MDCAT full mock (all subjects) key facts

Biology · Acellular life

Viruses: structure and classification

• Virion = one complete infective virus particle; nucleocapsid = capsid + the nucleic acid inside it
• Size: about 20 nm to 250 nm, so viruses pass through filters that hold back bacteria
• Capsid symmetry: helical (tobacco mosaic virus), polyhedral or icosahedral with 20 faces (poliovirus, adenovirus), complex (T₄ bacteriophage: polyhedral head + tail)
• By host: bacteriophages infect bacteria, plant viruses infect plants, animal viruses infect animals
• DNA viruses: smallpox, herpes, hepatitis B, T₄ phage. RNA viruses: polio, influenza, measles, mumps, HIV, tobacco mosaic virus
• Enveloped viruses: influenza, HIV, herpes. Naked (no envelope): poliovirus, tobacco mosaic virus, T₄ phage
• T₄ phage: head holds DNA; tail fibres attach to receptor sites on the bacterial wall

Viral replication and disease

• Lytic cycle order: attachment (adsorption) → penetration → replication of phage DNA and proteins → assembly → lysis
• Only the phage DNA enters the bacterium; the protein coat stays outside. Lysozyme from the phage tail makes the hole in the wall
• Virulent phage = lytic cycle only; temperate phage (for example lambda) = can follow either cycle
• Prophage = phage DNA inserted in the bacterial chromosome; the bacterium carrying it is lysogenic
• HIV in a helper T cell: attachment to the CD4 receptor → reverse transcription (RNA → DNA) → integration as a provirus → transcription and translation → assembly → budding
• HIV spreads by unprotected sexual contact, infected blood or shared needles, and from mother to child (placenta, birth, breast milk). It does not spread by touch, food, coughing or mosquitoes
• Prevention: screened blood, sterile needles, safe sexual behaviour, treatment of infected mothers. There is no cure and no vaccine; antiretroviral drugs slow the virus

Biology · Bioenergetics

Photosynthesis

• 6CO₂ + 12H₂O + light → C₆H₁₂O₆ + 6O₂ + 6H₂O; the oxygen released comes from water
• Chlorophyll absorbs mainly violet-blue and orange-red light and reflects green. Absorption spectrum = light absorbed at each wavelength; action spectrum = rate of photosynthesis at each wavelength
• Photolysis: 2H₂O → 4H⁺ + 4e⁻ + O₂, at photosystem II (P680)
• Non-cyclic photophosphorylation: both photosystems; electrons go water → PS II → PS I → NADP⁺; products ATP, NADPH and O₂
• Cyclic photophosphorylation: photosystem I (P700) only; electrons return to P700; product ATP only
• Calvin cycle: carbon fixation (CO₂ + RuBP → two 3-carbon PGA, enzyme RuBisCO) → reduction (PGA → G3P, uses ATP and NADPH) → regeneration of RuBP (uses ATP)
• Each CO₂ fixed uses 3 ATP and 2 NADPH. Limiting factors: light intensity, CO₂ concentration, temperature, water

Cellular respiration

• C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + energy (ATP)
• Glycolysis: glucose (6C) → 2 pyruvic acid (3C); 2 ATP used, 4 made, net 2 ATP; 2 NADH; no CO₂; no oxygen needed
• Link reaction (per pyruvic acid): pyruvic acid → acetyl CoA (2C) + CO₂ + NADH
• Krebs cycle (per turn): 2 CO₂, 3 NADH, 1 FADH₂, 1 ATP; oxaloacetic acid (4C) is regenerated; two turns per glucose
• Electron transport chain: each NADH gives 3 ATP, each FADH₂ gives 2 ATP; H⁺ is pumped into the intermembrane space and flows back through ATP synthase; oxygen is the final electron acceptor
• Per glucose: 10 NADH (30 ATP) + 2 FADH₂ (4 ATP) + 4 ATP made directly = 38 ATP; 36 ATP in most eukaryotic cells, because 2 ATP are spent moving the NADH of glycolysis into the mitochondrion
• Anaerobic: yeast, pyruvic acid → acetaldehyde + CO₂ → ethanol; muscle, pyruvic acid → lactic acid. Fats enter as acetyl CoA; amino acids are deaminated first

Biology · Biological molecules

Water and carbohydrates

• High specific heat capacity: keeps body and lake temperatures steady. High heat of vaporisation: sweating and transpiration cool. Ice is less dense than water, so it floats
• Monosaccharides by carbon number: triose 3C (glyceraldehyde), pentose 5C (ribose, deoxyribose, ribulose), hexose 6C (glucose, fructose, galactose). Aldose = aldehyde group (glucose); ketose = ketone group (fructose)
• Maltose = glucose + glucose; sucrose = glucose + fructose; lactose = glucose + galactose
• Starch = amylose (unbranched, α 1→4) + amylopectin (branched, α 1→4 and α 1→6): plant store
• Glycogen: α-glucose, more branched than amylopectin: animal store in liver and muscle
• Cellulose: β-glucose, β 1→4, straight unbranched chains: plant cell walls
• Iodine test: starch blue-black, glycogen red-brown, cellulose no colour. n monosaccharides in a chain have (n − 1) glycosidic bonds

Proteins and lipids

• About 20 kinds of amino acid occur in proteins; glycine is the simplest (R = H). A chain of n amino acids has (n − 1) peptide bonds
• Primary structure = sequence of amino acids. Secondary = α-helix or β-pleated sheet held by hydrogen bonds
• Tertiary = 3-D folding of one chain (ionic bonds, hydrogen bonds, disulphide bridges between cysteines, hydrophobic interactions). Quaternary = two or more chains together (haemoglobin has four)
• Fibrous proteins: long, insoluble, structural (keratin, collagen, silk). Globular proteins: compact, soluble (enzymes, antibodies, haemoglobin)
• Triglyceride = 1 glycerol + 3 fatty acids, 3 ester bonds, 3 water molecules released
• Saturated fatty acid: no C=C, higher melting point (fats). Unsaturated: one or more C=C, lower melting point (oils)
• Other lipids: waxes (waterproofing), steroids such as cholesterol (four fused rings), terpenoids such as carotenoids (isoprene units). Fat stores about twice as much energy per gram as carbohydrate

Nucleic acids

• Purines (two rings): adenine, guanine. Pyrimidines (one ring): cytosine, thymine, uracil. Nucleoside = sugar + base; nucleotide = sugar + base + phosphate
• Base pairing: A=T with 2 hydrogen bonds, G≡C with 3. So A = T, G = C and (A + G) = (T + C)
• Double helix: 10 base pairs per turn, 3.4 nm per turn, 0.34 nm between base pairs, 2 nm wide; nucleotides in one strand joined by phosphodiester bonds
• Replication: helicase unwinds; primase makes an RNA primer; DNA polymerase III adds nucleotides 5′ → 3′; leading strand continuous, lagging strand in Okazaki fragments joined by DNA ligase
• Transcription: RNA polymerase copies the template strand into mRNA; mRNA has the same sequence as the coding strand with U in place of T
• Genetic code: 64 triplet codons; 61 code for amino acids, 3 are stop codons (UAA, UAG, UGA); AUG is the start codon (methionine)
• mRNA carries the code, tRNA carries amino acids and has the anticodon, rRNA builds ribosomes

Biology · Cell structure and function

Cell membrane and transport

• Diffusion: net movement from higher to lower concentration; passive; no protein needed
• Facilitated diffusion: down the gradient through a specific channel or carrier protein; passive; rate levels off when all carriers are in use
• Osmosis: net movement of water through a selectively permeable membrane from higher water potential (dilute solution) to lower water potential (concentrated solution)
• Water potential (ψ): pure water = 0; any solution is negative; more solute = more negative
• Active transport: against the gradient, carrier protein + ATP; sodium–potassium pump moves 3 Na⁺ out and 2 K⁺ in per ATP
• Endocytosis (into the cell): phagocytosis = solid particles, pinocytosis = fluid droplets. Exocytosis: vesicles fuse with the membrane and release their contents. Both need ATP
• Animal cell: bursts in a hypotonic solution, shrinks in a hypertonic one. Plant cell: turgid in hypotonic, plasmolysed in hypertonic

Organelles

• Nucleus: double membrane (envelope) with pores; chromatin = DNA + histone proteins; nucleolus makes rRNA and ribosome subunits
• Rough ER: ribosomes attached, makes and transports proteins. Smooth ER: no ribosomes, makes lipids and steroids, detoxifies drugs
• Ribosomes: no membrane; 80S (60S + 40S) in eukaryotic cytoplasm, 70S (50S + 30S) in prokaryotes, mitochondria and chloroplasts
• Golgi apparatus: receives vesicles at the forming (cis) face, adds carbohydrate to make glycoproteins, releases vesicles and lysosomes from the maturing (trans) face
• Lysosomes: single membrane, hydrolytic enzymes; digest food particles and worn-out organelles (autophagy)
• Mitochondrion: inner membrane folded into cristae, matrix inside. Chloroplast: thylakoids stacked into grana, stroma around them
• Microtubules (tubulin): spindle, cilia, centrioles (9 triplets). Microfilaments (actin): cyclosis, amoeboid movement. Plant vacuole is bounded by the tonoplast; plant cell wall is cellulose and fully permeable

Prokaryotic and eukaryotic cells

• Prokaryote: 70S ribosomes, wall of peptidoglycan (murein), circular DNA without histones, no mitochondria (respiratory enzymes on the plasma membrane), divides by binary fission
• Eukaryote: 80S ribosomes in the cytoplasm, linear DNA with histones in a nucleus, membrane-bound organelles, divides by mitosis or meiosis
• Typical size: prokaryotic cell about 1 to 10 μm; eukaryotic cell about 10 to 100 μm
• Plant cell only: cellulose wall, chloroplasts and other plastids, large central vacuole. Animal cell only: centrioles; stores glycogen, not starch
• magnification = image size ÷ actual size; 1 mm = 1000 μm; 1 μm = 1000 nm
• Light microscope: lower resolution, can view living cells. Electron microscope: resolution to below 1 nm, specimen dead and in a vacuum; TEM shows internal detail, SEM shows 3-D surfaces
• Structure fits function: sperm has many mitochondria for movement; red blood cell has no nucleus, more room for haemoglobin; root hair has a large surface area

Biology · Coordination and control

Neurons and the nerve impulse

• Resting potential: sodium–potassium pump moves 3 Na⁺ out and 2 K⁺ in; the membrane leaks K⁺ out far more easily than Na⁺ in; large negative ions stay inside
• Depolarisation = Na⁺ diffuses in. Repolarisation = K⁺ diffuses out. The pump then restores the original ion balance
• All-or-none: below threshold there is no impulse; above it every impulse has the same size. A stronger stimulus gives more impulses per second, not bigger ones
• Myelin sheath (Schwann cells) insulates the axon; the impulse jumps between nodes of Ranvier (saltatory conduction), up to about 100 m s⁻¹
• Synapse: impulse arrives → Ca²⁺ enters the synaptic knob → vesicles release neurotransmitter → it diffuses across the cleft → binds receptors on the postsynaptic membrane → new impulse
• Acetylcholine is broken down by acetylcholinesterase. A synapse passes impulses in one direction only
• Sensory neuron cell bodies lie in the dorsal root ganglion; Nissl's granules are groups of ribosomes in the cell body

