Previous: Chapter 1. Conceptual Models of Statistical Mechanics
What could be solved in some cases in statistical mechanics is related to probability distribution functions. In this chapter, we explore the equilibrium state of ideal gas composed of polyatomic molecules. In a monatomic ideal gas, atoms have translational degrees of freedom only. However, a polyatomic molecule also has internal degrees of freedom, including rotational, and vibrational degrees of freedom.
We begin with a brief historical overview of the main events that led to the successful method to estimate the thermodynamic properties of polyatomic ideal gases. After that, we discuss meaning of probabilities in statistical mechanics. In the case of a monatomic ideal gas, the probabilities could be seen as statistics over atoms. In general, probabilities must be ascribed to the entire system in question, which raises the question of whether probabilities are related to human ignorance or the world.
The energy distribution of the Gibbs canonical ensemble opened the path to the final equation. The connection between the microcanonical and canonical probability distribution is also considered; this shows the connection with the Boltzmann entropy. At the end, the calculation of the thermodynamic properties of an ideal polyatomic gas is considered.
Special attention is paid to the limitations of the concept of entropy as the number of permutations. This metaphor is widely used and it is often presented in textbooks as an answer to the question of what entropy is.
- Key events in the development of statistical mechanics
- Probability distribution and phase space
- Microcanonical and canonical distribution
- Partition function for ideal polyatomic gas
Key events in the development of statistical mechanics
The ideas of the kinetic theory of gases became widespread after Clausius’s work ‘The Kind of Motion We Call Heat‘ (Über die Art der Bewegung, welche für Wärme nennen, 1857). Similar ideas had been expressed by other scientists earlier, but they were rejected by the scientific community. The law of energy conservation was an important reason for accepting the kinetic theory. The proof of the transformation of heat into work and work into heat led to the requirement to better understand what heat is.
One curious fact that shows how difficult to understand the scientific atmosphere of that time. In 1820, John Herapath proposed a paper with kinetic theory ideas for publication in the Proceedings of the Royal Society. Humphrey Davy, famous for his 1799 experiments against the caloric theory, and then the president of the Royal Society, strongly opposed this publication. In Herapath’s paper there were inaccuracies, but this was not the main reason. For reasons hard to understand from our time, Davy, an opponent of the caloric theory, didn’t like ideas of kinetic theory.
In 1860, from general considerations James Maxwell derived the equilibrium velocity distribution of atoms. In a series of papers beginning in 1866, Ludwig Boltzmann tried to prove that Maxwell’s distribution is obtained spontaneously from an arbitrary gas configuration. These works are examined in the next chapter, as this chapter considers equilibrium distributions only.
Maxwell’s distribution could be applied to translational motion and it did not include internal degrees of freedom. The discussion of vibrational and rotational motions within the kinetic theory led Maxwell and Boltzmann to the proof of the equipartition theorem. This theorem, in turn, led to theoretical predictions of gas heat capacities that differed from experimental values.
At the beginning of the work on the kinetic theory, only gas heat capacities for polyatomic gases were known. Mercury vapor turned out to be the first monatomic gas for which heat capacities were measured in 1871 (inert gases were discovered in the late 19th century). The heat capacity of mercury vapor was found to be in excellent agreement with the predictions of the kinetic theory, which only highlighted the problem with heat capacities of polyatomic molecules.
The general case of arbitrary interactions was considered by Gibbs (1902), who additionally emphasized the need for indistinguishability of particles to preserve the additivity of entropy. Thus, by this time, two features had been identified that demonstrated the inadequacy of classical mechanics for the motion of molecules: the discrepancy between predicted and experimental heat capacities for polyatomic gases and the indistinguishability of particles.
Max Planck in 1900 derived the distribution of energy density in black-body radiation and in this work there was the first quantum concept; a new fundamental constant, now known as the Planck constant, has appeared. Historians debate whether Planck fully understood the revolutionary nature of his ideas, but nevertheless, this paper laid the foundation for the development of quantum mechanics.
