linear $180^{\circ}$ · 3 trigonal planar $120^{\circ}$ · 4 tetrahedral $109.5^{\circ}$ · 5 trigonal bipyramidal $120^{\circ}/90^{\circ}$ · 6 octahedral $90^{\circ}$
Shapes with lone pairs§ 4.2 →
$\mathrm{AB_{2}E}$ bent ($\mathrm{SO_{2}}$) · $\mathrm{AB_{3}E}$ pyramidal ($\mathrm{NH_{3}}$ $107^{\circ}$) · $\mathrm{AB_{2}E_{2}}$ bent ($\mathrm{H_{2}O}$ $104.5^{\circ}$) · $\mathrm{AB_{4}E}$ see-saw ($\mathrm{SF_{4}}$) · $\mathrm{AB_{3}E_{2}}$ T ($\mathrm{ClF_{3}}$) · $\mathrm{AB_{2}E_{3}}$ linear ($\mathrm{XeF_{2}}$) · $\mathrm{AB_{5}E}$ square pyramidal ($\mathrm{BrF_{5}}$) · $\mathrm{AB_{4}E_{2}}$ square planar ($\mathrm{XeF_{4}}$)
Dipole of a bent AB₂ molecule§ 4.2 →
$\mu=2\mu_{b}\cos\tfrac{\theta}{2}$ ($\mu=0$ at $180^{\circ}$; $\mu=\mu_{b}$ at $120^{\circ}$)
Polarity and shape§ 4.2 →
symmetric $\mathrm{AB}_{n}$ ($\mathrm{CO_{2}}$, $\mathrm{BF_{3}}$, $\mathrm{CCl_{4}}$, $\mathrm{XeF_{4}}$): $\mu=0$; $\mu(\mathrm{NH_{3}})=1.47\gt\mu(\mathrm{NF_{3}})=0.23\ \mathrm{D}$
σ and π in multiple bonds§ 4.3 →
single = 1σ; double = 1σ + 1π; triple = 1σ + 2π; σ count = (atoms − 1) + rings
Hybridisation from electron pairs§ 4.3 →
σ bonds + lone pairs $=\tfrac12(V+M-c+a)$: 2 sp · 3 sp² · 4 sp³ · 5 sp³d · 6 sp³d²
Hybrid orbitals§ 4.3 →
sp linear $180^{\circ}$ (50% s) · sp² trigonal planar $120^{\circ}$ (33.3%) · sp³ tetrahedral $109.5^{\circ}$ (25%) · sp³d trigonal bipyramidal · sp³d² octahedral
Carbon–carbon bonds§ 4.3 →
$\mathrm{C{-}C}$ sp³ $154\ \mathrm{pm}$ · $\mathrm{C{=}C}$ sp² $134\ \mathrm{pm}$ · $\mathrm{C{\equiv}C}$ sp $120\ \mathrm{pm}$
The H₂ energy curve§ 4.3 →
minimum at $r=74\ \mathrm{pm}$, depth $435.8\ \mathrm{kJ\,mol^{-1}}$ $=7.24\times10^{-19}\ \mathrm{J}$ per molecule
Bond order (MO theory)§ 4.4 →
bond order $=\tfrac12(N_{b}-N_{a})$; $\gt0$ stable, $0$ does not exist; higher order → stronger, shorter bond
MO order up to N₂ (s–p mixing)§ 4.4 →
$\sigma1s\lt\sigma^{*}1s\lt\sigma2s\lt\sigma^{*}2s\lt\pi2p_{x}=\pi2p_{y}\lt\sigma2p_{z}\lt\pi^{*}2p_{x}=\pi^{*}2p_{y}\lt\sigma^{*}2p_{z}$
MO order for O₂, F₂, Ne₂§ 4.4 →
$\sigma1s\lt\sigma^{*}1s\lt\sigma2s\lt\sigma^{*}2s\lt\sigma2p_{z}\lt\pi2p_{x}=\pi2p_{y}\lt\pi^{*}2p_{x}=\pi^{*}2p_{y}\lt\sigma^{*}2p_{z}$
Diatomic bond orders§ 4.4 →
$\mathrm{H_{2}}$ 1 · $\mathrm{B_{2}}$ 1 (para) · $\mathrm{C_{2}}$ 2 · $\mathrm{N_{2}}$ 3 · $\mathrm{O_{2}}$ 2 (para) · $\mathrm{F_{2}}$ 1 · $\mathrm{He_{2}}$, $\mathrm{Be_{2}}$, $\mathrm{Ne_{2}}$ 0