The nervous system

• Cerebrum (cerebral cortex): thinking, memory, intelligence, voluntary movement, senses. Each hemisphere controls the opposite side of the body; the corpus callosum joins the two
• Thalamus: relays sensory impulses to the cortex. Hypothalamus: temperature, hunger, thirst, controls the pituitary. Amygdala and hippocampus: emotion and memory
• Cerebellum: balance, posture, coordination of movement. Medulla oblongata: breathing, heartbeat, blood pressure. Pons: links cerebellum, medulla and cerebrum
• Spinal cord: grey matter (cell bodies) inside in an H shape, white matter (myelinated axons) outside; the cerebrum has grey matter outside
• Reflex arc: receptor → sensory neuron (dorsal root) → interneuron in the spinal cord → motor neuron (ventral root) → effector
• Sympathetic system: prepares for emergency (faster heart, dilated pupils, slower digestion). Parasympathetic: rest (slower heart, constricted pupils, active digestion)
• Receptors: chemoreceptors (taste, smell), mechanoreceptors (touch, pressure, hearing), photoreceptors (light), thermoreceptors (temperature), nociceptors (pain). Disorders: Parkinson's (low dopamine, tremor), epilepsy (seizures), Alzheimer's (loss of memory)

Chemical coordination

• Anterior pituitary: growth hormone, TSH (thyroid), ACTH (adrenal cortex), FSH, LH, prolactin (milk production). Posterior pituitary releases ADH (water reabsorption in the kidney) and oxytocin (labour, milk ejection), both made in the hypothalamus
• Thyroid: thyroxine raises metabolic rate and needs iodine; calcitonin lowers blood Ca²⁺. Parathyroid: parathormone raises blood Ca²⁺
• Pancreas (islets of Langerhans): β cells, insulin lowers blood glucose (glucose → glycogen); α cells, glucagon raises it (glycogen → glucose)
• Adrenal medulla: adrenaline and noradrenaline (emergency). Adrenal cortex: cortisol (raises blood glucose in stress), aldosterone (Na⁺ reabsorption)
• Gonads: testosterone (testes); oestrogen and progesterone (ovaries)
• Growth hormone: too little in childhood = dwarfism; too much in childhood = gigantism; too much in adults = acromegaly
• Thyroxine: too little = cretinism (child), myxoedema (adult); too much = Graves' disease. Insulin lack = diabetes mellitus; ADH lack = diabetes insipidus; cortex hormones low = Addison's, high = Cushing's

Biology · Enzymes

Enzyme action

• Activation energy = energy of the peak − energy of the reactants. An enzyme lowers the peak only.
• Lock-and-key model (Fischer): the active site is rigid and the substrate (key) fits it exactly.
• Induced-fit model (Koshland): the active site changes shape slightly as the substrate binds, to fit it closely.
• Holoenzyme (active) = apoenzyme (protein part) + cofactor (non-protein part).
• Activator: a detachable inorganic ion, e.g. Mg²⁺, Cl⁻.
• Coenzyme: an organic cofactor that is loosely attached, e.g. NAD, FAD; many are made from vitamins.
• Prosthetic group: a non-protein part bound tightly (covalently) to the protein, e.g. haem.

Factors and inhibition

• Enzyme concentration: with excess substrate, rate is directly proportional to the amount of enzyme.
• Substrate concentration: rate rises, then levels off at a maximum when every active site is occupied (saturation).
• Optimum pH: pepsin about 2, salivary amylase about 6.8, pancreatic lipase about 9.
• Competitive inhibitor: resembles the substrate and binds at the active site; more substrate overcomes it, so the maximum rate is still reached.
• Non-competitive inhibitor: binds at another site and changes the shape of the active site; more substrate does not help, so the maximum rate is lower.
• Irreversible inhibitors (e.g. cyanide) bind by strong covalent bonds; the enzyme stays inactive.
• Feedback inhibition: the end product of a pathway slows an early enzyme of that pathway.

Biology · Evolution

Theories of evolution

• Lamarck: use and disuse of organs, and inheritance of acquired characters.
• Darwin: overproduction → struggle for existence → variation → survival of the fittest → inheritance of the favoured variations.
• Darwin published 'On the Origin of Species' in 1859; Malthus gave him the idea that populations outgrow their food supply.
• Fitness means leaving the most offspring that survive to breed, not being the strongest.
• Mutation is the only source of new alleles; crossing over and independent assortment only make new combinations.
• Only mutations in germ cells (which form gametes) can be inherited.
• Hardy-Weinberg: allele frequencies stay constant if the population is large, mates at random and has no mutation, migration or selection.

Evidence and speciation

• Fossils: in undisturbed sedimentary rock, deeper layers are older. Archaeopteryx links reptiles and birds.
• Homologous organs: same basic structure and origin, different function (human arm, bat wing, whale flipper); show divergent evolution from a common ancestor.
• Analogous organs: same function, different structure and origin (bird wing, insect wing); show convergent evolution.
• Vestigial organs: reduced and no longer used for their original function, e.g. coccyx, appendix.
• Embryology: early vertebrate embryos all have gill pouches and a tail.
• Biochemistry: the fewer the differences in a protein or in DNA, the more recent the common ancestor.
• Adaptive radiation: one ancestor gives many species suited to different ways of life, e.g. Galápagos finches.

Biology · Reproduction

Human reproductive systems

• Spermatogonium (2n) → primary spermatocyte (2n) → 2 secondary spermatocytes (n) → 4 spermatids (n) → 4 sperm.
• Oogonium (2n) → primary oocyte (2n) → secondary oocyte (n) + first polar body → ovum (n) + second polar body.
• One primary spermatocyte gives 4 sperm; one primary oocyte gives 1 ovum.
• Primary oocytes are held at prophase I from before birth; meiosis II is completed only after a sperm enters.
• Spermatogenesis begins at puberty and is continuous; oogenesis begins before birth and ends at menopause.
• In the male, FSH promotes spermatogenesis; LH stimulates the interstitial cells to secrete testosterone.
• Seminal vesicles secrete fructose, the energy source of sperm.

Reproductive cycle and development

• Order of events: menstruation (days 1 to 5) → follicle growth and oestrogen → LH surge and ovulation (about day 14) → corpus luteum and progesterone.
• Oestrogen and progesterone inhibit FSH and LH (negative feedback); this is how the contraceptive pill prevents ovulation.
• hCG from the embryo keeps the corpus luteum alive in early pregnancy; later the placenta makes the progesterone.
• Cleavage is mitosis with little growth: zygote → morula → blastocyst, which implants.
• Germ layers: ectoderm → nervous system and epidermis; mesoderm → muscle, skeleton, blood; endoderm → lining of gut and lungs.
• Placenta: maternal and foetal blood do not mix. The umbilical cord has two arteries (deoxygenated blood to the placenta) and one vein.
• STDs: gonorrhoea (Neisseria gonorrhoeae), syphilis (Treponema pallidum), AIDS (HIV), genital herpes (herpes simplex virus).

Biology · Support and movement

Skeleton and joints

• Vertebrae: 7 cervical, 12 thoracic, 5 lumbar, then the sacrum (5 fused) and coccyx (4 fused). Ribs: 12 pairs.
• Bone cells: osteoblasts build bone, osteocytes maintain it, osteoclasts dissolve it and release calcium.
• Cartilage: hyaline (ends of bones, nose, trachea), elastic (pinna, epiglottis), fibrocartilage (intervertebral discs, pubic symphysis).
• Joints: fibrous, immovable (skull sutures); cartilaginous, slightly movable (between vertebrae); synovial, freely movable.
• Synovial types: hinge (elbow, knee), ball and socket (shoulder, hip), pivot (atlas and axis).
• Osteoarthritis: articular cartilage wears away with age. Rheumatoid arthritis: the immune system attacks the joint lining.
• Osteoporosis: loss of bone mass, so bones become porous and break easily; common after menopause as oestrogen falls.

Muscles and contraction

• A band = full length of the myosin filaments (dark); I band = actin only (light); H zone = myosin only, in the middle of the A band.
• On contraction the A band stays the same; the I band and H zone get shorter; Z lines move closer.
• Ca²⁺ is released from the sarcoplasmic reticulum, binds troponin, and tropomyosin moves off the binding sites on actin.
• Myosin heads are ATPases. ATP is needed for the head to detach from actin; without ATP the muscle stays stiff (rigor mortis).
• Skeletal: striated, voluntary, many nuclei, unbranched. Cardiac: striated, involuntary, branched, with intercalated discs.
• Smooth: unstriated, involuntary, spindle-shaped cells with one nucleus; found in the gut and blood vessels.
• Tendon joins muscle to bone; ligament joins bone to bone. Origin is the fixed end, insertion the moving end.

Biology · Inheritance

Mendelian inheritance

• Monohybrid cross Aa × Aa: genotypes 1 AA : 2 Aa : 1 aa; phenotypes 3 dominant : 1 recessive.
• Test cross: unknown dominant phenotype × homozygous recessive. All dominant offspring → parent is AA; 1 : 1 → parent is Aa.
• Dihybrid cross AaBb × AaBb: phenotypes 9 : 3 : 3 : 1 (out of 16).
• Dihybrid test cross AaBb × aabb: 1 : 1 : 1 : 1.
• Number of kinds of gamete = 2ⁿ, where n is the number of heterozygous gene pairs.
• For independent genes, multiply the separate probabilities (product rule).

Beyond simple dominance

• Incomplete dominance, heterozygote × heterozygote: 1 : 2 : 1 for both phenotype and genotype.
• ABO: I^A and I^B are codominant and both are dominant to i. Group A = I^AI^A or I^Ai; B = I^BI^B or I^Bi; AB = I^AI^B; O = ii.
• Three alleles give 6 genotypes and 4 phenotypes.
• Group O has no A or B antigens (universal donor of red cells); group AB has no anti-A or anti-B antibodies (universal recipient).
• Epistasis changes 9 : 3 : 3 : 1 to 9 : 3 : 4 (recessive epistasis) or 12 : 3 : 1 (dominant epistasis).
• Rh factor: Rh-positive (D) is dominant to Rh-negative (d).
• Haemolytic disease of the newborn: Rh-negative mother, Rh-positive foetus, usually in a second or later such pregnancy.

Sex linkage and chromosomes

• A father gives his X to all his daughters and his Y to all his sons; a son's X always comes from his mother.
• Carrier mother × normal father: half the sons affected, half the daughters carriers, no daughter affected.
• Affected father × normal (non-carrier) mother: all daughters carriers, all sons normal.
• Haemophilia A is a lack of clotting factor VIII.
• Down syndrome: trisomy 21, 47 chromosomes. Turner syndrome: 45, XO, a sterile female. Klinefelter syndrome: 47, XXY, a sterile male.
• In humans the presence of a Y chromosome gives a male.
• Other systems: XO in grasshoppers (male has one X only); ZW in birds (the female is ZW and decides the sex).

Biology · Circulation

The heart

• Path: venae cavae → right atrium → tricuspid valve → right ventricle → pulmonary artery → lungs → pulmonary veins → left atrium → bicuspid (mitral) valve → left ventricle → aorta.
• The pulmonary artery carries deoxygenated blood; the pulmonary veins carry oxygenated blood.
• Impulse: SA node → AV node (short delay) → bundle of His → Purkinje fibres → ventricles contract from the apex upwards.
• Cardiac cycle at 75 beats per minute = 60 ÷ 75 = 0.8 s: atrial systole 0.1 s, ventricular systole 0.3 s, diastole of the whole heart 0.4 s.
• First sound 'lubb': the AV (tricuspid and bicuspid) valves close as the ventricles start to contract.
• Second sound 'dupp': the semilunar valves close as the ventricles start to relax.
• Sympathetic nerves speed up the SA node; the vagus (parasympathetic) nerve slows it.

