The Statistical Interpretation of Entropy Physical meaning of entropy Microstates and macrostates Statistical interpretation of entropy and Boltzmann equation Configurational entropy and thermal entropy Calculation of the equilibrium vacancy concentration Reading Chapter 4 of Gaskell Optional reading Chapter 1 5 8 of Porter and Easterling Any method involving the notion of entropy the very existence of which depends on the second law of thermodynamics will doubtless seem to many far fetched and may repel beginners as obscure and difficult of comprehension Willard Gibbs MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei What is the physical meaning of entropy Entropy is introduced in phenomenological thermodynamics based on the analysis of possible and impossible processes We know that heat flows from a hot region to a cold region of a system and that work can be irreversibly transferred into heat To describe the observations the entropy and the 2nd law stating that entropy is increasing in an isolated system have been introduced The problem with phenomenological thermodynamics is that it only tells us how to describe the empirical observations but does not tell us why the 2nd law works and what is the physical interpretation of entropy In statistical thermodynamics entropy is defined as a measure of randomness or disorder Intuitive consideration In a crystal atoms are vibrating about their regularly arranged lattice sites in a liquid atomic arrangement is more random Sliquid Ssolid Atomic disorder in gaseous state is greater than in a liquid state Sgas Sliquid S L G Does this agrees with phenomenological thermodynamics Melting at constant pressure requires absorption of the latent heat of melting q Hm therefore Sm Hm Tm the increase in the entropy upon melting correlates with the increase in disorder MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei How to quantify disorder Microstates and Macrostates A macroscopic state of a system can be described in terms of a few macroscopic parameters e g P T V The system can be also described in terms of microstates e g for a system of N particles we can specify coordinates and velocities of all atoms The 2nd law can be stated as follows The equilibrium state of an isolated system is the one in which the number of possible microscopic states is the largest Example for making it intuitive rolling dice Macrostate the total of the dice Each die have 6 microstates the system of 2 dices has 6 6 36 microstates a system of N dice has 6N microstates For two dice there are 6 ways microstates to get macrostate 7 but only one microstate that correspond to 2 or 12 The most likely macrostate is 7 For a big number N of dice the macrostate for which the number of possible microstates is a maximum is 3 5 N If you shake a large bag of dice and roll them it is likely that you get the total close to 3 5 N for which the number of ways to make it from individual dice is maximum An isolated thermodynamic system is similar thermal fluctuations do the shaking the macrostate corresponds to the largest number of microstates Actually the system of dice is closer to a quantum system with discrete states In the classical case the states form a continuum and we have to replace the sum over states by integrals over phase space MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei Statistical interpretation of entropy If we combine two systems the number of microstates multiply remember 6 6 36 for two dice At the same time we know that entropy is an extensive quantity SA B SA SB and if we want to relate enthalpy to the number of microstates we have to make sure that this equation is satisfied If we take logarithm of the number of microstates the logarithms adds when we put systems together The quantity maximized by the second law can be defined then by equation written on Ludwig Boltzmann s tombstone in Vienna S kB ln where is the number of microstates kB is the Boltzmann constant it is the same constant that relates kinetic energy to temperature but it was first introduced in this equation and S is the entropy The 2nd law can be restated again An isolated system tends toward an equilibrium macrostate with maximum entropy because then the number of microstates is the largest and this state is statistically most probable The entropy is related to the number of ways the microstate can rearrange itself without affecting the macrostate MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei Configurational entropy and vibrational entropy Vibrational or thermal entropy Sv Entropy associated with atomic vibrations the number of microstates can be thought as the number of ways in which the thermal energy can be divided between the atoms The vibrational entropy of a material increases as the temperature increases and decreases as the cohesive energy increases The vibrational entropy plays important role in many polymorphic transitions With increasing T the polymorphic transition is from a phase with lower Sv to the one higher Sv Configurational entropy Sc Entropy can be also considered in terms of the number of ways in which atoms themselves can be distributed in space Mixing of elements in two crystals placed in physical contact or gases in two containers mass transport leads to the increase of Sc and is similar to the heat transfer case when Sv is increasing T1 T2 Sv T1 T2 MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei Sc Heat flow and production of entropy Phenomenological thermodynamics transfer of energy from hot to cold is an irreversible process that leads to the production of entropy Consideration of probabilities of microstates can lead to the same conclusion U tot U A U B TA UA A tot A U A B U B TB UB B When the thermal contact is made between the two systems the number of microstates tot A B is not in the maximum and heat starts to flow increasing the value of tot The heat flows until the increase in A caused by the increase in UA is greater than the decrease in B caused by the decrease in UB The heat flow stops when tot reaches its maximum value i e ln A B 0 If we only have a heat exchange no work q qA qB q dS T S k B ln d ln A q A k BTA d ln B d ln A B d ln A d ln B qB q A k BTB k BTB q 1 1 k B TA TB ln A B 0 only when TA TB reversible heat transfer is only possible after temperatures are equal MSE 3050 Phase Diagrams and Kinetics Leonid Zhigilei Configurational entropy equilibrium vacancy concentration The configurational entropy of a crystal refers to the