Dioxygen species§ 4.4 →
$\mathrm{O_{2}^{+}}$ 2.5 $\gt$ $\mathrm{O_{2}}$ 2 $\gt$ $\mathrm{O_{2}^{-}}$ 1.5 $\gt$ $\mathrm{O_{2}^{2-}}$ 1 (only $\mathrm{O_{2}^{2-}}$ diamagnetic)
Hydrogen bonding§ 4.4 →
$\mathrm{X{-}H\cdots Y}$ (X, Y = F, O, N); b.p. $\mathrm{H_{2}O}\ 100\gt\mathrm{HF}\ 20\gt\mathrm{NH_{3}}\ {-33}\ ^{\circ}\mathrm{C}$; intramolecular in 2-nitrophenol
Chapter 5 · Thermodynamics
First law (IUPAC signs)§ 5.1 →
$\Delta U=q+w$; $q,w\gt0$ when energy enters the system; isolated system: $\Delta U=0$; cycle: $q=-w$
Work against a constant pressure§ 5.1 →
$w=-p_{ex}\Delta V$; free expansion ($p_{ex}=0$): $w=0$; $1\ \mathrm{L\,bar}=100\ \mathrm{J}$
Reversible isothermal work (ideal gas)§ 5.1 →
$w_{rev}=-2.303\,nRT\log\dfrac{V_{f}}{V_{i}}=-2.303\,nRT\log\dfrac{p_{i}}{p_{f}}$; any isothermal change of an ideal gas: $\Delta U=0$, $q=-w$
Adiabatic change§ 5.1 →
$q=0$: $\Delta U=w_{ad}$ (work on the gas warms it; expansion cools it)
Enthalpy§ 5.2 →
$H=U+pV$; $q_{p}=\Delta H$ and $q_{V}=\Delta U$ (only pV work); exothermic $\Delta H\lt0$
ΔH from ΔU (ideal gases, constant T)§ 5.2 →
$\Delta H=\Delta U+\Delta n_{g}RT$; $\Delta n_{g}$ = gaseous moles of products − reactants; $RT=2.478\ \mathrm{kJ\,mol^{-1}}$ at $298\ \mathrm{K}$
Heat and heat capacity§ 5.2 →
$q=C\Delta T=mc\Delta T=nC_{m}\Delta T$; $q_{V}=nC_{v}\Delta T=\Delta U$, $q_{p}=nC_{p}\Delta T=\Delta H$
Cp − Cv (ideal gas, molar)§ 5.2 →
$C_{p}-C_{v}=R$; monatomic: $C_{v}=\tfrac32R$, $C_{p}=\tfrac52R$
Calorimetry§ 5.2 →
$q_{r}=-q_{cal}=-C_{cal}\Delta T$; bomb (constant V) → $\Delta U=q_{r}/n$; coffee cup (constant p) → $\Delta H=q_{r}/n$
ΔrH° from enthalpies of formation§ 5.3 →
$\Delta_{r}H^{\circ}=\sum\nu\,\Delta_{f}H^{\circ}(\text{products})-\sum\nu\,\Delta_{f}H^{\circ}(\text{reactants})$; $\Delta_{f}H^{\circ}=0$ for elements in their reference states
Hess's law and thermochemical equations§ 5.3 →
$\Delta_{r}H^{\circ}=\sum$ steps; reverse → change sign; multiply equation → multiply $\Delta_{r}H^{\circ}$
Phase changes§ 5.3 →
$\Delta_{sub}H=\Delta_{fus}H+\Delta_{vap}H$ (same $T$); water: $6.01$ and $40.66\ \mathrm{kJ\,mol^{-1}}$
ΔrH° from bond enthalpies (gases only)§ 5.3 →
$\Delta_{r}H^{\circ}\approx\sum\text{BE}(\text{bonds broken})-\sum\text{BE}(\text{bonds formed})$