Blood and blood vessels

• Normal resting blood pressure is about 120/80 mm Hg (systolic/diastolic).
• Blood flows slowest in the capillaries, because their total cross-sectional area is the greatest.
• Blood returns up the leg veins by the squeezing of skeletal muscles, with valves stopping backflow.
• Red blood cells: biconcave, no nucleus, carry oxygen on haemoglobin; made in red bone marrow; live about 120 days.
• White blood cells (about 7 000 to 8 000 per mm³): neutrophils (most common, phagocytes), lymphocytes (immunity), monocytes (largest, become macrophages), eosinophils (allergy, worms), basophils (heparin and histamine).
• Platelets are cell fragments that start clotting: prothrombin → thrombin (needs Ca²⁺); thrombin turns fibrinogen → fibrin.
• Plasma proteins: albumin (osmotic pressure), globulins (antibodies), fibrinogen (clotting).

Lymphatic system and disorders

• Net filtration pressure = blood (hydrostatic) pressure − osmotic pressure of the plasma
• Arterial end: hydrostatic pressure is greater, so fluid leaves. Venous end: osmotic pressure is greater, so fluid returns
• Lymph nodes filter lymph and hold lymphocytes and macrophages; lacteals in the villi absorb digested fats
• Oedema = swelling from excess tissue fluid (low plasma protein, blocked lymph vessels or high blood pressure)
• Atherosclerosis = cholesterol-rich plaque (atheroma) in the artery wall; arteriosclerosis = hardening of the wall, with calcium deposits
• Thrombus = a clot fixed where it formed; embolus = a clot or fragment carried in the blood
• Hypertension = blood pressure that stays at about 140/90 mm Hg or above; normal is about 120/80 mm Hg

Biology · Immunity

Innate and adaptive immunity

• Phagocytosis: attachment to the microbe → engulfing into a vacuole (phagosome) → lysosomes fuse with it → enzymes digest the microbe
• Inflammation: mast cells release histamine → arterioles dilate and capillaries become leaky → redness, heat, swelling and pain
• Interferon is released by virus-infected cells and makes neighbouring cells resist viruses
• B cells mature in the bone marrow; T cells are made in the bone marrow and mature in the thymus
• An activated B cell divides to form plasma cells (which secrete antibodies) and memory cells
• Helper T cells activate B cells and killer T cells; killer (cytotoxic) T cells destroy infected body cells
• Antibody = 2 heavy chains + 2 light chains joined by disulphide bonds; the variable regions at the tips form 2 antigen-binding sites

Types of immunity and vaccination

• Natural active: recovering from an infection. Artificial active: vaccination
• Natural passive: antibodies across the placenta or in breast milk (colostrum). Artificial passive: an injection of antiserum or antitoxin
• A vaccine may be a live weakened (attenuated) microbe, a killed microbe, a toxoid (harmless toxin) or a single antigen
• Primary response: slow start, small amount of antibody, short-lived. Secondary response: quick, much more antibody, long-lasting
• Allergy = an excessive response to a harmless antigen (allergen); mast cells release histamine
• Autoimmune disease = the immune system attacks the body's own tissues, e.g. rheumatoid arthritis, type 1 diabetes, multiple sclerosis
• Edward Jenner introduced the first vaccine, against smallpox, in 1796

Biology · Respiration

Respiratory system and breathing

• Inspiration (active): diaphragm contracts and flattens; external intercostal muscles contract, ribs move up and out; volume rises, pressure falls, air enters
• Quiet expiration (passive): diaphragm and external intercostals relax; elastic recoil of the lungs; volume falls, pressure rises, air leaves
• Forced expiration is active: internal intercostal muscles and abdominal muscles contract
• Trachea and bronchi have cartilage; bronchioles have none, only smooth muscle. Goblet cells make mucus and cilia sweep it up to the pharynx
• Tidal volume = air breathed in or out in one quiet breath, about 500 cm³ (0.5 dm³)
• Vital capacity = tidal volume + inspiratory reserve volume + expiratory reserve volume
• Total lung capacity = vital capacity + residual volume; minute volume = tidal volume × breathing rate

Gas transport and disorders

• Hb + 4O₂ ⇌ Hb(O₂)₄ (oxyhaemoglobin); the oxygen dissociation curve is S-shaped (sigmoid)
• Bohr effect: more CO₂ or lower pH shifts the curve to the right (higher temperature does the same), so haemoglobin gives up more oxygen
• CO₂ transport: about 70% as hydrogencarbonate ions, most of the rest as carbaminohaemoglobin, a small amount dissolved in plasma
• In red cells: CO₂ + H₂O ⇌ H₂CO₃ ⇌ H⁺ + HCO₃⁻, speeded up by carbonic anhydrase
• Chloride shift: HCO₃⁻ diffuses out of the red cell and Cl⁻ moves in to keep the charges balanced
• Smoke: tar causes cancer and damages cilia; nicotine is addictive and raises heart rate and blood pressure; carbon monoxide cuts oxygen carriage
• Emphysema = alveolar walls break down, less surface; pneumonia = alveoli fill with fluid; tuberculosis = Mycobacterium tuberculosis, prevented by BCG vaccine

Biology · Digestion

Alimentary canal and enzymes

• Salivary amylase (salivary glands) and pancreatic amylase (pancreas): starch → maltose
• Pepsin (gastric glands; secreted as pepsinogen and activated by HCl): proteins → shorter polypeptides
• Trypsin (pancreas; secreted as trypsinogen and activated by enterokinase): proteins and polypeptides → smaller peptides
• Lipase (mainly pancreas): fats → fatty acids + glycerol
• Intestinal wall: maltase gives glucose; sucrase gives glucose + fructose; lactase gives glucose + galactose; peptidases give amino acids
• HCl kills microbes and activates pepsinogen; mucus protects the stomach wall; gastrin stimulates the release of gastric juice
• Bile is made in the liver and stored in the gall bladder; the colon absorbs water and its bacteria make vitamin K

Absorption, liver and disorders

• Glucose and amino acids: absorbed by diffusion and active transport (cells have many mitochondria) → blood capillaries → hepatic portal vein
• Fatty acids and glycerol → reformed into fats → lacteal → lymph; fat-soluble vitamins A, D, E and K are absorbed with fats and need bile
• Liver: makes bile, stores glucose as glycogen, deaminates excess amino acids and makes urea, detoxifies poisons, makes plasma proteins (albumin, fibrinogen, prothrombin), stores iron and vitamins
• Pancreas: exocrine part sends pancreatic juice into a duct; endocrine part (islets of Langerhans) sends insulin and glucagon into the blood
• Peptic ulcer = a sore where acid and pepsin erode the wall of the stomach or duodenum; often linked with Helicobacter pylori
• Dyspepsia = indigestion, with discomfort in the upper abdomen
• Food poisoning = illness from food carrying bacteria or their toxins, e.g. Salmonella, Campylobacter

Biology · Homeostasis

Osmoregulation and the kidney

• The afferent arteriole is wider than the efferent arteriole, so the blood pressure in the glomerulus is high
• Filtrate contains water, glucose, amino acids, salts and urea; blood cells and plasma proteins are too large to pass
• Proximal convoluted tubule: reabsorbs all the glucose and amino acids and most of the water and salts
• Descending limb of the loop is permeable to water; ascending limb moves salts out and is impermeable to water
• Less water in the blood → more ADH from the posterior pituitary → collecting ducts more permeable to water → small volume of concentrated urine
• Too little ADH = diabetes insipidus: large volumes of dilute urine with no glucose
• Kidney stones are mostly calcium oxalate (lithotripsy breaks them with shock waves); in kidney failure, dialysis removes urea by diffusion across a partially permeable membrane

Thermoregulation and excretion

• Too hot: vasodilation of skin arterioles, sweating (cooling by evaporation), hairs lie flat, lower metabolic rate
• Too cold: vasoconstriction of skin arterioles, shivering (heat from skeletal muscles), hairs raised to trap air, more thyroxine and adrenaline raise the metabolic rate
• Fever: pyrogens raise the set point of the hypothalamus
• Ammonia: very toxic, very soluble, needs most water; aquatic animals such as bony fishes (ammonotelic)
• Urea: less toxic, soluble, needs moderate water; mammals and adult amphibians (ureotelic)
• Uric acid: least toxic, almost insoluble, needs least water; birds, reptiles and insects (uricotelic)
• Urea cycle: ornithine → citrulline → arginine; arginase splits arginine into urea + ornithine. Overall: 2NH₃ + CO₂ → CO(NH₂)₂ + H₂O

Biology · Biotechnology

Recombinant DNA techniques

• Restriction enzymes (restriction endonucleases) cut DNA at specific palindromic sites, e.g. EcoRI at 5′-GAATTC-3′, often leaving sticky ends
• DNA ligase seals the sugar-phosphate backbone and joins the gene to the vector
• Cloning steps: isolate the gene → cut gene and plasmid with the same enzyme → join with ligase → transform host cells → select (antibiotic resistance marker) → grow
• One PCR cycle: denaturation at about 94 °C → primer annealing at about 50 to 60 °C → extension by Taq polymerase at about 72 °C
• Number of DNA molecules after n PCR cycles = starting number × 2ⁿ
• Electrophoresis: DNA is negatively charged and moves towards the positive electrode; smaller fragments travel further
• Sanger sequencing uses dideoxynucleotides, which lack a 3′-OH group and so stop the growing chain

Applications of biotechnology

• Human insulin from recombinant bacteria treats diabetes mellitus; recombinant growth hormone treats pituitary dwarfism
• Hepatitis B vaccine is a viral surface protein made in yeast; it has no whole virus, so it cannot cause the disease
• Gene therapy of body (somatic) cells is not inherited, because the eggs and sperm still carry the faulty allele
• Bt crops carry a gene from Bacillus thuringiensis for a protein that kills insect larvae; golden rice makes beta-carotene, which the body turns into vitamin A
• DNA fingerprinting compares repeated non-coding sequences (VNTRs or STRs); only identical twins share a profile, and every band in a child comes from the mother or the father
• Reproductive cloning by nuclear transfer (Dolly, 1996): the clone is genetically identical to the animal that gave the nucleus
• Ethical issues include gene flow to wild plants, long-term safety of GM food, privacy of genetic data and the cloning of humans

Chemistry · Fundamental concepts of chemistry

The mole

• moles = mass ÷ molar mass (n = m/M)
• number of particles = moles × 6.02 × 10²³
• volume of a gas at STP = moles × 22.4 dm³
• 1 amu = one-twelfth of the mass of one carbon-12 atom
• % of an element = (mass of that element in 1 mol ÷ molar mass) × 100
• Empirical formula: divide each % by the atomic mass, then divide every answer by the smallest one
• molecular formula = (empirical formula)_n, where n = molar mass ÷ empirical formula mass

