Jozsef Garai
Entropy is a Mathematical Formula
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Abstract
The microscopic explanation of entropy has been challenged from both experimental and theoretical point of view. The expression of entropy is derived from the first law of thermodynamics indicating that entropy or the second law of thermodynamics is not an independent law.
1. Introduction
The macroscopic determination of entropy first was expressed by Clausius in 1865. He postulated that the entropy change between two equilibrium states could be determined by the transferred reversible heat and the absolute temperature [T] as:
(1)Employing statistical mechanics in 1877 Boltzmann suggested a microscopic explanation for entropy. He stated that every spontaneous change in nature tends to occur in the direction of higher disorder, and entropy is the measure of that disorder. From the size of disorder entropy can be calculated as:
(2)where W is the number of microstates permissible at the same energy level, kB is the Boltzmann constant.
2. Experimental problems
The microscopic explanation of entropy has never been fully accepted1-3 since there is incomplete proof for equation 2 and there are counter examples where the increase of disorder cannot be justified. Among these counter examples the spontaneous crystallization of a super-cooled melt4 and the crystallization of a supersaturated solution5 are the most prominent challenges for the Boltzmann’s model. If a super-cooled melt is allowed to crystallize under adiabatic conditions then the entropy of the system increases. In a supersaturated solution there is a possibility of the deposition of crystalline solute. The deposition of crystalline solute is a spontaneous process with an increase of entropy. A crystallization of a liquid or a deposition of a crystalline solute does not imply any increase in disorder. I would like to add one additional examples to the above mentioned. The phase transformation of solid helium to liquid helium II does not require energy6. The lack of latent heat results the same entropy for both of these phases [Fig. 1], indicating the same level of disorder in solid and in liquid. These counter examples suggesting that the microscopic explanation of entropy is dubious.
3. Theoretical problems
The number of microstates for a given macrostate can be deduced from a combinatorial argument7 and can be calculated as
(4)where N is the total number of particles and n is the number of particles at the same energy level. The total number of particles and the total energy [E] can be written as
and 
(5)where Ej is the energy of the jth level.
(6)where 
is the energy difference between to consecutive states. As an example the number of microstates in the possible macrostates is given in Figure 2. The system consist 4 particles with the total energy of 
. Macrostate with the highest number of microstates is the most probable; however, macrostates with lower numbers for the microstates are also possible but their probability is lower. According to this model the number of microstates can vary while the total energy remains the same. Allowing to change the number of microstates without affecting the total energy of the system is in discrepancy with equation 1 and 2.
4. The formula of entropy
Assuming a quasi-static infinitesimal process and employing the first law of thermodynamics, the differential of the internal energy dU is the sum of the heat term dq and the mechanical work term dw.
(7)At constant volume the work is zero.
(8)where n is the number of moles, and cv is the molar heat capacity at constant volume. Assuming a monoatomic gas, where only translational energies are present the constant molar volume heat capacity is
(9)where R is the universal gas constant. Substituting R with 
from the equation of state [EoS] for ideal gasses:
(10)where V is volume and p is the pressure. Integrating the equation gives
(11)where subscript i represent the initial conditions while f represents the final conditions. Employing EoS and substituting pV with nRT and 1.5R with cv
(12)This equation contains the expression of entropy for constant volume.
(13)The internal energy then
(14)At constant temperature the heat term is zero and equation 7 is written.
(15)Substituting p with 
gives:
(16)Integrating the equation between the initial and final conditions:
(17)Equation 17 contains the expression of entropy for constant temperature.
(18)Substituting entropy into equation 17 gives
(19)In a general case, when both the temperature and the volume change, the internal energies should be summed.
(20)Since the heat transferred to the system supplies the energy for both the heat and the work the sign of the work changes.
(21)Summing the two parts of the entropy change
(22)gives the heat
(23)This equation is the same as equation 1 proposed by Clausius.
5. The second law of thermodynamics
According to Kelvin and Planck the second law of thermodynamics can be stated as "It is impossible to construct a heat engine that operating in a cycle, produces no other effect than the absorption of thermal energy from a reservoir and the performance of an equal amount of work" 8. This statement is the result of the conservation of energy described in the first law of thermodynamics.
since energy transfer involves waste
Either entropy or the Kelvin & Planck statement can be derived from the first law of thermodynamics indicating that the second law of thermodynamics is not an independent law.
5. Conclusions
The microscopic explanation of entropy, that is the entropy measures the disorder of a system, should be discredited since contradict with experiments and theory. The expression of entropy can be derived from the first law of thermodynamics suggesting that the second law of thermodynamics is not an independent law.
The formula for entropy most likely was introduced to provide a convenient way to calculate the changes in the internal energy of a system and to convert the thermal and mechanical energies into each other.
References
(1) Wright, P. G. Contemp. Phys. 1970 11, 581.
(2) Dingle, H. Bull. Inst. Phys. 1959 10, 218.
(3) Khinchin, A. I. Mathematical Foundations of Statistical Mechanics; English translation: New York, 1949.
(4) Bridgman, P. W. The Nature of Thermodynamics; Cambridge, Mass., 1941.
(5) McGlashan, M. L. J. Chem. Educ. 1966 43, 226.
(6) Swenson, C.A. Phys. Rev, 1950 79, 626.
(7) Schoepf, D.C. Am. J. Phys. 2002 70, 128.
(8) Serway, R.A., Physics for scientist and engineers with modern physics, Sounders College Publishing, 1990 p.589