Born–Haber cycle (MX)§ 5.3 →
$\Delta_{lattice}H=-\Delta_{f}H^{\circ}+\Delta_{sub}H+\tfrac12\Delta_{bond}H+\Delta_{i}H+\Delta_{eg}H$; NaCl: $788\ \mathrm{kJ\,mol^{-1}}$
Enthalpy of solution§ 5.3 →
$\Delta_{sol}H^{\circ}=\Delta_{lattice}H^{\circ}+\Delta_{hyd}H^{\circ}$
Entropy change§ 5.4 →
$\Delta S=\dfrac{q_{rev}}{T}$; phase change: $\Delta_{trs}S=\dfrac{\Delta_{trs}H}{T_{trs}}$
Second law§ 5.4 →
$\Delta S_{total}=\Delta S_{sys}+\Delta S_{surr}\gt0$ (spontaneous), $=0$ (equilibrium); $\Delta S_{surr}=-\Delta H_{sys}/T$ (constant $T$, $p$)
ΔrS° from standard entropies§ 5.4 →
$\Delta_{r}S^{\circ}=\sum\nu S^{\circ}(\text{P})-\sum\nu S^{\circ}(\text{R})$; $S=0$ for a perfect crystal at $0\ \mathrm{K}$
Gibbs energy criterion§ 5.4 →
$G=H-TS$; $\Delta G=\Delta H-T\Delta S=-T\Delta S_{total}$; $\Delta G\lt0$ spontaneous, $=0$ equilibrium; crossover $T=\Delta H/\Delta S$
ΔrG° and K§ 5.4 →
$\Delta_{r}G^{\circ}=-RT\ln K=-2.303RT\log K$; $\Delta_{r}G^{\circ}=\Delta_{r}H^{\circ}-T\Delta_{r}S^{\circ}$; $\Delta_{r}G^{\circ}\lt0\Leftrightarrow K\gt1$
Chapter 6 · Equilibrium
Law of chemical equilibrium§ 6.1 →
$a\mathrm{A}+b\mathrm{B}\rightleftharpoons c\mathrm{C}+d\mathrm{D}$: $K_{c}=\dfrac{[\mathrm{C}]^{c}[\mathrm{D}]^{d}}{[\mathrm{A}]^{a}[\mathrm{B}]^{b}}$ (equilibrium concentrations; depends only on $T$)
Units of $K_{c}$§ 6.1 →
$(\mathrm{mol\,L^{-1}})^{\Delta n}$, $\Delta n=(c+d)-(a+b)$; none if $\Delta n=0$
Changing the equation§ 6.1 →
reverse: $1/K$ · multiply by $n$: $K^{n}$ · add equations: $K_{1}K_{2}$
$K_{p}$ and $K_{c}$§ 6.1 →
$K_{p}=K_{c}(RT)^{\Delta n}$, $\Delta n$ = gaseous moles (products − reactants), $R=0.0831\ \mathrm{L\,bar\,mol^{-1}\,K^{-1}}$
Heterogeneous equilibria§ 6.1 →
pure solids and liquids omitted (must be present): $\mathrm{CaCO_{3}(s)\rightleftharpoons CaO(s)+CO_{2}(g)}$, $K_{p}=p_{\mathrm{CO_{2}}}$
Henry's law§ 6.1 →
mass of gas dissolved $\propto$ its pressure (constant $T$); solubility falls as $T$ rises
Extent of reaction§ 6.2 →
$K_{c}\gt10^{3}$ products predominate · $10^{-3}$ to $10^{3}$ both appreciable · $K_{c}\lt10^{-3}$ reactants predominate
Direction of reaction§ 6.2 →
$Q_{c}\lt K_{c}$ forward · $Q_{c}\gt K_{c}$ backward · $Q_{c}=K_{c}$ equilibrium
$\Delta G$, $Q$ and $K$§ 6.2 →
$\Delta_{r}G=\Delta_{r}G^{\circ}+RT\ln Q$; $\Delta_{r}G^{\circ}=-RT\ln K=-2.303RT\log K$; $\Delta_{r}G^{\circ}\lt0\Leftrightarrow K\gt1$