For this chapter it is important to mention the paper of Albert Einstein in 1907, which proposed the equation for the heat capacity of a solid based on the hypothesis of energy quantization. This approach allowed Walther Nernst in 1911 to explain the ‘freezing’ of vibrational degrees of freedom in polyatomic gases, thereby providing a solution to the problem of the heat capacities of polyatomic gases. In 1912, independently of each other, Otto Sakur and Hugo Tetrode proposed using the Planck constant to discretize the motion of a monatomic ideal gas. They have shown that with such discretization the Boltzmann entropy fitted the entropy of mercury vapor; thereby the importance of the Planck constant for all kinds of motion in a gas was shown.
The Sakur-Tetrode equation for the entropy of a monatomic gas predicted incorrect values for the entropy of the gas at very low temperatures. This problem was resolved in the 1920s by the development of quantum statistics (Bose-Einstein and Fermi-Dirac). As a result, the scope of applicability was found for the classical Maxwell-Boltzmann statistics.
In conclusion, I mention that statistical mechanics has also played a significant role in the study of the equation of state of real gases with intermolecular interactions. This topic however is not covered in this book, for more information I would recommend the book ‘Cohesion. Scientific History of Intermolecular Forces‘.
Probability distribution and phase space
Consider a monatomic ideal gas; each atom has three coordinates and three momenta. Imagine a six-dimensional space in which one atom is represented by a point, showing the current state of the atom (three coordinates plus three momenta). The state of the whole system is represented by a set of N points, with the motion of each point corresponding to the trajectory of one atom. This representation is called phase μ-space.
Maxwell’s velocity distribution represents the statistics of the motion of all atoms. At a given moment in time, the number of atoms moving within a given velocity interval, regardless of their location, makes a histogram. The probability of an atom having a velocity within the range of v±dv is equal to the ratio of the number of atoms with velocities within this interval to the total number of atoms. Thus, the probability for the Maxwell’s velocity distribution or the Maxwell-Boltzmann energy distribution has a simple and intuitive frequency interpretation.
The motion of atoms is determined by the Hamilton equations, but it is assumed that the statistics do not change in the equilibrium state, and thus Maxwell’s velocity distribution is independent of time. Moreover, it is assumed that this equilibrium distribution is achieved based from an arbitrarily chosen state, analogous to the Clausius inequality. In search of a proof, Ludwig Boltzmann proposed a statistical justification in the case of a monatomic gas by dividing the phase μ-space into discrete cells. We explore the history of Boltzmann’s work on the foundation of the kinetic theory in the next chapter.
In this chapter, I only present the modern form of the Boltzmann equation for the entropy of an ideal monatomic gas:
S = k ln W(E, V)
In this equation, k is the Boltzmann constant, and W(E, V) is the number of microstates compatible with a given macrostate, defined by the total energy (E) and volume (V).
For this equation, usually one detail is emphasized — the number of microstates is calculated as the number of possible permutations of atoms. This plays a prominent role for the statistical justification of the Clausius inequality in the case of an ideal monatomic gas, and also gives a vivid representation to explain entropy in this case. This explains the prevalence of the metaphor of entropy as the number of permutations of cards in the deck in the public consciousness.
However, it is often overlooked that even in the case of an ideal monatomic gas, the number of microstates is a function of energy and volume. Therefore, Boltzmann’s entropy is a function of external variables, which is not the case with the number of permutations of cards in a deck. Moreover, Boltzmann’s derivation is limited to an ideal monatomic gas. In general, when there is interaction between atoms, even in the case of an ideal polyatomic gas, Boltzmann’s cell method cannot be applied. This means that thermodynamic entropy as a number of permutations in the general case is just a metaphor.