Stoichiometry

• Coefficients in a balanced equation = ratio of moles
• Route: mass → moles → mole ratio → moles → mass or gas volume
• For gases at the same temperature and pressure, the volume ratio = the mole ratio
• Limiting reagent: divide the moles of each reactant by its coefficient; the smallest answer shows the limiting reagent
• Theoretical yield is calculated from the limiting reagent only
• percentage yield = (actual yield ÷ theoretical yield) × 100
• Mass is conserved: total mass of reactants used = total mass of products formed

Chemistry · Atomic structure

Particles, photons and the Bohr model

• neutrons = A − Z; electrons = Z − charge on the ion (so a negative ion has extra electrons)
• relative atomic mass = sum of (isotope mass × % abundance) ÷ 100
• Charge-to-mass ratio (e/m): electron > proton > alpha particle; a neutron has no charge and is not deflected
• energy of a photon = Planck's constant × frequency (E = hf = hc/λ); wave number = 1/λ
• Electron jump: ΔE = E₂ − E₁ = hf
• Hydrogen series: Lyman (to n = 1) ultraviolet; Balmer (to n = 2) visible; Paschen, Brackett and Pfund infrared
• Bohr radius of orbit n = 0.529 Å × n²; 1 Å = 10⁻¹⁰ m = 100 pm; 1 nm = 10 Å

Orbitals and electronic configuration

• n (principal) = 1, 2, 3 … gives the shell, its size and energy; l (azimuthal) = 0 to n − 1 gives the shape: s = 0, p = 1, d = 2, f = 3
• m (magnetic) = −l to +l gives the orientation; s (spin) = +½ or −½
• A shell has n² orbitals and holds up to 2n² electrons
• Aufbau: fill the subshell with the lower (n + l) first; if equal, the lower n first. Order: 1s 2s 2p 3s 3p 4s 3d 4p 5s
• Pauli: no two electrons in an atom have all four quantum numbers the same, so two electrons in one orbital have opposite spins
• Hund: orbitals of equal energy are each filled singly, with parallel spins, before any pairing
• Exceptions: Cr is [Ar] 3d⁵ 4s¹ and Cu is [Ar] 3d¹⁰ 4s¹

Chemistry · Gases

Gas laws

• Boyle's law: at constant T, P₁V₁ = P₂V₂
• Charles's law: at constant P, V₁/T₁ = V₂/T₂ (T in K); the V against T line reaches zero volume at −273 °C
• Avogadro's law: at the same T and P, equal volumes of gases contain equal numbers of molecules (V ∝ n)
• PV = nRT; R = 0.0821 dm³ atm K⁻¹ mol⁻¹ = 8.314 J K⁻¹ mol⁻¹; density d = PM/RT
• Dalton's law: total pressure = sum of the partial pressures; partial pressure = mole fraction × total pressure
• Graham's law: rate₁/rate₂ = √(M₂/M₁)
• K = °C + 273; °F = (9/5 × °C) + 32; 1 atm = 760 torr = 101 325 Pa; 1 cal = 4.18 J

Kinetic theory and real gases

• Postulates: the volume of the molecules is negligible; there are no forces between them; collisions are perfectly elastic; motion is random
• average kinetic energy ∝ absolute temperature (in K), and it is the same for all gases at the same temperature
• root mean square speed = √(3RT/M), so lighter molecules move faster at the same temperature
• Real gases are most ideal at low pressure and high temperature; they deviate most at high pressure and low temperature
• van der Waals equation: (P + n²a/V²)(V − nb) = nRT; a corrects for attractions, b for the volume of the molecules
• PV/nRT < 1: attractions dominate; PV/nRT > 1: the volume of the molecules dominates (very high pressure)
• Plasma examples: the Sun and stars, lightning, neon signs, fluorescent tubes

Chemistry · Liquids

Intermolecular forces

• London forces: the only force between non-polar molecules (Br₂, CH₄, noble gases); stronger with more electrons and a longer, less branched shape
• Dipole–dipole forces: between polar molecules such as HCl, propanone and ethanal
• Hydrogen bond: H joined to N, O or F is attracted to a lone pair on N, O or F of another molecule
• For molecules of similar size: hydrogen bond > dipole–dipole > London; all are far weaker than a covalent bond
• Hydrogen-bond strength: F–H···F > O–H···O > N–H···N; water still boils higher than HF because each molecule can form more hydrogen bonds
• Ice: each H₂O is hydrogen bonded to 4 others in an open tetrahedral structure, so ice is less dense than water; water is densest at 4 °C
• Carboxylic acids form hydrogen-bonded dimers; hydrogen bonds hold the two strands of DNA together

Properties of liquids

• Vapour pressure rises with temperature along a curve; it does not depend on the amount of liquid, the surface area or the size of the container
• Boiling point: the temperature at which vapour pressure = external pressure (100 °C for water at 760 torr); lower pressure gives a lower boiling point
• Stronger intermolecular forces: lower vapour pressure, higher boiling point, larger heat of vaporisation
• Evaporation is faster at a higher temperature, with a larger surface area and with weaker intermolecular forces
• heat for a change of state = moles × molar heat of fusion (or of vaporisation); melting, boiling and sublimation absorb heat; freezing and condensation release it
• Molar heat of vaporisation is larger than molar heat of fusion for the same substance
• Surface tension pulls a drop into a sphere; water rises in a glass capillary (adhesion > cohesion), mercury is pushed down (cohesion > adhesion)

Chemistry · Solids

Crystalline solids

• Unit cell: 6 parameters (edges a, b, c and angles α, β, γ); 7 crystal systems, e.g. cubic a = b = c, α = β = γ = 90°
• Ionic (NaCl): hard, brittle, high melting point, conducts only when molten or dissolved; in NaCl each ion has 6 nearest neighbours of the other kind
• Covalent network (diamond, SiO₂): very hard, very high melting point. Molecular (ice, I₂, dry ice): soft, low melting point, non-conductor
• Metallic: positive ions in a sea of free electrons; conducts as a solid, malleable and ductile
• Lattice energy is larger for smaller ions and for ions of higher charge
• Isomorphism: different substances, same crystal form (NaF and MgO, both cubic). Polymorphism: one compound, more than one crystal form. Allotropy: the same for an element (diamond and graphite)
• Radius ratio r⁺/r⁻: 0.225 to 0.414 gives coordination number 4; 0.414 to 0.732 gives 6; 0.732 to 1.0 gives 8

Chemistry · Chemical equilibrium

Equilibrium and Le Chatelier's principle

• For aA + bB ⇌ cC + dD: K_c = [C]^c[D]^d ÷ [A]^a[B]^b; K_p uses partial pressures in the same way; pure solids and liquids are left out
• K_p = K_c(RT)^Δn, where Δn = moles of gaseous products − moles of gaseous reactants; K_p = K_c when Δn = 0
• Reaction quotient Q: if Q < K the reaction moves forward; if Q > K it moves backward
• Adding a reactant or removing a product shifts the equilibrium forward; K does not change
• Higher pressure shifts towards the side with fewer moles of gas; higher temperature shifts in the endothermic direction
• A catalyst speeds up both directions equally: equilibrium is reached sooner, with no change in position or in K
• Haber process: N₂ + 3H₂ ⇌ 2NH₃, ΔH = −92 kJ mol⁻¹; high pressure (about 200 atm), about 400 to 450 °C as a compromise, iron catalyst

Ionic equilibria

• Arrhenius: an acid gives H⁺ in water, a base gives OH⁻. Brønsted–Lowry: an acid donates a proton, a base accepts one. Lewis: an acid accepts an electron pair, a base donates one
• pH = −log[H⁺]; pOH = −log[OH⁻]; pH + pOH = pK_w = 14 at 25 °C (K_w = 1.0 × 10⁻¹⁴ mol² dm⁻⁶)
• K_a = [H⁺][A⁻] ÷ [HA]; pK_a = −log K_a; a larger K_a (smaller pK_a) means a stronger acid
• Weak acid of concentration c: [H⁺] = √(K_a × c)
• K_sp for A_xB_y = [A]^x[B]^y; with solubility s: AB gives s², AB₂ or A₂B gives 4s³
• A precipitate forms only when the ionic product is greater than K_sp
• Buffer: pH = pK_a + log([salt] ÷ [acid]); blood is kept at pH 7.35 to 7.45, mainly by H₂CO₃ / HCO₃⁻

Chemistry · Reaction kinetics

Rate and rate law

• rate = change in concentration ÷ time taken; unit mol dm⁻³ s⁻¹
• For aA → bB: rate = −(1/a) × Δ[A]/Δt = (1/b) × Δ[B]/Δt
• Rate law: rate = k[A]^x[B]^y; overall order = x + y; k is the rate when every concentration is 1 mol dm⁻³ and changes only with temperature
• Units of k: zero order mol dm⁻³ s⁻¹; first order s⁻¹; second order dm³ mol⁻¹ s⁻¹
• Half-life: first order t_½ = 0.693 ÷ k, constant; zero order t_½ ∝ starting concentration; second order t_½ ∝ 1 ÷ starting concentration
• After n half-lives of a first-order reaction, the fraction left = (½)ⁿ
• Methods: colour (spectrometry), electrical conductivity, optical rotation, volume change (dilatometry), or titration of samples; a reactant in large excess gives a pseudo-first-order reaction

Activation energy and catalysis

• ΔH = E_a(forward) − E_a(reverse). Exothermic: the forward barrier is the smaller one. Endothermic: the forward barrier is the larger one.
• Higher concentration (or gas pressure) and larger surface area: more collisions per second, so a faster rate.
• Higher temperature: a much larger fraction of molecules have energy of at least E_a. The rate roughly doubles for each 10 °C rise.
• Lower E_a means a faster reaction at the same temperature. ΔH does not decide the rate.
• Arrhenius equation: k = Ae^−E_a/RT. A graph of ln k against 1/T is a straight line of slope −E_a/R.
• Homogeneous catalyst: same phase as the reactants. Heterogeneous catalyst: different phase. Autocatalysis: a product catalyses the reaction (Mn²⁺ in the KMnO₄ and oxalic acid reaction).
• An inhibitor (negative catalyst) slows a reaction. A catalyst never changes ΔH or the position of equilibrium.

Chemistry · Thermochemistry and energetics

Enthalpy changes

• Formation, ΔH°_f: one mole of a compound is formed from its elements in their standard states. For an element in its standard state ΔH°_f = 0.
• Combustion, ΔH°_c: one mole of a substance burns completely in excess oxygen. It is always negative.
• Atomisation: one mole of gaseous atoms is formed from the element. It is always positive.
• Neutralisation: one mole of water is formed from an acid and a base; about −57 kJ mol⁻¹ for any strong acid with a strong base.
• ΔH = (sum of bond energies of bonds broken) − (sum of bond energies of bonds formed).
• Heat = mass × specific heat capacity × temperature change (q = mcΔT); for a whole calorimeter q = CΔT with C in J K⁻¹. Then ΔH = −q ÷ moles, in kJ mol⁻¹.
• Food energy: fat gives about 38 kJ g⁻¹, carbohydrate and protein about 17 kJ g⁻¹ each.

