Physical chemistry

where k is Boltzmann’s constant, Ei is the energy of level i, and q is known as the partition function and is given by . The summation is over all quantum states and the partition function can be interpreted as the number of quantum states with significant populations at a given temperature T.
For any two energy levels separated by energy gap DE the relative populations can be obtained without knowing the partition function from

Statistical definition of entropy
A true definition of entropy has nothing to do with disorder. More correctly it is a measure of available microstates through the relationship S = klnW, where W is the number of possible microstates yielding the same overall state of the system.
Chemical kinetics
Rate laws
The rate of any reaction xA+ yB + zC products can be written in terms of a rate law
rate = kAmbnCp
where k is the rate constant and m, n, and p are known as the order of reaction with respect to reagents A, B, and C, respectively. Note that the orders of reaction are quantities that must be determined from experiment and need not be the same as the stoichiometries x, y, and z.
Temperature dependence of rate constants
An increase in temperature usually increases the rate constant, and therefore the rate of reaction. The temperature dependence is given by the Arrhenius equation where Ea is the activation energy and A is known as the pre-exponential factor. The activation energy is a measure of the size of the energy barrier to reaction. A has a more complex interpretation but is dependent on the size of the colliding molecules and is small if a strict orientation for the colliding molecules is required.
In order to determine Ea and A the logarithmic form of the Arrhenius equation, , is more useful. If k is measured at various temperatures then Ea and A can be determined from the slope and the y-axis intercept of a plot of versus 1/T.
Elementary reactions
An elementary reaction (also known as an elementary step) is a reaction which occurs literally as written, e.g. if A + 2B products is an elementary reaction then it occurs via collision of one molecule of A with two molecules of B. Molecularity refers to the number of molecules reacting in an elementary reaction. Important cases are unimolecular, bimolecular and termolecular reactions. The rate law for an elementary reaction can be written down immediately since it reflects the molecularity, e.g. for the above example rate = k a b2.
Kinetics of complex (multistep) reactions - mechanism
Actual reactions may occur by a sequence of elementary reactions known as the mechanism. In each elementary reaction the molecularity must not exceed 3, and more usually each step is bimolecular. The rate of reaction is often limited by a particular elementary reaction, and this slowest step is known as the rate-determining step. The orders of reaction in the rate law then reflect the reactant molecularities in the rate-determining step.
Equilibrium electrochemistry
Redox reactions can be separated into half-reactions, one involving oxidation and the other reduction. The direction of chemical change for a redox reaction is determined by the cell potential E, which can be determined the standard potentials for each half-reaction coupled with the Nernst equation

where ared and aox are the activities of the reduced and oxidised species, respectively. The direction of chemical change follows from the Gibbs energy change

Quantisation and spectroscopy
At the atomic and molecular scale particles show wave-particle duality with a wavelength l linked to the particle momentum p by the de Broglie relationship l = h/p.
Confinement of these waves gives rise to the quantum states of atoms and molecules. In particular, molecules can possess rotational, vibrational, and electronic energy, all of which is quantised.
Rotational levels – for linear molecules the rotational energy is where J is the rotational quantum number (0, 1, 2, 3, …, etc.) and B is the rotational constant for the particular molecule. For a diatomic AB,

where r = bond length and m = reduced mass = mAmB/( mA + mB)
Vibrational levels – for diatomics the simple harmonic oscillator (SHO) model predicts vibrational energies given by where v = vibrational quantum number (0, 1, 2, 3, …etc.) and is the harmonic vibrational frequency. An important relationship is

where k = bond force constant and m = reduced mass.
For polyatomic molecules, there are 3N-6 vibrational modes if the molecule is non-linear and 3N-5 if linear. The SHO approximation usually suffices and the vibrational energy in each mode is given by where is the harmonic vibrational frequency for the ith mode.
Electronic levels – generally do not follow a simple pattern and so we cannot write down a formula, as was done above for vibrations and rotations, for electronic energies.
Spectroscopic transitions
Pure rotational (microwave) spectroscopy
Only possible if the molecule possesses a permanent electric dipole moment. Transitions between rotational levels are governed by the selection rule . Rotational constants, and therefore bond lengths, can be determined to high precision from microwave spectroscopy.
IR spectroscopy
IR radiation is required to excite transitions between vibrational levels. For a vibration to be IR active, it must give rise to an oscillating electric dipole moment. For heteronuclear diatomic molecules (but not homonuclear diatomics), this condition is always satisfied. For polyatomic molecules some modes may be IR active and others not.
The selection rule applies to spectroscopic transitions for each vibrational mode.
Rotational structure may be resolved in the IR spectra of molecules in the gas phase. This is due to simultaneous vibration-rotation transitions. For diatomic molecules the rotational selection rule is and this gives rise to P and R branches. The rotational constant can be determined from this structure.
Raman spectroscopy
Raman spectroscopy is based on inelastic light scattering, instead of direct absorption or emission. In vibrational Raman spectroscopy, for a vibration to be Raman active it must give rise to an oscillating polarisability. Both homonuclear and heteronuclear diatomic molecules satisfy this criterion. In polyatomics, some modes satisfy this criterion, others don’t – that is determined by the symmetry of the vibration.
Electronic spectroscopy
Electronic transitions must satisfy the spin selection rule. In addition, transitions between vibrational levels in different electronic states satisfy the Franck-Condon principle. This states that the most probable transition is a vertical transition involving no change in equilibrium nuclear positions.