Young's modulus (Y) measures resistance to change in length (longitudinal stress/strain). Bulk modulus (B) measures resistance to change in volume (volume stress/strain) under uniform pressure. Shear (rigidity) modulus (G) measures resistance to change in shape (shearing stress/strain).

Stefan's (Stefan-Boltzmann) law: the total energy radiated per unit area per unit time by a black body is proportional to the fourth power of its absolute temperature, E = sigma T^4. Wien's displacement law: the wavelength at which the emission is maximum is inversely proportional to the absolute temperature, lambda_max * T = constant (= 2.9 x 10^-3 m K).

E = sigma T^4 ; lambda_max T = b

The mean free path is the average distance a gas molecule travels between two successive collisions. It increases when the number density of molecules is low and when the molecular diameter is small; that is, it is inversely proportional to the number density and to the square of the molecular diameter.

lambda = 1 / (sqrt(2) pi d^2 n)

The centre of mass is the point at which the entire mass of the system can be considered to be concentrated, and where an applied external force produces the same acceleration as on the whole system. For two particles of masses m1 and m2 at positions x1 and x2, x_cm = (m1 x1 + m2 x2) / (m1 + m2).

x_cm = (m1 x1 + m2 x2) / (m1 + m2)

With depth d, g decreases as g' = g (1 - d/R), assuming uniform density, because only the mass within the smaller inner sphere attracts the body. At the centre (d = R) the value becomes zero, since the mass is symmetrically distributed all around.

g' = g (1 - d/R)

Resolution of a vector is the process of splitting it into two or more components, usually along mutually perpendicular directions. For a vector A at angle theta to the x-axis, the rectangular components are Ax = A cos(theta) along the x-axis and Ay = A sin(theta) along the y-axis, with A = sqrt(Ax^2 + Ay^2).

Ax = A cos(theta) ; Ay = A sin(theta)

A quasi-static process is one carried out infinitely slowly so that the system stays in thermal and mechanical equilibrium with its surroundings at every stage. It is important because only for such (reversible) processes can the state variables (P, V, T) be defined throughout, and the work done can be represented as an area on a P-V diagram.

Every particle of matter attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them, directed along the line joining them: F = G m1 m2 / r^2, where G is the universal gravitational constant (6.67 x 10^-11 N m^2 / kg^2).

F = G m1 m2 / r^2

Law of conservation of mass: matter can neither be created nor destroyed in a chemical reaction; the total mass of reactants equals the total mass of products. Law of definite proportions: a given chemical compound always contains the same elements combined in the same fixed proportion by mass, irrespective of its source or method of preparation.

Molar mass of CO2 = 12 + 2(16) = 44 g/mol. Number of moles = given mass / molar mass = 11 / 44 = 0.25 mol.

n = m / M

One mole is the amount of a substance that contains as many elementary entities (atoms, molecules, ions) as there are atoms in exactly 12 g of carbon-12. This number of entities is Avogadro's number, N_A = 6.022 x 10^23 mol^-1.

1 mol = 6.022 x 10^23 particles

Molarity (M) is the number of moles of solute dissolved per litre of solution; unit mol/L (or M). Molality (m) is the number of moles of solute dissolved per kilogram of solvent; unit mol/kg (or m). Molality is independent of temperature since it uses mass, whereas molarity changes with temperature because volume changes.

M = n_solute / V(L); m = n_solute / mass_solvent(kg)

Mole fraction of a component = number of moles of that component / total number of moles of all components in the solution; it is dimensionless and the sum of mole fractions equals 1. Mass percent of a component = (mass of the component / total mass of solution) x 100.

x_A = n_A / n_total; mass% = (mass component / total mass) x 100

Wavelength = 4000 Angstrom = 4000 x 10^-10 m = 4 x 10^-7 m. Energy E = hc/lambda = (6.626 x 10^-34 x 3 x 10^8) / (4 x 10^-7) = 1.988 x 10^-25 / 4 x 10^-7 = 4.97 x 10^-19 J.

E = hc / lambda

It is impossible to determine simultaneously and with absolute accuracy both the position and the momentum (or velocity) of a microscopic particle such as an electron. The product of the uncertainties in position (delta x) and momentum (delta p) is at least of the order of h/4pi.

delta x . delta p >= h / 4pi

Wave number (nu bar) = R (1/n1^2 - 1/n2^2) = 1.097 x 10^7 (1/2^2 - 1/3^2) = 1.097 x 10^7 (1/4 - 1/9) = 1.097 x 10^7 x (5/36) = 1.523 x 10^6 m^-1. This is the H-alpha line of the Balmer series.

nu bar = R (1/n1^2 - 1/n2^2)

Pauli's exclusion principle: no two electrons in an atom can have the same set of all four quantum numbers; an orbital can hold at most two electrons with opposite spins. Hund's rule of maximum multiplicity: electron pairing in orbitals of the same subshell (degenerate orbitals) does not occur until each orbital is singly occupied, and all singly filled orbitals have parallel spin.

Aufbau principle: in the ground state of an atom, electrons are filled into orbitals in order of increasing energy, i.e. the lowest energy orbital is filled first. (n + l) rule: the orbital with the lower (n + l) value has lower energy and is filled first; if two orbitals have the same (n + l) value, the one with the lower n is filled first (e.g. 4s (n+l=4) is filled before 3d (n+l=5)).

energy order by increasing (n + l)

Modern periodic law: the physical and chemical properties of elements are a periodic function of their atomic numbers. Mendeleev's law was based on atomic mass, whereas the modern law is based on atomic number (number of protons), which removed anomalies such as the position of argon and potassium.

Nitrogen has the configuration 1s2 2s2 2p3 with a half-filled 2p subshell, which is extra stable due to symmetry and exchange energy, so removing an electron requires more energy. Oxygen (1s2 2s2 2p4) has one paired electron in 2p; removing it relieves electron-electron repulsion and gives a stable half-filled configuration, so less energy is needed. Hence IE1 of nitrogen > oxygen.