On the generalized validity of the Maxwell-Boltzmann distribution and the zeroth Law of Thermodynamics

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1 On the generalized validity of the Maxwell-Boltzmann distribution and the zeroth Law of Thermodynamics, Medellin, Colombia ORCID : doi: /RG Abstract The Maxwell-Boltzmann distribution has been a very useful statistical distribution for understanding the molecular motion of ideal gases. In this work, it will be shown that it is also possible to obtain a generalized Maxwell-Boltzmann distribution valid for any macroscopic system of any composition and having any arbitrary state of aggregation. This distribution is the result of the large number of collisions and molecular interactions taking place in such macroscopic system. On the other hand, in order to better understand the concept of thermal equilibrium, a mathematical interpretation of the zeroth law of Thermodynamics is included. Keywords Generalized Maxwell-Boltzmann distribution, Macroscopic systems, Relative molecular speed, Thermal equilibrium, Zeroth law of Thermodynamics 1. Introduction Although our eyes are not able to watch individual molecules, we know that they are in constant motion. For example, if we consider the diffusion of smoke in the air, or a certain colored substance in water, we can evidence that the particular behavior observed is the result of random molecular motion. We can also feel the random motion of the molecules by means of our thermoreceptors,[1] which allow our brains to get an idea of the temperature of an external body, relative to our own body. In the 19 th century James Clerk Maxwell [2] imagined random molecular motion as the result of perfectly elastic collisions between hard spherical molecules. This way, he was able to derive a useful mathematical expression for the molecular speed distribution, which is nowadays known as the Maxwell-Boltzmann distribution.[3] Given that in his derivation, Maxwell did not consider long-range molecular interactions (he only 21/03/2017 Reports (1 / 11)

2 considered short-range interactions taking place during collisions), it has been assumed that the Maxwell-Boltzmann distribution is only valid for perfect or ideal gases.[4] The purpose of the present work is the analysis of molecular speed distributions of a system in any arbitrary aggregation state, and the interpretation of thermal equilibrium and the zeroth law of thermodynamics for such generalized molecular speed distribution. Before continuing, it is important to introduce some concepts regarding molecular motion. In general, it is possible to describe the velocity vector of a certain molecule in a system at any arbitrary aggregation state as: where (1.1) (1.2) is the average velocity vector of the system, is the relative molecular velocity vector of molecule, and is the number of molecules in the system. In other words, represents the macroscopic motion of the system, whereas represents individual molecular velocities randomly fluctuating, both in direction and magnitude. Such fluctuations can be expressed mathematically as: where is the magnitude of the relative velocity vector (random molecular speed relative to the macroscopic motion of the system), and is a random unit direction vector, such that: (1.3) (1.4) (1.5) (1.6) Eqs. (1.4) - (1.6) indicate that all directions are equally probable and that the magnitude of the unit vector is always the unit. From these properties of the unit direction vector, and considering that the direction is independent of the relative speed, it can also be concluded that: (1.7) (1.8) (1.9) 21/03/2017 Reports (2 / 11)

3 That is, the fluctuating behavior of the relative molecular velocity is determined by the particular fluctuation of the relative molecular speed. The total kinetic energy of such molecule of mass is given by (using Eq. (1.1)): (1.10) Thus, there are three components of the total kinetic energy of the molecule: The contribution of the molecule to the macroscopic kinetic energy (1.11) A fluctuating relative kinetic energy correction term (1.12) where, and The thermal kinetic energy (1.13) In a previous report,[5] it was shown that only the thermal kinetic energy term is relevant to the concept of temperature, and therefore, that temperature is independent of the macroscopic motion of the system: (1.14) where represents a conversion factor between energy and temperature (known as the Boltzmann constant ) which is equal to. The conclusion obtained from (1.14) is that the macroscopic temperature of the system and the distribution of relative molecular speeds are related. 2. Molecular velocity distribution in macroscopic systems Let us consider a macroscopic system at any arbitrary state of aggregation (solid, liquid or gas). Because of the macroscopic nature of the system, any molecule present in it will quickly participate in a large amount of interactions with their neighbor molecules. It is because of those multiple interactions, that the relative velocity vector of the molecule fluctuates both in 21/03/2017 Reports (3 / 11)