Dissociation of $\mathrm{N_{2}O_{4}}$ at total pressure $P$§ 6.2 →
$K_{p}=\dfrac{4\alpha^{2}P}{1-\alpha^{2}}$; for $\mathrm{PCl_{5}}$: $K_{p}=\dfrac{\alpha^{2}P}{1-\alpha^{2}}$
Le Chatelier summary§ 6.2 →
add reactant → forward · compress → fewer gas moles · inert gas (constant $V$) → no shift · raise $T$ → endothermic direction ($K$ changes) · catalyst → no shift
Ionic product of water§ 6.3 →
$K_{w}=[\mathrm{H^{+}}][\mathrm{OH^{-}}]=1.0\times10^{-14}$ at $298\ \mathrm{K}$; $\mathrm{pH}+\mathrm{pOH}=14.00$
pH§ 6.3 →
$\mathrm{pH}=-\log[\mathrm{H^{+}}]$, $[\mathrm{H^{+}}]=10^{-\mathrm{pH}}$; neutral at $298\ \mathrm{K}$: pH 7
Weak acid (or base) of concentration $c$§ 6.3 →
$K_{a}=\dfrac{c\alpha^{2}}{1-\alpha}$; if $\alpha\lt0.05$: $\alpha\approx\sqrt{K_{a}/c}$, $[\mathrm{H^{+}}]\approx\sqrt{K_{a}c}$ (same with $K_{b}$, $[\mathrm{OH^{-}}]$)
Conjugate pair§ 6.3 →
$K_{a}K_{b}=K_{w}$; $\mathrm{p}K_{a}+\mathrm{p}K_{b}=14.00$ ($298\ \mathrm{K}$)
Diprotic acid $\mathrm{H_{2}A}$§ 6.3 →
pH from $K_{a_{1}}$ alone; $[\mathrm{A^{2-}}]\approx K_{a_{2}}$; $[\mathrm{A^{2-}}]=\dfrac{K_{a_{1}}K_{a_{2}}[\mathrm{H_{2}A}]}{[\mathrm{H^{+}}]^{2}}$
Common-ion effect§ 6.3 →
weak acid + its salt: $[\mathrm{H^{+}}]=K_{a}\dfrac{[\mathrm{HA}]}{[\mathrm{A^{-}}]}$
pH of salt solutions (298 K)§ 6.4 →
WA + SB: $\mathrm{pH}=7+\tfrac12(\mathrm{p}K_{a}+\log c)$ · SA + WB: $\mathrm{pH}=7-\tfrac12(\mathrm{p}K_{b}+\log c)$ · WA + WB: $\mathrm{pH}=7+\tfrac12(\mathrm{p}K_{a}-\mathrm{p}K_{b})$
Hydrolysis constant§ 6.4 →
$K_{h}=K_{w}/K_{a}$ (anion) or $K_{w}/K_{b}$ (cation); $h\approx\sqrt{K_{h}/c}$
Henderson equation§ 6.4 →
$\mathrm{pH}=\mathrm{p}K_{a}+\log\dfrac{[\text{salt}]}{[\text{acid}]}$; $\mathrm{pOH}=\mathrm{p}K_{b}+\log\dfrac{[\text{salt}]}{[\text{base}]}$; best near $\mathrm{pH}=\mathrm{p}K_{a}\pm1$
$K_{sp}$ and solubility§ 6.4 →
$\mathrm{M}_{x}\mathrm{A}_{y}$: $K_{sp}=x^{x}y^{y}S^{x+y}$; AB: $S^{2}$, $\mathrm{AB_{2}}$: $4S^{3}$, $\mathrm{AB_{3}}$: $27S^{4}$
Precipitation and common ion§ 6.4 →
$Q_{sp}\gt K_{sp}$ → precipitate; in $c$ of common ion: AB $S\approx K_{sp}/c$
Chapter 7 · Redox Reactions
Oxidation and reduction§ 7.1 →
Oxidation: loss of $\mathrm{e^{-}}$, ON rises. Reduction: gain of $\mathrm{e^{-}}$, ON falls. Oxidising agent is reduced; reducing agent is oxidised.
