3. Methodology
3.2 Statistical thermodynamics
In physics, there are two types of mechanics usually examined: classical
mechanics and quantum mechanics. Whereas ordinary mechanics only considers the behavior of a single state, in quantum statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states.
As a branch of mathematical physics, statistical mechanics is using probability theory, the average behavior of a mechanical system where the state of the system is
uncertain. A common use of statistical mechanics is in explaining the thermodynamic behavior of large systems.
Classical thermodynamics is difficult to explain concepts in microscopic level, however statistical mechanics shows how these concepts arise from the natural uncertainty that arises about the state of a system when that system is prepared in practice. The benefit of using statistical mechanics is that it provides exact methods to connect thermodynamic quantities to microscopic behavior. Statistical mechanics also makes it possible to extend the laws of thermodynamics to cases which are not
considered in classical thermodynamics, for example microscopic systems and other mechanical systems with few degrees of freedom. With statistical mechanics, which treats and extends classical thermodynamics is known as statistical thermodynamics.
Applied these laws to chemical problems, it bridged the gap between the
microscopic realm of atoms and molecules and the macroscopic realm of classical thermodynamics. Statistical thermodynamics demonstrates how the thermodynamic parameters of a system, such as temperature and pressure, are related to microscopic behaviors of such constituent atoms and molecules.
3.2.1 Description in microscopic - molecular model
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About microscopic model, there is a basic definition: we cannot distinguish (or observe) behaviors of molecular without perturbing their energies; and once we have done this they are no longer degenerate and the problem is changed. Such property about indistinguishability is a “microscopically separate state” or microstate. All of these individual microstates produce the same "macroscopically observable state" or macrostate.
One of the first successful attempts to use a microscopic model to explain
macroscopic behavior was the kinetic molecular theory of gases, associated with such names as Maxwell and Boltzmann. This model was based on as simple a molecular picture as could be imagined.
• Molecules are points (or at most, hard spheres)
• The only energy that molecules possess is kinetic energy of motion;
moreover, any kinetic energy is possible
• Molecules do not exert any influence upon one another except at the moment of impact; when they do collide, they do so elastically
Actually, all three of these assumptions are wrong. Truth is more complicated:
• Molecules are grossly misshapen objects
• The possess many and complex forms of internal energy; moreover, the available energy states are quantized
• Molecules exert long-range forces on one another, for which “collision” is interactions that do not lead to reaction. Elastic collision is even more unrealistic.
3.2.2 Basic definition of statistical mechanics
Assume a collection of molecules – gas, liquid or solid – each having available to it
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various molecular states describable by wave functions Økof various energies and spatial distributions. Where a wave function (also named a state function) in quantum mechanics describes the quantum state of a system of one or more particles, and contains all the information about the system. The measurable property of the bulk system will depend upon how many molecules are occupying which of these states. If we know the distribution of molecules among their possible states, we can calculate these properties. And, with this distribution it is possible to calculate properly the quantities we are seeking.
Large numbers of these states will have so nearly the same energy that they can be grouped into one energy level and treated together. Assume now that we have nj
molecules of energy ɛj. Let the number of wave functions having an energy ɛj be gj. This gj is known as the degeneracy or the statistical weight of energy level ɛj , as shown as Figure 4.
Source: Adapted from R. E.. Dickerson, “Molecular Thermodynamics”, W. A. Benjamin (1969)
The fundamental principle of statistical mechanics is that, energy considerations aside, Figure 4 Degenerate states in a quantum system
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one microstate is just as probable as any other one. Therefore, the probability of occurrence of any given macrostate ( )
nj
P is proportional to the number of possible microstates ( )
3.2.3 Entropy in statistic thermodynamics
We shortly define the entropy S of an arrangement of particles, to be a measure of the uniqueness of the arrangement. Since a given distribution is proportional to the number of ways of obtaining that distribution
) ( ) (nj Wnj
P [2]
To form the probability of a distribution we should divide the number of ways of producing the given distribution by the total number of ways of producing all possible distributions. Take poker as an example, there are 3,744 different ways of obtaining a full house, and, there are 2,598,960 different possible five-card hands. That is, the relative probability of being dealt a full house is 0.00144.
Since we would like to measure simply by the number of different ways by which the distribution can be obtained, W nj . We shall define a quantity, known as another definition under statistical mechanics of entropy S, proportional to the logarithm of the number of ways of obtaining a given distribution
nj k W nj
S ln [3]
The constant k in Equation [3] can be any quantity, but for convenience, it would be assigned as Boltzmann's constant, 1.3806488 × 10-23 m2 kg s-2 K-1. And the standard unit of entropy is
1 cal/deg mole = 1 entropy = 1 e.u.
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There is only one way of constructing a perfectly ordered system, W = 1, that is, entropy = 0. At the other extreme, the number of ways of obtaining a totally
disordered system tends toward infinity for system containing many particles. As the number of particles increases, the entropy of such a system also tends to infinity.
Clearly, entropy can never be negative, because entropy in negative is meaningless in terms of probabilities.
The entropy can be calculated in a form of “configurational entropy”, the entropy of mixing. Since the configurational entropy of the pure substances before mixing is zero and the mole fractions of components are always less than one. So, the entropy of mixing is always positive. In any mixing operation, the entropy of the disorder of the system increases, so that mixing always leads to more probable states. This fact leads to the conclusion that mixing should be irreversible.
3.2.4 Dilute systems
Systems in which the degeneracy of each level far exceeds the number of objects in that level, or which gj >> nj , are known as dilute systems.
Derived from Maxwell-Boltzmann particle,
n
Because the exponential term is always greater than one, then we can say ....
In other words, in the most probable distribution, the higher up the energy ladder one
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goes, the more states are available per particle or the greater the dilution. If the dilution ratio of the ground state is great enough that one can forget about different types of distribution properties and use an single simplfied statistics, the the same will also be true for all the higher energy levels.
3.2.5 Conditions for applicability
As mass increases like Argon, it will have more states then Helium in lower mass because its energy levels are more closely spaced than are those of Helium. In general, heavier particles will have more closely spaces energy levels as illustrated in Figure 5.
Source: Adapted from R. E.. Dickerson, “Molecular Thermodynamics”, W. A. Benjamin (1969)
Thinking in entropy, entropy increases as molecular weight increase since it leads to the energy level to be more closely spaced (image of the particle in a box), and both dilution ratio and W increase.
Figure 5 The effect of increased mass on energy level spacings
He (lower mass) Ar (higher mass)
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As the temperature is lowered, the total energy of the particles is lowered, thus forcing them to trickle down from upper energy level and fill up the bottom ones, as illustrated in Figure 6.
Source: Adapted from R. E.. Dickerson, “Molecular Thermodynamics”, W. A. Benjamin (1969)
Thinking in entropy, a temperature increase results in an increase in molecular motions, a condition that enhances the chances of particle mixing, that is, entropy also rises.
As the pressure is increased, the holding of total energy constant results in a
decrease in volume. A decrease in volume causes an increase in the spacing between energy levels, so that under conditions of constant total energy, the particles are forced to drop down into lower quantum states, as illustrated in Figure 7.
High Temperature Low Temperature
Figure 6 The effect of lowering the temperature on the distribution of particles among energy levels
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Source: Adapted from R. E.. Dickerson, “Molecular Thermodynamics”, W. A. Benjamin (1969)
Thinking in entropy, since volume decreased, there is less room available to molecules, the less different arrangements are possible. Therefore, the energy level moving upwards, the dilution ration decreases and, with less states now available to the particles, W decreases as well.