First law and Hess's law

• ΔE = q + w. q is positive when the system absorbs heat; w is positive when work is done on the system.
• Work of a gas at constant pressure: w = −PΔV. Expansion gives a negative w. 1 kPa × 1 dm³ = 1 J.
• At constant volume no work is done, so q_v = ΔE. At constant pressure q_p = ΔH, where H = E + PV and ΔH = ΔE + PΔV.
• Hess's law: if an equation is reversed, change the sign of ΔH; if it is multiplied, multiply ΔH by the same number; then add.
• ΔH° = ΣΔH°_f(products) − ΣΔH°_f(reactants), each multiplied by its coefficient in the equation.
• Born–Haber cycle: ΔH°_f = atomisation of the metal + ionisation energy + atomisation of the non-metal + electron affinity + lattice energy.
• Lattice energy is the energy released when one mole of an ionic solid forms from its gaseous ions, so it is negative. Atomisation and ionisation are endothermic.

Chemistry · Electrochemistry

Oxidation numbers and redox

• A free element is 0. A simple ion equals its charge. The numbers add up to 0 in a compound and to the charge in a polyatomic ion.
• Group 1 metals are +1, group 2 metals +2, fluorine always −1. Hydrogen is +1, but −1 in metal hydrides such as NaH.
• Oxygen is −2, except −1 in peroxides (H₂O₂), −½ in superoxides (KO₂) and +2 in OF₂.
• Balancing in acid: balance the main atoms, add H₂O for oxygen, H⁺ for hydrogen and electrons for charge; then multiply the half-equations so the electrons cancel.
• An element in its highest state can only be an oxidising agent (Mn in MnO₄⁻); in its lowest state only a reducing agent (I⁻); in between it can be both (H₂O₂, SO₂).
• Disproportionation: the same element is oxidised and reduced in one reaction.

Electrode potentials and cells

• SHE: hydrogen gas at 1 atm bubbles over platinum coated with platinum black, in 1 M H⁺ at 298 K. Its E° is 0.00 V by convention.
• Standard conditions: ion concentration 1 M, temperature 298 K, gases at 1 atm.
• E°_cell = E°(cathode) − E°(anode), using reduction potentials. A positive value means the reaction is feasible.
• In both kinds of cell, oxidation is at the anode and reduction is at the cathode. Galvanic: anode is negative. Electrolytic: anode is positive.
• A metal with a negative E° displaces hydrogen from dilute acid, and a metal displaces from solution the ions of any metal with a more positive E°.
• The salt bridge (|| in cell notation) lets ions move between the half-cells, keeping the solutions neutral and completing the circuit.
• Lead accumulator: anode Pb, cathode PbO₂, electrolyte dilute H₂SO₄. Electrolysis is used to refine copper, extract aluminium and electroplate.

Chemistry · Chemical bonding

Ionic and covalent bonding

• An ionic bond forms most easily between a metal of low ionisation energy and a non-metal that releases much energy on gaining an electron.
• Coordinate bond: the donor needs a lone pair and the acceptor an empty orbital, as in NH₄⁺, H₃O⁺ and Al₂Cl₆.
• Electronegativity difference: 0 is non-polar covalent; up to about 1.7 is polar covalent; above 1.7 is mainly ionic.
• Dipole moment = charge × distance between the charges (μ = q × r), unit debye (D). In symmetrical molecules (CO₂, BF₃, CH₄, CCl₄) the bond dipoles cancel, so μ = 0.
• Bond energy: the energy needed to break one mole of a bond in gaseous molecules, in kJ mol⁻¹.
• More shared pairs give a shorter, stronger bond: C–C is longer and weaker than C=C, which is longer and weaker than C≡C. Bond length ≈ sum of the two covalent radii.
• Fajans' rules: a small, highly charged cation with a large anion gives more covalent character (AlCl₃ > MgCl₂ > NaCl).

Shapes of molecules

• Repulsion: lone pair–lone pair > lone pair–bond pair > bond pair–bond pair.
• 2 regions: linear, 180°, sp (BeCl₂, CO₂). 3 regions: trigonal planar, 120°, sp² (BF₃). 4 regions: tetrahedral, 109.5°, sp³ (CH₄).
• 5 bond pairs: trigonal bipyramidal, 90° and 120° (PCl₅). 6 bond pairs: octahedral, 90° (SF₆).
• Four pairs with lone pairs: NH₃ (one lone pair) is trigonal pyramidal, 107.5°; H₂O (two lone pairs) is bent, 104.5°.
• s-character: sp 50%, sp² 33%, sp³ 25%.
• Single bond = 1σ; double bond = 1σ + 1π; triple bond = 1σ + 2π. A σ bond is stronger because head-on overlap is greater; π electron density lies above and below the line joining the nuclei.
• Molecular orbital theory: O₂ has two unpaired electrons in antibonding π orbitals, so it is paramagnetic, with bond order 2.

Chemistry · s and p block elements

Periodic trends and the s block

• Across a period: atomic radius decreases; ionisation energy, electronegativity and the energy released on gaining an electron generally increase. Down a group: the opposite.
• Exceptions: first ionisation energy Al < Mg (3p electron, shielded by 3s) and S < P (paired 3p electron is repelled). Cl releases more energy than F on gaining an electron. Adding a second electron (O⁻ → O²⁻) absorbs energy.
• Period 3 melting points: rise Na < Mg < Al (more delocalised electrons), highest at Si (giant covalent), then low for the small molecules: S₈ > P₄ > Cl₂ > Ar.
• A positive ion is smaller than its atom and a negative ion is larger. For ions with the same number of electrons, more protons means a smaller ion.
• Burning in excess oxygen: Li gives Li₂O, Na gives Na₂O₂ (peroxide), K gives KO₂ (superoxide). Only Li in group 1 forms a nitride, Li₃N.
• Down group 2: hydroxides become more soluble and more basic; sulphates become less soluble (BaSO₄ least); carbonates, nitrates and hydroxides need stronger heating to decompose. BeO is amphoteric.
• Flame colours: Li crimson, Na golden yellow, K lilac, Ca brick red, Sr crimson, Ba apple green. Be and Mg give no colour.

The p block

• F₂ pale yellow gas, Cl₂ greenish-yellow gas, Br₂ red-brown liquid, I₂ grey-black solid. Oxidising power: F₂ > Cl₂ > Br₂ > I₂.
• A halogen displaces a less reactive halogen from its halide: Cl₂ + 2KBr → 2KCl + Br₂.
• Acid strength: HF < HCl < HBr < HI (the H–X bond gets weaker), and HOCl < HClO₂ < HClO₃ < HClO₄ (more oxygen atoms).
• Cold dilute alkali: Cl₂ + 2NaOH → NaCl + NaOCl + H₂O. Hot concentrated alkali: 3Cl₂ + 6NaOH → 5NaCl + NaClO₃ + 3H₂O.
• Period 3 oxides: Na₂O and MgO basic; Al₂O₃ amphoteric; SiO₂, P₄O₁₀, SO₃ and Cl₂O₇ acidic. Chlorides: NaCl neutral; AlCl₃, SiCl₄ and PCl₅ are hydrolysed, giving acidic solutions with HCl.
• Concentrated HNO₃ turns yellow in light: 4HNO₃ → 4NO₂ + 2H₂O + O₂. Concentrated H₂SO₄ (oil of vitriol) is a dehydrating and oxidising agent. Blue vitriol is CuSO₄·5H₂O.
• Aluminium: bauxite (Al₂O₃·2H₂O) is purified to alumina, dissolved in molten cryolite (Na₃AlF₆) to lower the working temperature, and electrolysed.

Chemistry · Transition elements

Properties of transition elements

• General configuration: (n−1)d^1–10 ns^1–2. Exceptions: Cr is [Ar] 3d⁵ 4s¹ and Cu is [Ar] 3d¹⁰ 4s¹, because half-filled and full d subshells are extra stable.
• Zn and Zn²⁺ both have a full 3d¹⁰ subshell, so zinc is not a typical transition element: its ion is colourless and diamagnetic.
• Highest oxidation state from Sc to Mn = number of 4s + 3d electrons: Cr +6, Mn +7.
• More unpaired electrons means stronger paramagnetism: Mn²⁺ and Fe³⁺ (3d⁵) have five. Ions with no unpaired electron (Sc³⁺, Cu⁺, Zn²⁺) are diamagnetic and colourless.
• Catalysts: Fe in the Haber process, V₂O₅ in the Contact process, Ni for hydrogenation of oils.
• Brass = Cu + Zn; bronze = Cu + Sn. Interstitial compounds (C in Fe) are harder and stronger than the pure metal. Binding energy rises to the middle of the 3d series, then falls to its lowest at Zn.
• 2CrO₄²⁻ (yellow) + 2H⁺ ⇌ Cr₂O₇²⁻ (orange) + H₂O. Acid gives orange, alkali gives yellow. Cr is +6 in both, so it is not redox.

Complexes

• Monodentate ligands (one donor atom): H₂O, NH₃, Cl⁻, CN⁻, OH⁻. Bidentate (two): ethylenediamine (en), oxalate C₂O₄²⁻.
• Coordination number = number of donor atoms bonded to the metal, not the number of ligands: [Co(en)₃]³⁺ has 6.
• Oxidation number of the metal = charge on the complex ion − total charge of the ligands.
• Naming: cation before anion; ligands in alphabetical order with di, tri, tetra, hexa (aqua, ammine, chloro, cyano, hydroxo); then the metal with its oxidation state in Roman numerals. In a complex anion the metal ends in -ate (ferrate, cuprate).
• Shapes: coordination number 2 linear; 4 tetrahedral (sp³) or square planar (dsp²); 6 octahedral (d²sp³ or sp³d²).
• In an octahedral complex the five d orbitals split into three of lower energy and two of higher energy. Ions with d⁰ or d¹⁰ are colourless.
• Stability is greater for a smaller, more highly charged metal ion (Co³⁺ > Co²⁺) and for chelating ligands; for 3d ions Mn²⁺ < Fe²⁺ < Co²⁺ < Ni²⁺ < Cu²⁺.

Chemistry · Fundamental principles of organic chemistry

Classification and isomerism

• Skeleton: open chain (aliphatic) or cyclic. Cyclic compounds are alicyclic (cyclohexane), aromatic (benzene, naphthalene) or heterocyclic, with N, O or S in the ring (pyridine, furan, thiophene).
• Functional groups: alcohol –OH, aldehyde –CHO, ketone >C=O, carboxylic acid –COOH, ester –COOR, ether R–O–R, amine –NH₂, thiol –SH.
• Members of a homologous series differ by CH₂. A carbon atom is primary, secondary, tertiary or quaternary when it is bonded to 1, 2, 3 or 4 other carbon atoms.
• Structural isomerism: chain, position, functional group (ethanol and dimethyl ether), metamerism (different alkyl groups on each side of the same group) and tautomerism (shift of an H atom, keto and enol forms).
• Cis-trans isomerism needs a C=C bond (no free rotation) with two different groups on each of its carbon atoms.
• Optical isomerism needs a chiral carbon: one bonded to four different groups. The two mirror-image forms rotate plane-polarised light in opposite directions.
• Fullerene C₆₀: a hollow cage of 20 hexagons and 12 pentagons; each carbon is bonded to three others. It is a molecular form (allotrope) of carbon.