4 direction and magnitude. Let us consider the general equation of relative motion for a molecule in a system at any arbitrary state of aggregation: (2.1) where is the net relative force acting on molecule Given that is the sum of the relative interaction forces between molecule and all other molecules in the system, according to the central limit theorem,[6] the net relative force can be considered as a normal random variable, independently of the distribution of the individual interaction forces. Furthermore, the expected value of the net relative force is, considering that all force directions are equally probable. Thus, the net force on each Cartesian coordinate can then be expressed as a function of the standard normal random variable as: where (2.2) is the standard deviation of the magnitude of the net force acting on molecule in the Cartesian coordinate Please notice that, the unit direction vector, is no longer a random variable. Now, given that the force is a normal random variable, then the relative molecular velocity behaves as a Markovian random variable, and can be expressed in terms of the standard Markovian random variable :[7] (2.3) Again, the expected value of the relative velocity in any coordinate for any molecule in the system is. is a characteristic time of the process, in this case related to the time between two consecutive collisions. represents the standard deviation in the relative speed of molecule along the Cartesian coordinate Replacing Eqs. (2.2) and (2.3) in Eq. (2.1), it can be found that: [ ] (2.4) Considering that, and neglecting the term, as it is expected to be much smaller than : (2.5) Then, Eq. (2.3) becomes: 21/03/2017 Reports (4 / 11)

5 (2.6) Now, given that and, then (2.7) (2.8) An interesting characteristic of the standard Markovian random variable is that after many time steps (in this case, after a large number of collisions), and according to the central limit theorem, the distribution will resemble a normal distribution. Thus, if the molecules in the system are allowed to collide for a significant time (in the order of seconds, for example), then: Given that even for anisotropic molecules, in the absence of any external field there is no preferential orientation, then: And therefore, Eq. (2.9) becomes: (2.9) (2.10) (2.11) Please, notice that this is a general result obtained for any macroscopic system at any arbitrary state of aggregation and of any composition. 3. Molecular speed distribution in macroscopic systems Let us now consider the behavior of the relative molecular speeds of the arbitrary system discussed in Section 2. The relative molecular speed in any direction will be, from Eq. (2.11): whereas the overall relative molecular speed will be: (3.1) (3.2) 21/03/2017 Reports (5 / 11)

6 The value inside the square root corresponds to a χ 2 distribution with 3 degrees of freedom.[8] Thus, we can say that (3.3) Squaring both sides of Eq. (3.3): (3.4) Thus, the corresponding probability distribution function for a single molecule is:[8] Now, given that from probability theory: (3.5) (3.6) then, the probability distribution function for the relative molecular speed of molecule is: (3.7) which corresponds to the mathematical expression of the probability density function for a Maxwell-Boltzmann distribution. This result indicates that the Maxwell-Boltzmann distribution is valid for describing the relative molecular speed distribution of any molecule present in any arbitrary macroscopic system, and not only for perfect ideal gases. From the properties of the distribution, it is also possible to conclude that (from Eq. 3.4): (3.8) Therefore: (3.9) Recalling the definition of the thermal kinetic energy (Eqs ): 21/03/2017 Reports (6 / 11)

7 (3.10) Now, if there are identical molecules in the system (each with mass ), belonging to a certain compound, then it is possible to say that the distribution of relative molecular speeds for all molecules can be described by the following probability density function: (3.11) and that the average thermal kinetic energy for the set of identical molecules with mass is:[5] (3.12) where is the local temperature determined only for the molecules of compound present in the system. Therefore, from Eq. (3.12) and (2.5) the following relationships hold true: (3.13) (3.14) Thus, replacing Eq. (3.13) in Eq. (3.11) results in: (3.15) On the other hand, the average relative speed of the molecules of compound is: (3.16) Since the relative molecular speed for compound is described by a Maxwell-Boltzmann distribution, then it is possible to express its relative molecular speed in terms of the standard Maxwell-Boltzmann random variable as: [3] (3.17) 21/03/2017 Reports (7 / 11)