Oxidation-number rules (in priority order)§ 7.1 →
Element $0$; monatomic ion = charge; F $-1$; group 1 $+1$, group 2 $+2$, Al $+3$; O $-2$ (peroxide $-1$, superoxide $-\tfrac12$, $\mathrm{OF_{2}}$ $+2$); H $+1$ (metal hydride $-1$); Cl, Br, I $-1$ in halides
Sum rule§ 7.1 →
$\sum(\text{ON}\times\text{number of atoms})=$ charge of the species ($0$ if neutral); e.g. $\mathrm{Cr_{2}O_{7}^{2-}}$: $2x-14=-2$, $x=+6$
Range of oxidation numbers (groups 13–17)§ 7.1 →
highest $=$ group $-10$; lowest $=$ group $-18$ (N: $-3$ to $+5$; S: $-2$ to $+6$; Cl: $-1$ to $+7$)
Activity series§ 7.1 →
K, Na, Ca, Mg, Al, Zn, Fe, Ni, Sn, Pb, H, Cu, Hg, Ag, Au — a metal displaces the ions of any metal below it
Half-reaction method — order of steps§ 7.2 →
atoms (not O, H) → O with $\mathrm{H_{2}O}$ → H with $\mathrm{H^{+}}$ → (basic: add $\mathrm{OH^{-}}$ to both sides) → charge with $\mathrm{e^{-}}$ → equalise $\mathrm{e^{-}}$, add, cancel, check
Oxidation-number method§ 7.2 →
total increase in ON $=$ total decrease; then charge with $\mathrm{H^{+}}$ / $\mathrm{OH^{-}}$, H with $\mathrm{H_{2}O}$, check O
Key half-reactions (acidic)§ 7.2 →
$\mathrm{MnO_{4}^{-}+8H^{+}+5e^{-}\rightarrow Mn^{2+}+4H_{2}O}$$\mathrm{Cr_{2}O_{7}^{2-}+14H^{+}+6e^{-}\rightarrow 2Cr^{3+}+7H_{2}O}$
Iodine and thiosulphate§ 7.2 →
$\mathrm{I_{2}+2S_{2}O_{3}^{2-}\rightarrow 2I^{-}+S_{4}O_{6}^{2-}}$ (starch: blue disappears)
Redox titration stoichiometry§ 7.2 →
$a\,\mathrm{Ox}+b\,\mathrm{Red}$: $\dfrac{M_{\mathrm{Ox}}V_{\mathrm{Ox}}}{a}=\dfrac{M_{\mathrm{Red}}V_{\mathrm{Red}}}{b}$ ($\mathrm{MnO_{4}^{-}}:\mathrm{Fe^{2+}}=1:5$)
Standard cell EMF§ 7.3 →
$E^{\circ}_{\text{cell}}=E^{\circ}_{\text{cathode}}-E^{\circ}_{\text{anode}}$ (reduction potentials; never multiplied by coefficients); $E^{\circ}_{\text{cell}}\gt0$ ⇒ spontaneous
Daniell cell§ 7.3 →
$\mathrm{Zn(s)\,|\,Zn^{2+}(aq)\,\|\,Cu^{2+}(aq)\,|\,Cu(s)}$, $E^{\circ}_{\text{cell}}=0.34-(-0.76)=1.10\ \mathrm{V}$; anode (−) on the left
Standard hydrogen electrode§ 7.3 →
$\mathrm{2H^{+}(aq,1\ M)+2e^{-}\rightarrow H_{2}(g,1\ bar)}$; $E^{\circ}=0.00\ \mathrm{V}$ by convention (Pt electrode)
Using the electrochemical series§ 7.3 →
higher $E^{\circ}$: stronger oxidised form (oxidant); lower $E^{\circ}$: stronger reduced form (reductant); an oxidant oxidises any reductant whose couple lies below it; metals with $E^{\circ}\lt0$ displace $\mathrm{H_{2}}$ from dilute acid
Chapter 8 · Organic Chemistry – Some Basic Principles and Techniques
Hybridisation of carbon§ 8.1 →
4 σ bonds → $sp^{3}$, $109.5^{\circ}$ · 3 σ + 1 π → $sp^{2}$, $120^{\circ}$ · 2 σ + 2 π → $sp$, $180^{\circ}$
s character and bond length§ 8.1 →
$s$ character $25\%$ / $33\%$ / $50\%$; $\mathrm{C{-}C}\ 154$, $\mathrm{C{=}C}\ 134$, $\mathrm{C{\equiv}C}\ 120\ \mathrm{pm}$; electronegativity $sp\gt sp^{2}\gt sp^{3}$