However, Boltzmann’s entropy played an important role in the development of statistical and quantum mechanics. In his 1900 paper, Planck used Boltzmann’s equation to derive the distribution of black-body radiation, although Planck’s calculation of the number of permutations differed from Boltzmann’s approach. Additionally, the Sakur-Tetrode equation played a significant role in the development of statistical mechanics.
Gibbs proposed a general solution for systems with interactions by going to a multidimensional Γ-space. In this case, the dimensionality of the phase space is equal to 6*N, and the state of the entire system is represented by a single point. The movement of this point in the Γ-space is associated with changes in the coordinates and momenta of all particles, representing the trajectory of the entire system. In the Γ-space, the probability of the system being in a specific region is considered, but the interpretation of this probability becomes pretty complex.
To preserve the frequency interpretation, Gibbs introduced the concept of an ensemble – a collection of a huge number of systems that all are in the same macrostate but they have different microstates. In an ensemble, the frequency interpretation is preserved, as the probability of finding a system in a small region of Γ-space is equal to the ratio of the number of systems in that region to the total number of systems in the ensemble. However, this interpretation may lead to a misconception that the Gibbs method is not suitable for studying a single system.
In the case of a single system, one can say the following. The system is deterministic, and the motion of all interacting atoms is governed by the Hamilton equations, but we do not know the initial state of the system. The measured properties of this macroscopic system represent time averages, and the use of probabilities relies on the assumption that time averages coincide with phase space averages (the ergodic hypothesis). In other words, the average is over the distribution of system initial states, which we do not know.
In any case, the introduction of the probability of the system being in the Γ-space in the Gibbs method leads to the general question of whether the probabilities refer to the observer ignorance or to the world. This was one of the reasons for the subjective interpretation of entropy – entropy is related to the observer ignorance.
Microcanonical and canonical distribution
For this book, it is sufficient to consider two Gibbs ensembles: the microcanonical ensemble and the canonical ensemble. The usual presentation of statistical mechanics begins with an isolated system, which corresponds to the microcanonical ensemble. We know the energy and volume of the system, but we do not know the specific microstate — where in phase space the system is located at a given moment in time. The equilibrium probability distribution related to the principle of equal a priori probability of all states compatible with the given energy and volume. This assumes that the system of the microcanonical ensemble is in the energy interval E+dE, and all states from this layer of the Γ-space are considered equally probable.
The equation for the entropy in the case of the microcanonical ensemble resembles the Boltzmann equation:
S = k ln W(E, dE, V)
where W is the so-called statistical weight. The difference from the Boltzmann equation is connected with the interpretation of the statistical weight, which is now a characteristic of the entire system. The visual representation of the statistical weight is given by the surface area of the classical Γ-space corresponding to a given energy. To use classical statistical mechanics in practice, it is necessary to perform discretization using the Planck constant and introduce the factor N! as an adjustment for the indistinguishability of particles.
A numerical calculation of the statistical weight is only possible for an ideal monatomic gas; in this sense, the above equation adds little to the Boltzmann equation. However, this step is used to introduce temperature as the derivative of energy over entropy and after that to switch to the canonical distribution.
The Gibbs canonical ensemble corresponds to the state of the system in thermal equilibrium with a thermostat at a given temperature; energy can be exchanged between the system and the thermostat. Thus, temperature and volume become the external parameters.
It is possible to obtain the canonical energy distribution directly from general considerations, but it is most often derived from the microcanonical distribution. For this purpose, the system in the microcanonical distribution is partitioned; one part becomes the system in question, the other the thermostat. Thus, the energy probability distribution (the Gibbs distribution) is obtained as a consequence of the equal a priori probability in the microcanonical ensemble.
The most interesting thing is that the equation for thermodynamic functions contain only the normalizing factor for the probability distribution, which is the partition function. It is easier to write it down for quantum statistical mechanics and then discuss the transition to classical components. Note that in the partition function there are no probabilities, and therefore the probabilities will not appear in the equations for thermodynamic properties:
The sum is taken over all energy states of the system (the solution of the stationary Schrödinger equation with the wave function of the entire system), where the energies are given in respect to the ground state of the system E0. In the case of degenerate energy states, the sum includes all of them, i.e., the term with this energy is repeated multiple times.