Bonding and reactivity in carbon compounds

• sp³ carbon: 4 σ bonds, tetrahedral, 109.5° (methane, ethane)
• sp² carbon: 3 σ bonds + 1 π bond, trigonal planar, 120° (ethene, benzene, C=O)
• sp carbon: 2 σ bonds + 2 π bonds, linear, 180° (ethyne, the carbon of C≡N)
• Homolytic fission: A–B → A• + B•. It is favoured by non-polar bonds, ultraviolet light or heat
• Heterolytic fission: A–B → A⁺ + B⁻. It is favoured by polar bonds and polar solvents; the more electronegative atom takes both electrons
• Electrophiles are positive or electron-deficient: H⁺, NO₂⁺, Br⁺, AlCl₃, BF₃, SO₃
• Nucleophiles have a lone pair to give: OH⁻, CN⁻, Cl⁻, NH₃, H₂O

Chemistry · Hydrocarbons

Alkanes and alkenes

• Initiation: Cl₂ → 2Cl• (light). Propagation: Cl• + CH₄ → •CH₃ + HCl, then •CH₃ + Cl₂ → CH₃Cl + Cl•. Termination: two radicals join
• Reactivity of halogens with alkanes: F₂ > Cl₂ > Br₂ > I₂
• Markovnikov's rule: when HX adds to an unsymmetrical alkene, H goes to the carbon that already has more H atoms, because the more stable carbocation forms
• Reactivity of HX with alkenes: HI > HBr > HCl (the weakest H–X bond breaks most easily)
• Tests for C=C: bromine turns from red-brown to colourless; cold, dilute, alkaline KMnO₄ (Baeyer's reagent) loses its purple colour and a diol forms
• Alkenes are made by dehydrating alcohols (concentrated H₂SO₄ at about 170 °C, or hot Al₂O₃), by heating alkyl halides with alcoholic KOH, or from vicinal dihalides with zinc dust
• More branching gives a lower boiling point; a cycloalkane boils higher than the straight-chain alkane with the same number of carbon atoms

Alkynes and benzene

• Terminal alkynes give a white precipitate with ammoniacal AgNO₃ and a reddish-brown precipitate with ammoniacal Cu₂Cl₂; they also react with Na or NaNH₂
• Hydration of an alkyne (dilute H₂SO₄, HgSO₄): ethyne gives ethanal, all other alkynes give ketones
• Benzene: all C–C bonds 139 pm (between C–C and C=C), bond angle 120°, resonance energy about 150 kJ mol⁻¹
• Nitration: concentrated HNO₃ + concentrated H₂SO₄, electrophile NO₂⁺. Sulphonation: fuming H₂SO₄, electrophile SO₃
• Halogenation: Cl₂ or Br₂ with FeCl₃, FeBr₃ or AlCl₃. Friedel–Crafts: RCl or RCOCl with anhydrous AlCl₃ (a Lewis acid that makes R⁺ or RCO⁺)
• –CH₃ and –OH activate the ring and direct to positions 2 and 4; –NO₂ deactivates the ring and directs to position 3
• Faraday discovered benzene in 1825; Kekulé proposed the ring structure in 1865. Toluene = methylbenzene, cumene = (1-methylethyl)benzene

Chemistry · Alkyl halides

Alkyl halides

• Reactivity: R–I > R–Br > R–Cl > R–F (C–F is the strongest bond)
• S_N2: one step, rate = k[RX][Nu⁻], attack from the back, inversion of configuration. Order: primary > secondary > tertiary
• S_N1: two steps, slow step forms a carbocation, rate = k[RX], product is mostly racemic. Order: tertiary > secondary > primary
• E2: one step, rate depends on both halide and base. E1: two steps through a carbocation, rate depends on the halide only
• Aqueous KOH favours substitution (alcohol); hot concentrated alcoholic KOH favours elimination (alkene). Tertiary halides eliminate most easily
• R–X + Mg in dry ether → R–Mg–X (Grignard reagent); with water it gives the alkane R–H. Wurtz: 2R–X + 2Na → R–R + 2NaX
• Boiling points: R–I > R–Br > R–Cl for the same R, and branching lowers them. Freons (CFCs) were refrigerants and aerosol propellants; their Cl atoms destroy ozone

Chemistry · Alcohols and phenols

Alcohols

• Oxidation with acidified K₂Cr₂O₇ (orange to green): primary → aldehyde → carboxylic acid; secondary → ketone; tertiary → no oxidation
• Lucas reagent (concentrated HCl + anhydrous ZnCl₂): tertiary turns cloudy at once, secondary in 5 to 10 minutes, primary stays clear at room temperature
• Concentrated H₂SO₄: about 170 °C with excess acid gives an alkene; about 140 °C with excess alcohol gives an ether
• Esterification: RCOOH + R′OH ⇌ RCOOR′ + H₂O (concentrated H₂SO₄). The acid loses –OH and the alcohol loses H
• 2ROH + 2Na → 2RONa + H₂. ROH + SOCl₂ → RCl + SO₂ + HCl. ROH + PCl₅ → RCl + POCl₃ + HCl
• Iodoform test (I₂ + NaOH, yellow precipitate of CHI₃): positive for CH₃CH(OH)–, so ethanol and propan-2-ol react but methanol does not
• Made by: hydration of alkenes, hydrolysis of alkyl halides with aqueous NaOH, Grignard reagent + carbonyl compound, reduction of aldehydes, ketones and acids, fermentation (ethanol)

Phenols and ethers

• Acid strength: carboxylic acid > phenol > water > alcohol. K_a of phenol is about 1 × 10⁻¹⁰
• Phenol reacts with NaOH to give sodium phenoxide and water, but it gives no CO₂ with Na₂CO₃ or NaHCO₃
• Electron-withdrawing groups (–NO₂) make a phenol more acidic; electron-donating groups (–CH₃) make it less acidic
• Bromine water at room temperature: white precipitate of 2,4,6-tribromophenol. Dilute HNO₃: 2- and 4-nitrophenol. Concentrated HNO₃ + H₂SO₄: 2,4,6-trinitrophenol (picric acid)
• Phenol + zinc dust (heat) → benzene + ZnO. Phenol + methanal → Bakelite. Phenol + CH₃COCl → phenyl ethanoate + HCl
• Phenol (carbolic acid): colourless crystals, melting point 41 °C, absorbs moisture from air, turns pink in air by oxidation. Catechol, resorcinol and quinol are benzene-1,2-, 1,3- and 1,4-diol
• Ethers are named alkoxyalkanes, with the smaller group as alkoxy: CH₃OC₂H₅ is methoxyethane. Oxygen is sp³, so C–O–C is bent, about 110°. Williamson: RONa + R′X → ROR′ + NaX

Chemistry · Aldehydes and ketones

Aldehydes and ketones

• General formula C_nH_2nO. An aldehyde and a ketone with the same formula are functional group isomers (propanal and propanone)
• 2,4-DNPH: yellow or orange precipitate with both aldehydes and ketones (test for C=O)
• Tollens' reagent: silver mirror with aldehydes only. Fehling's solution: brick-red Cu₂O with aliphatic aldehydes only (not benzaldehyde, not ketones)
• Iodoform test (I₂ + NaOH, yellow CHI₃): positive for CH₃CO–, so ethanal and methyl ketones. Alkaline sodium nitroprusside: red colour with ketones
• HCN (with a little NaCN) adds to give a hydroxynitrile (cyanohydrin); the attacking species is CN⁻. Saturated NaHSO₃ gives white crystals with aldehydes and small methyl ketones
• Reduction (NaBH₄, LiAlH₄ or H₂/Ni): aldehyde → primary alcohol, ketone → secondary alcohol. Zn/Hg + concentrated HCl (Clemmensen) changes C=O to CH₂
• Dilute NaOH: aldol condensation if there is an α-hydrogen. Concentrated NaOH: Cannizzaro reaction if there is no α-hydrogen (methanal, benzaldehyde)

Chemistry · Carboxylic acids

Carboxylic acids and derivatives

• Acid strength: ClCH₂COOH > HCOOH > C₆H₅COOH > CH₃COOH > CH₃CH₂COOH. A larger K_a (smaller pK_a) means a stronger acid
• Made by: oxidising primary alcohols or aldehydes; hydrolysing nitriles (RCN → RCOOH); Grignard reagent + CO₂, then dilute acid; oxidising the side chain of an alkylbenzene to benzoic acid
• Acids react with metals, alkalis and carbonates; CO₂ from Na₂CO₃ or NaHCO₃ is the test for –COOH
• RCOOH + PCl₅ → RCOCl + POCl₃ + HCl. RCOOH + SOCl₂ → RCOCl + SO₂ + HCl. 3RCOOH + PCl₃ → 3RCOCl + H₃PO₃
• Reactivity towards nucleophiles: acid chloride > anhydride > ester > amide
• Ester hydrolysis: with dilute acid it is reversible; with NaOH it goes to completion, giving the sodium salt of the acid and the alcohol
• LiAlH₄ reduces RCOOH to RCH₂OH. Decarboxylation: RCOONa + NaOH (soda lime, heat) → RH + Na₂CO₃

Chemistry · Macromolecules

Polymers

• Addition polymers: polyethene from CH₂=CH₂; PVC from CH₂=CHCl; polystyrene from C₆H₅CH=CH₂; Teflon (PTFE) from CF₂=CF₂; polyacrylonitrile from CH₂=CHCN
• To find the monomer of an addition polymer, take the two-carbon repeating unit and put the double bond back between the two chain carbons
• Nylon-6,6 (a polyamide): hexanedioic acid + hexane-1,6-diamine, joined by amide links –CO–NH–, water lost
• Terylene or Dacron (a polyester): benzene-1,4-dicarboxylic acid + ethane-1,2-diol, joined by ester links –CO–O–, water lost
• Bakelite: phenol + methanal, a condensation polymer with cross-links between chains
• Thermoplastics (polyethene, PVC, nylon) soften on heating and can be remoulded; thermosetting plastics (Bakelite) are cross-linked and cannot be remoulded
• Number of monomer units in an addition polymer chain = M_r of polymer ÷ M_r of monomer

Biological macromolecules

• Fibrous proteins are insoluble and structural (collagen, keratin). Globular proteins are folded and soluble (enzymes, haemoglobin, insulin, albumin)
• Simple proteins give only amino acids on hydrolysis; conjugated proteins also have a non-protein part (haemoglobin has haem)
• Primary structure = sequence of amino acids, held by peptide bonds. The α-helix is held by hydrogen bonds between C=O and N–H groups
• Competitive inhibitor: looks like the substrate, blocks the active site, overcome by adding more substrate. Non-competitive inhibitor: binds elsewhere, not overcome by more substrate
• Cofactors: metal ions or organic coenzymes such as NAD⁺. A zymogen is an inactive form of an enzyme (pepsinogen, trypsinogen) that becomes active when part of its chain is cut off
• Reducing sugars (glucose, fructose, maltose, lactose) reduce Fehling's solution; sucrose does not. Starch and cellulose are polymers of glucose
• Iodine number = grams of iodine taken up by 100 g of fat; a high value means many C=C bonds. A nucleotide = pentose sugar + phosphate group + nitrogenous base

Chemistry · Industrial chemistry

Industrial chemistry

• Urea: 2NH₃ + CO₂ → NH₂COONH₄ (ammonium carbamate), then NH₂COONH₄ → CO(NH₂)₂ + H₂O. Urea is about 46 to 47% nitrogen by mass (28 ÷ 60)
• Ammonium nitrate: NH₃ + HNO₃ → NH₄NO₃
• Single superphosphate: Ca₃(PO₄)₂ + 2H₂SO₄ → Ca(H₂PO₄)₂ + 2CaSO₄
• Triple superphosphate: Ca₃(PO₄)₂ + 4H₃PO₄ → 3Ca(H₂PO₄)₂ (no CaSO₄, so more phosphorus per kg)
• Cracking: a large alkane → a smaller alkane + an alkene, for example C₁₀H₂₂ → C₈H₁₈ + C₂H₄
• Octane number: 2,2,4-trimethylpentane (iso-octane) = 100, heptane = 0. Reforming changes straight chains into branched, ring and aromatic molecules to raise it
• Ethene from cracking gives polyethene, ethanol and ethane-1,2-diol

