Contents
General
Physical Chemistry is the combined science of physics, chemistry, thermodynamics, and quantum mechanics which functions to provide molecular-level interpretations of observed macroscopic phenomena1. Typically, changes in temperature, pressure, volume, heat, and work of systems in the solid, liquid, and or gas phase are correlated to microscopic atomic and molecular interactions. Most cite Willard Gibbs as the founder of physical chemistry as stemming from his 1876 paper: "On the
Equilibrium of Heterogeneous Substances", wherein such cornerstones as free energy, chemical potential, and phase rule were developed.
Chemical potential
The precise meaning of the term chemical potential depends on the context in which it is used.
When speaking of thermodynamic systems, chemical potential refers to the thermodynamic chemical potential.
Theoretical chemists and physicists often use the term chemical potential in reference to the electronic chemical potential, which is related to the functional derivative of the density functional
(sometimes called the energy functional)
Physicists sometimes use the term chemical potential in the description of relativistic systems.
Thermodynamic Chemical Potential
Conjugate variables of thermodynamics
Pressure
Volume
Entropy
Temperature
Chem. potential
Particle no.
The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume held fixed. If a system contains more than one species of particle, there is a separate chemical potential associated with each species, defined as the change in energy when the number of particles of that species is increased by one. The chemical potential is a fundamental parameter in thermodynamics and it is conjugate to the particle number.
Precise definition

Consider a thermodynamic system containing n constituent species. Its total internal energy U is postulated to be a function of the entropy S, the volume V, and the number of particles of each species N1,..., Nn:
By referring to U as the internal energy, it is emphasized that the energy contributions resulting from the interactions between the system and external objects are excluded. The chemical potential of the i-th species, µi is defined as the partial derivative where the subscripts simply emphasize that the entropy, volume, and the other particle numbers are to be kept constant.
In real systems, it is usually difficult to hold the entropy fixed, since this involves good thermal insulation. It is therefore more convenient to use the Helmholtz free energy A, which is a function of the temperature T, volume, and particle numbers:
In terms of the Helmholtz free energy, the chemical potential is
Laboratory experiments are often performed under conditions of constant temperature and pressure. Under these conditions, the chemical potential is the partial derivative of the Gibbs free energy with respect to number of particles
A similar expression for the chemical potential can be written in terms of partial derivative of the enthalpy (under conditions of constant entropy and pressure).
Relativistic Chemical Potential
For relativistic systems (systems in which the rest mass is much smaller than the equivalent thermal energy) the chemical potential is related to symmetries and charges.
Electronic Chemical Potential
The electronic chemical potential is the functional derivative of the density functional with respect to the electron density.
Values and trends
Generally speaking, atomic ionization energies decrease down a group of the periodic table, and increase left-to-right across a period. Ionization energy exhibits a strong negative correlation with

4,950
4,565
Third
7,730
2,744
6,270
6,950
Fourth
11,600
4,354
3,228
2,905
3,375
21,200
8,490
Fifth
16,100 atomic radius. Particularly dramatic increases occur after any given block of atomic orbitals is exhausted. Some values for elements of the third period are given in the following table:
Seventh
11,000
Sixth

1,060
999
Element
Na
Mg
Al
2,260
First
496
738
577
786
Second
4,560
1,450
1,816
1,577
1,890
S
Si
P
Electrostatic explanation
* Atomic ionization energy can be predicted by a simple analysis using electrostatic potential and the
Bohr model of the atom, as follows.
* Consider an electron of charge -e, and an ion with charge +ne, where n is the number of electrons missing from the ion.

Quantum-mechanical explanation
According to the more sophisticated theory of quantum mechanics, the location of an electron is best described as a "cloud" of likely locations that ranges near and far from the nucleus. The energy can be calculated by integrating over this cloud. This cloud corresponds to a wavefunction or, more specifically, to a linear combination of Slater determinants, i.e., according to Pauli exclusion principle, antisymmetrized products of atomic or molecular orbitals. This linear combination is called a configuration interaction expansion of the electronic wavefunction.
Examples
Fermat's principle in geometrical optics
The principle of least action in mechanics, electromagnetic theory, and quantum mechanics, where the dimension is action.
The Einstein equation also involves a variational principle, according to Stephen Wolfram, (A New
Kind of Science, p. 1052.), as a constraint on the Einstein-Hilbert Action.
Variational principle in quantum mechanics
For a hamiltonian H that describes the studied system and any normalizable function ? with arguments appropriate for the unknown wave function of the system, we define the functional
The variational principle states that where E0 is the lowest energy eigenstate (ground state) of the hamiltonian
if and only if ? is exactly equal to the wave function of the ground state of the studied system.
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