The partition function is related to the Helmholtz energy (A) of the system in respect the ground state energy:
A − E0 = −kT ln Z
The Helmholtz energy is a characteristic function, and its derivatives allow us to determine all other thermodynamic properties, including entropy:
It is this approach that has made it possible in a number of cases to compute the partition function, that is, to compute the thermodynamic properties of a system from molecular constants.
Partition function for ideal polyatomic gas
In the case of a polyatomic ideal gas, the calculation of the partition function is simplified, since the energy of the system is equal to the sum of the energies of the molecules. Thus, it is possible to calculate the partition function of a single molecule (Q), and the partition function of the entire system is the product of the molecular partition functions:
Z = QN/N!
The factor N! appears due to the indistinguishability of particles in quantum mechanics, as the product of molecular partition functions takes into account the same energy levels of the system many times. In the presence of intermolecular interactions, a configuration integral appears in the equation above, and it allows us to obtain the equation of state for real gases.
The next step is to split the energy levels of a molecule into translational, rotational, vibrational, and electronic levels:
E = Epost + Ehr + Ekol + Eel
This in turn leads to the factorization of the molecular partition function into components:
Q = Qpost Qhr Qkol Qel
In the final step, the translational and rotational partition functions are replaced by integrals, which corresponds to the transition to classical mechanics. At the same time, the electronic partition function always remains as an energy spectrum, and the vibrational partition function under normal conditions also requires the use of discrete vibrational energy levels. The transition to the integral for the vibrational partition function is only possible at high temperatures.
These equations to calculate the thermodynamic properties of gases from molecular constants were a great success of equilibrium statistical mechanics. The experimental values of gas entropies determined from experiments using classical thermodynamics were in good agreement with the entropies obtained from molecular constants.
In conclusion, I would like to point out that statistical mechanics is more difficult to study than classical thermodynamics. A brief review in this chapter shows that understanding statistical mechanics requires knowledge of classical and quantum mechanics, as well as the corresponding mathematical formalism. This also applies to understanding entropy in statistical mechanics.
The simplified image of statistical entropy as a number of permutations from an ideal monatomic gas is not suitable even for an ideal polyatomic gas. Below, there are entropies of gases at two temperatures from the thermodynamics tables, where they were calculated using statistical mechanics from molecular constants:
| O2 | CO2 | N2 | |
| S°(298.15 K), J/(mol K) | 205.0 | 213.7 | 191.5 |
| S°(2518 K), J/(mol K) | 277.5 | 323.2 | 260.3 |
As an exercise, try to relate these numerical values of the entropy to the number of permutations. It would also be useful to try to interpret these values in the spirit of another popular entropy-as-disorder metaphor.
At the same time, at the level of equilibrium statistical mechanics, there are no contradictions with the entropy in classical mechanics. Entropy remains an objective property of substance, and equilibrium statistical mechanics allows us in some cases to calculate this property from molecular constants.
Next: Chapter 3. The Arrow of Time in Statistical Mechanics
References
Ya. M. Gelfer, History and Methodology of Thermodynamics and Statistical Physics (in Russian), 2nd ed., 1981.
W. Grimus, 100th anniversary of the Sackur–Tetrode equation. Annalen der Physik, 2013, 525: A32-A35.
A. Ya. Borshchevsky, Physical Chemistry, Volume 2, Statistical Thermodynamics, 2023.
J. S. Rowlinson, Cohesion. Scientific History of Intermolecular Forces, 2002
V. V. Eremin, S. I. Kargov, N. E. Kuzmenko, Problems in Physical Chemistry (in Russian), Part I. Chemical Thermodynamics, 11, Partition function and Its Properties, 2000.
The link provides the equations for the partition function in the case of a polyatomic ideal gas.