Counting σ and π bonds§ 8.1 →
single = 1σ; double = 1σ + 1π; triple = 1σ + 2π; total σ = (atoms − 1) + rings
Reading a bond-line formula§ 8.1 →
every vertex / line end = C; H on that C $=4-(\text{bonds drawn})$
General formulae of homologous series§ 8.1 →
$\mathrm{C}_{n}\mathrm{H}_{2n+2}$ alkanes · $\mathrm{C}_{n}\mathrm{H}_{2n}$ alkenes · $\mathrm{C}_{n}\mathrm{H}_{2n-2}$ alkynes · $\mathrm{C}_{n}\mathrm{H}_{2n+1}\mathrm{OH}$ alkanols · $\mathrm{C}_{n}\mathrm{H}_{2n}\mathrm{O}$ aldehydes/ketones · $\mathrm{C}_{n}\mathrm{H}_{2n}\mathrm{O_{2}}$ acids/esters; successive members differ by $\mathrm{CH_{2}}$ ($14\ \mathrm{u}$)
Degree of unsaturation (JEE)§ 8.1 →
$\text{DU}=c-\tfrac{h}{2}-\tfrac{x}{2}+\tfrac{n}{2}+1$ (O, S ignored); ring or double bond = 1, triple bond = 2
Naming a saturated chain§ 8.2 →
longest chain (tie: more substituents) → lowest locant set (first point of difference) → prefixes in alphabetical order (ignore di-, tri-) → tie: lower locant to the prefix first in the alphabet
Word roots§ 8.2 →
1 meth · 2 eth · 3 prop · 4 but · 5 pent · 6 hex · 7 hept · 8 oct · 9 non · 10 dec
Priority of functional groups§ 8.2 →
$\mathrm{{-}COOH}\gt\mathrm{{-}SO_{3}H}\gt\mathrm{{-}COOR}\gt\mathrm{{-}COCl}\gt\mathrm{{-}CONH_{2}}\gt\mathrm{{-}CN}\gt\mathrm{{-}CHO}\gt\mathrm{\gt C{=}O}\gt\mathrm{{-}OH}\gt\mathrm{{-}NH_{2}}\gt\mathrm{C{=}C},\ \mathrm{C{\equiv}C}$; halo, nitro, alkoxy always prefixes
Numbering with a functional group§ 8.2 →
lowest locant: principal group → multiple bonds (tie: C=C lower) → prefixes; $\mathrm{{-}CHO}$, $\mathrm{{-}COOH}$, $\mathrm{{-}CN}$ carbon is C1
Disubstituted benzenes§ 8.2 →
1,2- ortho (o-) · 1,3- meta (m-) · 1,4- para (p-)
Conditions for stereoisomerism§ 8.2 →
$\mathrm{abC{=}Cxy}$ cis/trans if $\mathrm{a}\ne\mathrm{b}$ and $\mathrm{x}\ne\mathrm{y}$; chiral carbon = 4 different groups; at most $2^{n}$ stereoisomers for $n$ chiral carbons
Bond fission§ 8.3 →
homolysis $\mathrm{A{-}B\rightarrow A^{\bullet}+{}^{\bullet}B}$ (half-headed arrows); heterolysis $\mathrm{A{-}B\rightarrow A^{+}+B^{-}}$ (full arrow, B more electronegative)
Reactive intermediates§ 8.3 →
$\mathrm{R_{3}C^{+}}$: 6 e⁻, $sp^{2}$, planar · $\mathrm{R_{3}C^{\bullet}}$: 7 e⁻, ≈ $sp^{2}$ · $\mathrm{R_{3}C^{-}}$: 8 e⁻, $sp^{3}$, pyramidal; cations and radicals $3^{\circ}\gt2^{\circ}\gt1^{\circ}\gt\mathrm{CH_{3}}$; carbanions $\mathrm{CH_{3}^{-}}\gt1^{\circ}\gt2^{\circ}\gt3^{\circ}$
Inductive effect§ 8.3 →
+I: $\mathrm{(CH_{3})_{3}C{-}\gt(CH_{3})_{2}CH{-}\gt C_{2}H_{5}{-}\gt CH_{3}{-}}$; −I: $\mathrm{NO_{2}}$, $\mathrm{CN}$, $\mathrm{COOH}$, halogens $\mathrm{F\gt Cl\gt Br\gt I}$, $\mathrm{OH}$
Resonance (R) effect§ 8.3 →