Physics · Vectors and equilibrium

Vectors and components

• Components of A at angle θ to the positive x-axis: A_x = A cos θ, A_y = A sin θ
• Adding by components: R_x = A_x + B_x + ..., R_y = A_y + B_y + ...
• magnitude of resultant: R = √(R_x² + R_y²)
• direction of resultant: tan θ = R_y/R_x, then pick the quadrant from the signs of R_x and R_y
• Two vectors A and B with angle θ between them: R = √(A² + B² + 2AB cos θ)
• Greatest resultant = A + B (same direction); least resultant = A − B (opposite directions)
• Weight mg on a slope at angle θ to the horizontal: mg sin θ down the slope, mg cos θ into the slope

Products of vectors and torque

• scalar product: A · B = AB cos θ = A_xB_x + A_yB_y + A_zB_z; A · A = A²
• vector product: size of A × B = AB sin θ, direction perpendicular to the plane of A and B (right-hand rule); B × A = −(A × B)
• Unit vectors: i · i = 1, i · j = 0, i × i = 0, i × j = k
• tan θ = (size of A × B) ÷ (A · B)
• torque = force × perpendicular distance from the pivot (τ = rF sin θ), unit N m
• torque of a couple = one force × perpendicular distance between the two forces
• Equilibrium: first condition ΣF = 0; second condition Στ = 0 (clockwise torques = anticlockwise torques)

Physics · Force and motion

Kinematics

• average velocity = total displacement ÷ total time; average speed = total distance ÷ total time
• v = u + at
• s = ut + ½at² and s = ½(u + v)t
• v² − u² = 2as
• From rest with uniform acceleration, distances in the 1st, 2nd, 3rd, 4th seconds are in the ratio 1 : 3 : 5 : 7
• Displacement–time graph: slope = velocity. Velocity–time graph: slope = acceleration, area = displacement
• Area below the time axis counts as negative for displacement but is added for distance

Newton's laws and momentum

• First law: no net force means no change in velocity; inertia depends on mass only
• Second law: net force = mass × acceleration (F = ma) = rate of change of momentum (F = Δp/Δt)
• Third law: action and reaction are equal, opposite and act on different bodies, so they never cancel
• momentum = mass × velocity (p = mv), unit kg m s⁻¹ or N s
• impulse = force × time = change in momentum = area under a force–time graph
• Conservation of momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ for an isolated system
• Elastic collision: kinetic energy also conserved, speed of approach = speed of separation. Inelastic collision: some kinetic energy is lost

Projectile motion

• Horizontal: velocity = v cos θ (constant), distance = v cos θ × t
• Vertical: starting velocity = v sin θ, acceleration = g downwards
• time of flight: T = 2v sin θ ÷ g
• maximum height: H = v² sin² θ ÷ 2g
• range on level ground: R = v² sin 2θ ÷ g; greatest at θ = 45°, where R_max = v²/g
• At the top: vertical velocity = 0, speed = v cos θ, acceleration = g, and velocity is at 90° to acceleration
• Air resistance reduces the range and the maximum height

Physics · Work, power and energy

Work and power

• work = force × displacement × cos θ (W = Fd cos θ), unit joule (J) = N m
• θ = 90°: work is zero. θ greater than 90°: work is negative
• Variable force: work = area under the force–displacement graph
• Spring: work to stretch from 0 to x = ½kx²; from x₁ to x₂ = ½k(x₂² − x₁²)
• power = work ÷ time = force × velocity (P = W/t = Fv), unit watt (W) = J s⁻¹
• 1 horsepower = 746 W; 1 kW h = 3.6 × 10⁶ J
• efficiency = useful output ÷ total input × 100%

Energy and its conservation

• kinetic energy = ½ × mass × speed² (KE = ½mv² = p²/2m)
• gravitational potential energy = mgh, with h measured from the chosen reference level
• Work–energy principle: net work = change in KE (Fd = ½mv² − ½mu²)
• No friction: loss of PE = gain in KE, so from rest v = √(2gh)
• With friction: loss of PE = gain in KE + work done against friction (f × distance moved)
• elastic potential energy of a spring = ½kx²

Physics · Circular and rotational motion

Angular motion and centripetal force

• angle in radians = arc length ÷ radius (θ = s/r); 2π rad = 360° = 1 revolution
• angular velocity ω = Δθ/Δt in rad s⁻¹; ω = 2π/T = 2πf
• angular acceleration α = Δω/Δt in rad s⁻²
• Linear and angular: s = rθ, v = rω, a = rα
• ω_f = ω_i + αt; θ = ω_it + ½αt²; θ = ½(ω_i + ω_f)t; ω_f² − ω_i² = 2αθ
• centripetal acceleration = v²/r = rω², directed towards the centre
• centripetal force = mv²/r = mrω²

Rotational dynamics

• moment of inertia I = Σmr², unit kg m²; torque = Iα
• Ring or hoop: I = MR². Disc or solid cylinder: I = ½MR². Solid sphere: I = (2/5)MR². Thin rod about its centre: I = (1/12)ML²
• rotational kinetic energy = ½Iω²
• Rolling without slipping: total KE = ½mv² + ½Iω² (disc: ¾mv²; hoop: mv²)
• angular momentum L = Iω = mvr, unit kg m² s⁻¹ (J s), directed along the axis by the right-hand rule
• Conservation of angular momentum: I₁ω₁ = I₂ω₂ when no external torque acts
• Satellite: orbital speed v = √(GM/r); close to the surface v = √(gR), about 7.9 km s⁻¹ for the Earth; T² ∝ r³

Physics · Fluid dynamics

Viscosity and drag

• Viscosity η measures the internal friction of a fluid; unit N s m⁻² (Pa s)
• Stokes' law: drag force = 6πηrv for a small sphere of radius r moving at speed v
• Terminal velocity: drag = weight, net force = 0, acceleration = 0, speed constant
• terminal velocity v_t = 2gr²ρ/9η, where ρ is the density of the sphere
• v_t ∝ r² and v_t ∝ 1/η: larger drops fall faster, thicker fluids slow the fall
• Laminar flow: steady, in layers, streamlines never cross; streamlines close together mean higher speed
• Turbulent flow: irregular and noisy, begins when the speed passes a critical value

Continuity and Bernoulli

• Continuity: area × speed is constant (A₁v₁ = A₂v₂)
• volume flow rate = Av, in m³ s⁻¹
• Bernoulli: P + ½ρv² + ρgh = constant
• Horizontal pipe: P₁ − P₂ = ½ρ(v₂² − v₁²), so higher speed means lower pressure
• Torricelli's theorem: speed of efflux from a hole at depth h below the surface is v = √(2gh)
• force from a pressure difference = pressure difference × area
• Conversions: 1 cm² = 10⁻⁴ m²; 1 litre = 10⁻³ m³

Physics · Waves and oscillations

Simple harmonic motion

• Condition for SHM: a = −ω²x, with ω = 2πf = 2π/T
• displacement: x = x₀ sin ωt, where x₀ is the amplitude
• speed: v = ω√(x₀² − x²); greatest speed = ωx₀ at the mean position
• greatest acceleration = ω²x₀ at the extreme positions
• Energy: PE = ½kx², KE = ½k(x₀² − x²), total = ½kx₀²
• Mass–spring system: T = 2π√(m/k), does not depend on g or on the amplitude
• Simple pendulum: T = 2π√(l/g), does not depend on the mass or on a small amplitude

Progressive waves

• wave speed = frequency × wavelength (v = fλ), and f = 1/T
• Phase difference = 2π × distance apart ÷ wavelength (Δφ = 2πx/λ), in radians
• Speed of a transverse wave on a string: v = √(T/μ), with tension T in N and mass per unit length μ in kg m⁻¹
• Coherent sources have the same frequency and a constant phase difference
• Sources in phase: maximum where path difference = nλ, minimum where path difference = (n + ½)λ
• Intensity is proportional to amplitude², so I_max/I_min = (A₁ + A₂)²/(A₁ − A₂)²
• A compression and the nearest rarefaction are half a wavelength apart

Stationary waves and sound

• String fixed at both ends, or pipe open at both ends: f_n = nv/(2L), n = 1, 2, 3, …
• Pipe closed at one end: f_n = nv/(4L), n = 1, 3, 5, … only
• String vibrating in n loops: L = nλ/2; its fundamental is f₁ = (1/(2L)) × √(T/μ)
• Speed of sound in a gas: v = √(γP/ρ) = √(γRT/M), so v is proportional to √T (T in kelvin) and to 1/√M
• Beat frequency = difference of the two frequencies (f₁ − f₂)
• Doppler, moving source: f′ = f × v/(v − v_s) when approaching, f × v/(v + v_s) when moving away
• Doppler, moving listener: f′ = f × (v + v_o)/v when approaching, f × (v − v_o)/v when moving away

Physics · Thermodynamics

Kinetic theory and the first law

• Ideal gas: PV = nRT, and P = ⅓ρ⟨v²⟩ = ⅔ × (N/V) × ⟨½mv²⟩
• Average translational kinetic energy of a molecule = (3/2)kT, with T in kelvin; v_rms = √(3RT/M), proportional to √T
• First law: Q = ΔU + W (Q = heat supplied to the gas, W = work done by the gas)
• Isothermal: ΔU = 0, so Q = W
• Adiabatic: Q = 0, so W = −ΔU (compression heats the gas, expansion cools it)
• Isochoric: W = 0, so Q = ΔU
• Isobaric: W = PΔV = nRΔT

Heat engines and entropy

• efficiency = work output ÷ heat input: η = W/Q₁ = 1 − Q₂/Q₁
• Carnot engine: η = 1 − T₂/T₁, and Q₂/Q₁ = T₂/T₁ (T₁ source, T₂ sink, both in kelvin)
• A real engine is always less efficient than a Carnot engine between the same temperatures; 100% would need a sink at 0 K
• Entropy change ΔS = Q/T, unit J K⁻¹; positive for heat gained, negative for heat lost
• Total entropy increases in an irreversible process and stays constant in a reversible one; it never decreases
• Melting and boiling raise the entropy of a substance; freezing and condensing lower it

Physics · Electrostatics

Coulomb's law and electric field

• Coulomb's law in a vacuum: F = kq₁q₂/r², with k = 1/(4πε₀) = 9 × 10⁹ N m² C⁻²
• In a medium: F_medium = F_vacuum ÷ ε_r, so ε_r = F_vacuum/F_medium
• electric field strength = force ÷ charge (E = F/q), unit N C⁻¹ or V m⁻¹
• Field of a point charge: E = kq/r²
• Uniform field between parallel plates: E = V/d
• Field lines start on positive charges and end on negative charges, never cross, and are closer together where the field is stronger

