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Statistical mechanics is one of the strands of modern physics. It is necessary for the fundamental analysis of any physical system that has a large number of degrees of freedom. The strategy is based on statistical methods, probability theory and the microscopic physical laws.

It can be used to explain the thermodynamic behaviour of large systems. This branch of statistical mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics.

Statistical mechanics illustrates how the concepts from macroscopic observations (such as temperature and pressure) are related to the description of microscopic state that fluctuates around an average state. It links together thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.

Statistical mechanics can also be used to analyze systems that are out of equilibrium. An important subbranch known as non-equilibrium statistical mechanics deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes are chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge attained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.

Brownian motion arises due to the random motion of atoms in a liquid of gas.

Boltzman defined entropy as a function of system's microstates. In statistical mechanics, the entropy S of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of W, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy E):

## Boltzman equation

Boltzman equation describes statistical behavior of a thermodynamic system that is not in the state of equilibrium. Boltzman was first to discover such system. The probability density function f is defined such thatrepresents the number of molecules which all have positions lying inside the volume element dr^3 about r and momenta lying inside momentum space dp^3 about p. Then Boltzman equation states that

## Partition function

We briefly explore the connection between the Euclidean path integral and statistical
mechanics. Consider an ensemble system at thermodynamic equilibrium with ordered
12 D.V. Perepelitsa
energy microstates {En} for n = 1, 2, . . . . The partition function Z of statistical
mechanics encodes probabilistic information about the system.

Z =
Σ(n=1, α)
e
−βEn

Above, β = 1/kbT be the inverse temperature of the system at a given temperature T
with Boltzmann’s constant kb. T

### IONIC BONDS

From one point of view a molecule is a stable arrangement of a group of nuclei and electrons. The exact arrangement is determined by electromagnetic forces and the laws of quantum mechanics. This concept of a molecule is a natural extension of the concept of an atom. Another view regards a molecule as a stable structure formed by the association of two or more atoms. In this view the atoms retain their identity whereas in the first-mentioned view they do not. Of course, both views are useful and there are situations wherein each is directly applicable. In general, however, the structure and properties of molecules are best described by a combination of both views. When a molecule is formed from two atoms, the inner shell electrons of each atom remain tightly bound to the original nucleus and are barely disturbed at all. The outermost loosely bound electrons, known as the valence electrons, are influenced by all the particles (ions + electrons) of the system. Their wave functions are significantly modified when the atoms are brought together. Indeed, it is this very interaction that leads to binding, i.e., to a lower total energy, when the nuclei or ions are close together.This interaction, called the interatomic force, is of electromagnetic origin. Hence, we see that valence electrons play the central role in molecular binding. There are two principal types of molecular binding, the ionic bond and the covalent bond. The NaCl molecule is an example of ionic binding and the H2 molecule an example of covalent binding. Consider the formation of a NaCl molecule from an atom of Na and an atom of Cl which are far apart initially. Figure 9-15 shows that to remove the outermost 3s electron from Na and form the Na + ion requires an ionization energy of 5.1 eV. The atomic binding in the alkali Na is relatively weak because its filled inner subshells are effective in shielding the valence electron electrically from the nucleus so that it moves in a weakened field at an outlying position. If now we attach this electron to the halogen Cl atom it will complete a previously unfilled 3p shell in Cl to form a Cl ion. The halogen has a relatively high electron affinity; that is, the closed shell ion is more stable than the neutral atom, its energy being lower by 3.8 eV. Hence, at the cost of 1.3 eV of energy (5.1 eV — 3.8 eV), we have formed two distinct separate ions, Na + and Cl ; but these ions exert attractive Coulomb forces on one another, and the energy of attraction is greater than 1.3 eV. Now, since the mutual Coulomb potential energy of the ions is negative, the potential energy of the combined system initially decreases as the separation of the ions is steadily reduced. As the ions are brought still closer together the electron charge distributions begin to overlap. This has two effects, each of which increases the potential energy: (1) the nuclei are not as well shielded from one another as before and they begin to repel one another and (2) at small internuclear separation we effectively have a single system to which the exclusion principle applies, and some electrons must be in higher energy states than before to avoid violating this principle. The potential energy curve therefore yields a repulsive force at small interatomic separations and an attractive force at large separations. There is a separation at which this energy is a minimum, the energy being 4.9 eV lower at this proximity than for distantly separated ions. Hence, compared to two neutral atoms, Na + Cl, the combined system NaCl is lower in energy by 3.6 eV (that is, E = 1.3 eV — 4.9 eV = — 3.6 eV) so that a bound state is energetically favored, as illustrated in Figure 12-1. The equilibrium nuclear separation in NaC1 is 2.4 A.

### COVALENT BONDS

Let us consider now the formation of the H2 molecule. If in the case of H2 we were to calculate the energy required to form positive and negative hydrogen ions by moving an electron from one hydrogen atom to the other, and then added to this the energy of the Coulomb interaction of the ions, we would find that there is no distance of separation at which the total energy is negative. That is, ionic bonding does not result in a bound H2 molecule. The fact that H2 is bound is explained quantum mechanically by the behavior of the electronic eigenfunction describing the charge distribution of the system, as two hydrogen atoms approach one another. As we shall see soon, the resulting charge distribution does lead to electrostatic attraction, but it is a charge distribution that can be interpreted as a sharing of electrons by both atoms. The binding is called covalent.Miscellaneous pages

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