+R: $\mathrm{{-}OH,\ {-}OR,\ {-}NH_{2},\ {-}NHCOCH_{3},\ {-}X}$; −R: $\mathrm{{-}NO_{2},\ {-}CN,\ {-}CHO,\ {-}COR,\ {-}COOH,\ {-}COOR}$
Hyperconjugation§ 8.3 →
number of hyperconjugative structures = number of α-H: $\mathrm{(CH_{3})_{3}C^{+}}$ 9, $\mathrm{(CH_{3})_{2}CH^{+}}$ 6, $\mathrm{C_{2}H_{5}^{+}}$ 3, $\mathrm{CH_{3}^{+}}$ 0
Types of organic reaction§ 8.3 →
substitution · addition · elimination · rearrangement (nucleophilic, electrophilic or free-radical)
Retardation factor§ 8.4 →
$R_{f}=\dfrac{\text{distance moved by substance}}{\text{distance moved by solvent front}}$, $\ 0\lt R_{f}\lt1$
Steam distillation§ 8.4 →
boils when $p_{\text{organic}}+p_{\text{water}}=p_{\text{atm}}$ (below $373\ \mathrm{K}$)
Carbon and hydrogen (Liebig)§ 8.4 →
$\%\mathrm{C}=\tfrac{12}{44}\times\tfrac{m(\mathrm{CO_{2}})}{m}\times100$; $\ \%\mathrm{H}=\tfrac{2}{18}\times\tfrac{m(\mathrm{H_{2}O})}{m}\times100$; $\%\mathrm{O}$ by difference
Nitrogen (Dumas, Kjeldahl)§ 8.4 →
Dumas: $\%\mathrm{N}=\tfrac{28}{22\,400}\times\tfrac{V_{0}(\mathrm{mL,\ STP})}{m}\times100$; Kjeldahl: $\%\mathrm{N}=\tfrac{1.4\,n(\mathrm{NH_{3}},\ \mathrm{mmol})}{m(\mathrm{g})}=\tfrac{1.4\times M\times2(V-V_{1}/2)}{m}$
Halogens, S, P (Carius and related)§ 8.4 →
$\%\mathrm{X}=\tfrac{A_{r}(\mathrm{X})}{M(\mathrm{AgX})}\times\tfrac{m(\mathrm{AgX})}{m}\times100$ (143.5, 188, 235); $\ \%\mathrm{S}=\tfrac{32}{233}\times\tfrac{m(\mathrm{BaSO_{4}})}{m}\times100$; $\ \%\mathrm{P}=\tfrac{62}{222}\times\tfrac{m(\mathrm{Mg_{2}P_{2}O_{7}})}{m}\times100$
Lassaigne's test results§ 8.4 →
N: Prussian blue · S: black PbS / violet (nitroprusside) · N + S: blood red · X: AgCl white, AgBr pale yellow, AgI yellow
Chapter 9 · Hydrocarbons
General formulae of hydrocarbons§ 9.1 →
alkane $\mathrm{C_{n}H_{2n+2}}$; alkene, cycloalkane $\mathrm{C_{n}H_{2n}}$; alkyne $\mathrm{C_{n}H_{2n-2}}$
Complete combustion of an alkane§ 9.1 →
$\mathrm{C_{n}H_{2n+2}+\tfrac{3n+1}{2}O_{2}\rightarrow nCO_{2}+(n+1)H_{2}O}$
Preparing alkanes: carbon count§ 9.1 →
Wurtz $\mathrm{2RX+2Na\rightarrow R{-}R}$; soda lime $\mathrm{RCOONa\rightarrow RH}$; Kolbe $\mathrm{2RCOO^{-}\rightarrow R{-}R+2CO_{2}}$ (anode)
Free-radical chlorination of methane§ 9.1 →
$\mathrm{Cl_{2}\xrightarrow{h\nu}2Cl^{\bullet}}$; $\mathrm{Cl^{\bullet}+CH_{4}\rightarrow {}^{\bullet}CH_{3}+HCl}$; $\mathrm{{}^{\bullet}CH_{3}+Cl_{2}\rightarrow CH_{3}Cl+Cl^{\bullet}}$; reactivity $\mathrm{F_{2}\gt Cl_{2}\gt Br_{2}\gt I_{2}}$, H: 3° $\gt$ 2° $\gt$ 1°
Conformations of ethane§ 9.1 →
$E(\phi)\approx\tfrac12E_{0}(1+\cos3\phi)$, $E_{0}\approx12.5\ \mathrm{kJ\,mol^{-1}}$; staggered $\phi=60^{\circ}$ most stable
Condition for cis–trans isomerism§ 9.2 →
$\mathrm{abC{=}Cxy}$ has cis and trans forms only if $\mathrm{a\ne b}$ and $\mathrm{x\ne y}$; cis: higher μ and b.p.; trans: higher m.p.