Potential and capacitors

• potential difference = work done ÷ charge (V = W/q); energy gained by a charge = qV
• Point charge: V = kq/r; field strength is the potential gradient, E = −ΔV/Δr, so for a point charge E = V/r
• 1 eV = 1.6 × 10⁻¹⁹ J
• capacitance = charge ÷ potential difference (C = Q/V); 1 F = 1 C V⁻¹
• Parallel-plate capacitor: C = ε₀ε_rA/d
• Parallel: C = C₁ + C₂ + …; series: 1/C = 1/C₁ + 1/C₂ + …
• Energy stored = ½QV = ½CV² = Q²/(2C)

Physics · Current electricity

Resistance and Ohm's law

• current = charge ÷ time (I = Q/t), and Q = ne with e = 1.6 × 10⁻¹⁹ C
• Ohm's law: V = IR, with R constant at constant temperature
• resistance = resistivity × length ÷ area (R = ρL/A); ρ in Ω m; A = πd²/4
• Temperature: R_t = R₀(1 + αΔt); α is positive for metals
• Series: R = R₁ + R₂ + …
• Parallel: 1/R = 1/R₁ + 1/R₂ + …; for two resistors R = R₁R₂/(R₁ + R₂)
• n equal resistors R in parallel give R/n

Circuits, EMF and power

• Terminal p.d. = EMF − lost volts: V = E − Ir
• Current from a cell: I = E/(R + r)
• power = p.d. × current: P = VI = I²R = V²/R
• energy = power × time; 1 kW h = 3.6 × 10⁶ J
• Kirchhoff's first rule: total current into a junction = total current out
• Kirchhoff's second rule: round any closed loop, sum of EMFs = sum of IR drops; EMFs that oppose are subtracted
• Balanced Wheatstone bridge: P/Q = R/S

Physics · Electromagnetism

Magnetic fields and forces

• Long straight wire: B = μ₀I/(2πr), with μ₀ = 4π × 10⁻⁷ T m A⁻¹
• Inside a solenoid: B = μ₀nI, where n = N/L is the number of turns per metre
• force on a wire = B × I × L × sin θ (F = BIL sin θ), θ between the wire and the field
• Direction of the force: Fleming's left-hand rule (first finger field, second finger current, thumb force)
• Parallel wires: force per unit length F/L = μ₀I₁I₂/(2πd), the same size on both wires
• Fields from two wires add as vectors: between wires with opposite currents they add; between wires with the same currents they subtract
• 1 T = 1 N A⁻¹ m⁻¹

Charges in magnetic fields

• force on a moving charge = q × v × B × sin θ (F = qvB sin θ), θ between velocity and field
• Circular path: qvB = mv²/r, so r = mv/(qB) = √(2mE_k)/(qB)
• Period T = 2πm/(qB), which does not depend on the speed or the radius
• Electrons accelerated through V: eV = ½mv², which gives e/m = 2V/(B²r²)
• Direction: Fleming's left-hand rule for a positive charge; reverse it for an electron
• Ammeter: shunt in parallel, R_s = I_gR_g/(I − I_g)
• Voltmeter: series resistance R_h = V/I_g − R_g

Physics · Electromagnetic induction

Faraday's and Lenz's laws

• magnetic flux = B × A × cos θ (Φ = BA cos θ), θ between the field and the normal to the loop; unit weber, 1 Wb = 1 T m²
• Faraday's law: EMF = −N × ΔΦ/Δt
• Motional EMF of a rod cutting the field at right angles: EMF = BLv; induced current I = BLv/R
• Lenz's law: the induced current opposes the change in flux that produces it (the minus sign in Faraday's law)
• A north pole approaching a coil makes the near face a north pole (repels); a north pole moving away makes it a south pole (attracts)
• Seen from outside, a face with anticlockwise current is a north pole and a face with clockwise current is a south pole

Inductance, generators and transformers

• Self-induced EMF = −L × ΔI/Δt; L in henry, 1 H = 1 V s A⁻¹
• Mutual induction: EMF in secondary = −M × ΔI_p/Δt
• Energy stored in an inductor = ½LI²
• Generator: EMF = NBAω sin θ, θ between the field and the normal to the coil; peak EMF E₀ = NBAω, with ω = 2πf
• Transformer turns ratio: V_s/V_p = N_s/N_p
• Ideal transformer: V_pI_p = V_sI_s, so I_s/I_p = N_p/N_s
• Laminated core reduces eddy currents; soft iron reduces hysteresis loss

Physics · Alternating current

AC quantities

• V = V₀ sin ωt, with ω = 2πf and T = 1/f
• rms value = peak value ÷ √2 (V_rms = V₀/√2 = 0.707 V₀; I_rms = I₀/√2)
• peak-to-peak value = 2 × peak value
• average power = rms voltage × rms current × power factor (P = V_rms I_rms cos φ)
• For a resistor: P = I_rms²R = V_rms²/R = V₀²/2R
• A sine wave goes from zero to its peak in a quarter of a period (T/4)
• Mains values such as 220 V are rms values

AC circuits

• inductive reactance = 2π × frequency × inductance (X_L = ωL = 2πfL), in Ω
• capacitive reactance = 1 ÷ (2π × frequency × capacitance) (X_C = 1/ωC = 1/(2πfC)), in Ω
• impedance of a series RLC circuit: Z = √(R² + (X_L − X_C)²), and I_rms = V_rms/Z
• supply voltage in a series circuit: V = √(V_R² + (V_L − V_C)²)
• phase angle: tan φ = (X_L − X_C)/R; current lags if X_L > X_C, leads if X_C > X_L
• resonant frequency: f₀ = 1/(2π√(LC))
• At series resonance: Z = R, current is maximum and in phase with the supply voltage

Physics · Physics of solids and electronics

Properties of solids

• stress = force ÷ area (σ = F/A), in Pa or N m⁻²
• strain = extension ÷ original length (ε = ΔL/L), no unit
• Young's modulus = stress ÷ strain (Y = FL/(AΔL)), in Pa
• Order on the stress-strain curve: proportional limit, elastic limit, yield point, ultimate tensile strength (highest stress), breaking point
• elastic energy stored = ½ × force × extension (within the proportional limit)
• Pentavalent impurity (e.g. arsenic) gives n-type: electrons are majority carriers. Trivalent impurity (e.g. gallium) gives p-type: holes are majority carriers
• A pure semiconductor at 0 K is an insulator; any doped piece is electrically neutral as a whole

Electronics

• Potential barrier: about 0.7 V for silicon, 0.3 V for germanium. Forward current = (supply voltage − barrier) ÷ series resistance
• Half-wave rectifier: one diode, uses half of each cycle. Full-wave bridge: four diodes, two conduct in each half cycle
• emitter current = base current + collector current (I_E = I_B + I_C)
• current gain = collector current ÷ base current (β = I_C/I_B)
• Amplifier biasing: base-emitter junction forward biased, collector-base junction reverse biased
• AND: output 1 only if all inputs are 1. OR: output 1 if any input is 1. NOT: output is the opposite of the input
• NAND = AND followed by NOT; NOR = OR followed by NOT; XOR gives 1 when the two inputs are different; XNOR gives 1 when they are the same

Physics · Dawn of modern physics

Photons and the photoelectric effect

• photon energy = Planck's constant × frequency (E = hf = hc/λ), h = 6.6 × 10⁻³⁴ J s
• Einstein's equation: hf = φ + KE_max, with work function φ = hf₀ = hc/λ₀
• stopping potential: eV₀ = KE_max; 1 eV = 1.6 × 10⁻¹⁹ J
• Graph of KE_max against f: straight line, slope h, intercept on the frequency axis f₀
• Compton shift: Δλ = (h/m₀c)(1 − cos θ); zero at θ = 0°, greatest at θ = 180°
• Pair production needs a photon of at least 1.02 MeV (2 × 0.51 MeV); any extra becomes kinetic energy of the pair
• Annihilation of an electron and a positron at rest gives two 0.51 MeV photons moving in opposite directions

Wave nature of particles

• de Broglie wavelength = Planck's constant ÷ momentum (λ = h/p = h/mv)
• In terms of kinetic energy: λ = h/√(2mE_k); for a charge q accelerated through V, λ = h/√(2mqV)
• Uncertainty principle: Δx Δp ≈ h and ΔE Δt ≈ h
• Time dilation: t = t₀/√(1 − v²/c²) (longer)
• Length contraction: L = L₀√(1 − v²/c²) (shorter, only along the motion)
• Relativistic mass: m = m₀/√(1 − v²/c²)
• Mass-energy: E = mc², so energy released = mass lost × c²

Physics · Atomic spectra

Bohr model and spectra

• angular momentum: mvr = nh/2π, with n = 1, 2, 3, ...
• radius: r_n = n²r₁, with r₁ = 0.053 nm
• energy: E_n = −13.6/n² eV (−13.6, −3.40, −1.51, −0.85 eV for n = 1 to 4); ionisation energy from level n = 13.6/n² eV
• photon energy: hf = E_n − E_p
• 1/λ = R_H(1/p² − 1/n²), with R_H = 1.1 × 10⁷ m⁻¹
• Series by lower level p: Lyman 1 (ultraviolet), Balmer 2 (visible), Paschen 3, Brackett 4, Pfund 5 (all infrared)
• Number of possible lines from level n down to the ground state = n(n − 1)/2

X-rays and lasers

• maximum photon energy = electron charge × tube voltage (eV = hf_max = hc/λ_min), so λ_min = hc/eV
• Higher tube voltage: shorter λ_min, more penetrating X-rays. Larger filament current: more electrons, greater intensity only
• K_α line: an electron drops from the L shell into a vacancy in the K shell; K_β: from the M shell
• Dense materials of high atomic number (bone, lead) absorb X-rays more than soft tissue. Uses: radiography, CT scans, cancer therapy, studying crystals
• Metastable state lasts about 10⁻³ s; an ordinary excited state about 10⁻⁸ s
• Laser light is monochromatic, coherent, intense and in a narrow beam
• Helium-neon laser: excited helium atoms pass energy to neon atoms by collision; neon emits red light of 632.8 nm

Physics · Nuclear physics

Nucleus and radioactivity

• mass defect: Δm = Zm_p + Nm_n − m_nucleus
• binding energy = Δm × c²; 1 u = 931 MeV; binding energy per nucleon = binding energy ÷ A
• α decay: A falls by 4, Z falls by 2. β⁻ decay: A unchanged, Z rises by 1. γ emission: A and Z unchanged
• decay law: N = N₀e^−λt; activity A = λN, in becquerel (1 Bq = 1 decay per second)
• half-life: T_½ = 0.693/λ
• After n half-lives: fraction left = (½)ⁿ, fraction decayed = 1 − (½)ⁿ

Nuclear energy and radiation

• In every nuclear equation the mass numbers and the charge numbers balance on both sides
• energy released = mass lost × c²; 1 u = 931 MeV
• Moderator (graphite, heavy water) slows fast neutrons; control rods (cadmium, boron) absorb neutrons; coolant carries heat out of the core
• Fast breeder reactor: uranium-238 absorbs a neutron and becomes fissile plutonium-239
• Detectors: Geiger-Müller counter (one particle starts an avalanche, giving one pulse), Wilson cloud chamber (tracks of droplets), solid-state detector (reverse-biased p-n junction)
• Units: activity in becquerel (Bq); absorbed dose in gray (1 Gy = 1 J kg⁻¹); equivalent dose in sievert = dose in Gy × RBE
• Somatic effects harm the exposed person; genetic effects appear in their children. Iodine-131 is used for the thyroid, cobalt-60 γ-rays for cancer treatment