8 and its relative molecular velocity as: On the generalized validity of the Maxwell-Boltzmann distribution (3.18) If the system contains whole system will be: different compounds, then the average thermal kinetic energy of the (3.19) where is the proportion of molecules of compound present in the system, also known as the mole fraction of. The macroscopic temperature of the whole system will then be expressed by: (3.20) Finally, the relative molecular speed distribution for the whole system would be: (3.21) which is a generalized expression of the Maxwell-Boltzmann distribution for a system of any composition with any arbitrary aggregation state in terms of the local temperatures for each compound. 4. Interpretation of the Zeroth Law of Thermodynamics The zeroth law of Thermodynamics states that: If a body A is in thermal equilibrium with two other bodies B and C, bodies B and C will then be in thermal equilibrium with each other. [9] The zeroth law of thermodynamics relates to the principle of thermal equilibrium, indicating that no differences in temperature are expected in different bodies at thermal equilibrium between them. Let us now consider two single-component systems, each containing a different compound, with molecular masses and, and with average relative molecular speeds and, respectively. The temperature of the systems will be: 21/03/2017 Reports (8 / 11)

9 (4.1) (4.2) Let us now assume that the first system containing molecules is mixed with the second system containing molecules. Please notice that after mixing and, that is the local temperatures for each compound in the mixture are equivalent to the macroscopic temperatures of the single-component systems before the mixture. Thus, Eq. (3.20) becomes: (4.3) given that. It is therefore possible to conclude that, since, only if will, independently of the values of and, and, or and. If systems 1 and 2 are in thermal equilibrium, mixing, joining or contacting them will result in a third system also in thermal equilibrium with 1 and 2. The relative molecular speed distribution of the molecules in the mixture will be (from Eq. 3.21): (4.4) From Eq. (4.1) and (4.2) and after doing some algebra, Eq. (4.4) becomes: [ ] (4.5) Therefore, the expected value of the relative molecular speed of the mixture is: [ ] [ ] [ ] [ ] (4.6) 21/03/2017 Reports (9 / 11)

10 That is, the average relative molecular speed of the mixture is just the sum of the average speed of each compound multiplied by their fraction of molecules. And the expected value of the thermal kinetic energy of the system will then be: [ ] [ ] From Eq. (4.1)-(4.2) and (3.20): (4.7) (4.8) which is a result consistent with the zeroth law of Thermodynamics. Therefore, if a system of any composition with any arbitrary aggregation state is allowed to reach thermal equilibrium between all of its components ( ) then the relative molecular speed distribution for the whole system would be: (4.9) which is the generalized expression of the Maxwell-Boltzmann distribution for any arbitrary system when all its components are in thermal equilibrium. 5. Conclusion It has been shown that the relative speed of molecules in a macroscopic system follow the generalized Maxwell-Boltzmann distribution presented in Eq. (4.9), as a result of the fluctuating intermolecular forces with neighboring molecules. This generalized Maxwell- Boltzmann distribution is a function of the composition of the system, as well as the macroscopic temperature. It was also shown that this macroscopic temperature is related to the composition of the system and the local temperature of each component (Eq. 3.20), and this equation is the basis of thermal equilibrium, or equivalently, of the zeroth law of Thermodynamics. 21/03/2017 Reports (10 / 11)

11 Acknowledgments The author gratefully acknowledges the helpful discussions with Prof. Jaime Aguirre (Universidad Nacional de Colombia). This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] Schepers, R. J., & Ringkamp, M. (2009). Thermoreceptors and thermosensitive afferents. Neuroscience & Biobehavioral Reviews, 33(3), [2] Maxwell, J. C. (1860). Illustrations of the dynamical theory of gases. - Part I. On the motions and collisions of perfectly elastic spheres. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 19(124), [3] Hernandez, H. (2017). Standard Maxwell-Boltzmann distribution: Definition and properties. Reports doi: /RG [4] Mattis, D. C. (2004). Statistical mechanics made simple: a guide for students and researchers. Singapore: World Scientific Publishing Co. [5] Hernandez, H. (2017). Analysis of Temperature fluctuations in ideal gases - From the macroscopic to the molecular scale. Reports doi: /RG [6] Durrett, R. (2010). Probability: theory and examples. Cambridge: Cambridge University Press. [7] Hernandez, H. (2016). Variance algebra applied to dynamical systems, Reports, , doi: /RG [8] Lancaster, H. O. and Seneta, E. (2005). Chi-Square Distribution. Encyclopedia of Biostatistics. 2. John Wiley & Sons. [9] Turner, L. A. (1961). Zeroth law of Thermodynamics. American Journal of Physics, 29(2), /03/2017 Reports (11 / 11)