Markovnikov and peroxide effect§ 9.2 →
$\mathrm{CH_{3}CH{=}CH_{2}+HBr\rightarrow CH_{3}CHBrCH_{3}}$; with peroxide (HBr only): $\mathrm{CH_{3}CH_{2}CH_{2}Br}$
Preparing alkenes§ 9.2 →
alkyne + $\mathrm{H_{2}}$/Lindlar → cis; Na/liq. $\mathrm{NH_{3}}$ → trans; $\mathrm{RX}$ + alc. KOH; vic-dihalide + Zn; alcohol + conc. $\mathrm{H_{2}SO_{4}}$ (443 K for ethanol)
Ozonolysis§ 9.2 →
$\mathrm{R_{2}C{=}CR'_{2}\xrightarrow{O_{3};\ Zn/H_{2}O}R_{2}C{=}O+O{=}CR'_{2}}$
Permanganate oxidation of alkenes§ 9.2 →
cold dil. alk. $\mathrm{KMnO_{4}}$ → 1,2-diol; hot acidic: $\mathrm{{=}CR_{2}\rightarrow R_{2}CO}$, $\mathrm{{=}CHR\rightarrow RCOOH}$, $\mathrm{{=}CH_{2}\rightarrow CO_{2}+H_{2}O}$
Ethyne from calcium carbide§ 9.3 →
$\mathrm{CaO+3C\rightarrow CaC_{2}+CO}$; $\mathrm{CaC_{2}+2H_{2}O\rightarrow Ca(OH)_{2}+C_{2}H_{2}}$
Alkyne from a vicinal dihalide§ 9.3 →
$\mathrm{RCHX{-}CH_{2}X\xrightarrow{\text{alc. KOH}}\text{vinylic halide}\xrightarrow{NaNH_{2}}RC{\equiv}CH}$
Acidity and s character§ 9.3 →
sp (50% s) $\gt$ sp² (33%) $\gt$ sp³ (25%): $\mathrm{HC{\equiv}CH\gt C_{2}H_{4}\gt C_{2}H_{6}}$; only ≡C–H reacts with Na, $\mathrm{NaNH_{2}}$
Hydration of alkynes§ 9.3 →
$\mathrm{HC{\equiv}CH+H_{2}O\xrightarrow{Hg^{2+}/H^{+},\ 333\,K}CH_{3}CHO}$; $\mathrm{RC{\equiv}CH\rightarrow RCOCH_{3}}$
Cyclic polymerisation§ 9.3 →
$\mathrm{3HC{\equiv}CH\xrightarrow{\text{Fe, }873\,K}C_{6}H_{6}}$
Hückel's rule§ 9.4 →
aromatic ⇔ cyclic, planar, fully conjugated, $(4n+2)$ π electrons: 2, 6, 10, …
Nitration of benzene§ 9.4 →
$\mathrm{HNO_{3}+2H_{2}SO_{4}\rightleftharpoons NO_{2}^{+}+H_{3}O^{+}+2HSO_{4}^{-}}$; $\mathrm{C_{6}H_{6}\rightarrow C_{6}H_{5}NO_{2}}$ (323–333 K)
Friedel–Crafts reactions§ 9.4 →
$\mathrm{C_{6}H_{6}+RCl\xrightarrow{AlCl_{3}}C_{6}H_{5}R+HCl}$; $\mathrm{C_{6}H_{6}+RCOCl\xrightarrow{AlCl_{3}}C_{6}H_{5}COR+HCl}$
Directive influence§ 9.4 →
o/p, activating: –NH₂, –OH, –OCH₃, –NHCOCH₃, –R; o/p, deactivating: halogens; m, deactivating: –NO₂, –CN, –CHO, –COR, –COOH, –SO₃H
Combustion of any hydrocarbon§ 9.4 →
$\mathrm{C_{x}H_{y}+(x+\tfrac{y}{4})O_{2}\rightarrow xCO_{2}+\tfrac{y}{2}H_{2}O}$