Entropy is a Mathematical Expression

1. Entropy isa Mathematical Expression Therefore there is no need to invent any special physical explanation Jozsef Garai Dept. of Earth Sciences

2. The macroscopic determination of entropy first was expressed by Clausius in 1865. S = Entropy Q = Heat T = Absolute Temperature

3. The microscopic explanation was suggested by Boltzmann in 1877 Employing statistical mechanics 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.

4. From the size of disorder entropy can be calculated as: W = the number of microstates permissible at the same energy level is the Boltzmann constant

5. The microscopic explanation of entropy has never been fully accepted since there is incomplete proof for the Boltzmann equation and there are counter examples where the increase of disorder cannot be justified. • spontaneous crystallization of a super-cooled melt • crystallization of a supersaturated solution

6. Name of the Mineral Composition Standard Entropy Citation [J/Kmol] Almandine 336.00 [10] Andradite 316.35 [11] Annite 440.91 [12] Anthophyllite 535.19 [12] Antigorite 3672.80 [12] Clinochlorite 421.00 [13] Cordierite 410.88 [11; 14] Cummingtonite 483.06 [12] Ferrocordierite 410.88 [12] Glaucophane 535.00 [13] Grunerite 714.60 [12] Muscovite 306.40 [11] Phlogopite 334.60 [15] Tremolite 550.00 [13] The standard entropy of hydrogen is 130.68 [J/Kmol]

7. The much higher entropy values of solids indicating that the disorder in these solids should be much higher then the disorder of hydrogen. This conclusion is against any common sense! It is impossible that molecules in solid phase can be more disordered than molecules in gas phase.

8. For a monoatomic gas, where only translational energies are present the internal energy can be calculated as: = Avogadro’s number n = the number of moles R = the universal gas constant. If heat is transferred to the system and the volume kept constant then the change in the internal energy of the system is:

9. Substituting R with from the equation of state [EoS]: V = volume p = pressure Integrating the equation: subscript i represent the initial conditions while f represents the final conditions

10. From EoS therefore: For general case, when rotational and vibrational energies are also present, the internal energy change can be written as: = the heat capacity for a mol quantity at constant volume This equation contains the expression of entropy for constant volume!

11. Because we can write: Investigating the effect of heat on the internal energy at constant temperature and pressure.

12. Integrating the equation: Part of this equation is equivalent with the well known expression of entropy for constant temperature! Calculating the heat for the energy changes leads to the Clausius equation.

13. Deriving entropy from existing thermodynamical expressions suggest that entropy is not a new independent physical parameter. Thus the original parameters of the manipulated expressions should fully explain the physics of entropy.

14. What makes the expression of entropy so powerful and immiscible for thermodynamic calculations? The change of the internal energy of a system at constant volume can be determined by integrating the equation: There is a hidden problem here. Heat capacity is not an independent physical parameter because it contains the variable T

15. Multiplying the numerator and the denominator by T Leads to: It can be seen that the expression of entropy allows us to integrate and determine the change in the internal energy of a system in a convenient way.

16. The other mathematical trick what the expression of entropy allows is to convert heat and mechanical energies into each other. Let’s have an ideal gas system described by the EoS as: Changing the temperature of the gas at constant pressure will induce a volume change, which can be written as: Dividing this equation with the original expression of EoS leads to:

17. Entropy incorporates this expression

18. Conclusion • entropy is a variable representing an expression extracted from the equations of internal energy and work. • the physical parameters contained in these expressions should fully explain the physics of entropy • the expression of entropy allows a simple mathematical way to calculate the changes in the internal energy of a system and to convert the thermal and mechanical energies into each other • most likely these mathematical advantages led to the introduction of the formula of entropy