Classical Thermodynamics

Statistical Thermodynamics

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Entropy

### Thermodynamics

 Classical Thermodynamics

Definition of Entropy

 ``` 1. Entropy: The second law of Thermodynamics is related to practical devices suchs as heat machine and refrigerators. It exist another statement phrase for this law that involves a quantity called the entropy. 2. Entropy is a variable of state : That is, it depends only on the initial and final state of the system. A system, at the themodynamic equilibrium, is governed by its equation of state (PV = nRT = NkT for an ideal gas). In other words, It is characterized by the values of its variables of state, such as P, V, n, and T. Recall that, for the ideal gaz that the internal energy depends only on the temperature. Thus, we can write U(T). recall also the change is null for a variable state for a cycle; for instance ΔU = 0, ΔT = 0, ΔV = 0, Δn = 0, for a cycle. Ui and Ui are independant of the path connecting the states "i" to "f". In contrast Q(NET heat) or W(NET work) involved during the cycle depend on the state of the system before it falls in the equilibrium situation. 3. Macroscopic definition of the entropy : We can state a mathematical definition only for a reversible process. The related equation is: dS = dQ/T Between two states, we have : ΔS = ∫dQ/T [from i → f] In the case if an irreversible process, it is always possible to device a reversible process (starting in "i" and ending in "f") that mach the starting and the ending of the irreversible process (mostly an isothermal + adiabatic processes in a Carnot cycle: [ + dQ1, T1, - dQT2, T2]. For the cycle, we have: + dQ1/T1 = dQ2/T2, that is S2 = S1; or ΔS = 0 ΔS = 0 [for a cycle] wchich infer that the entropy S is a variable of state Heat goes from a hot source to a cold one; then dQ >=0. Foretheremore, entropy always increases. S >= 0 - At the equilibrium, S = 0, - At the irreversible case, S > 0 If we write Universe = System + Surroundings, we have always SUniverse >= 0 4. Microscopic definition of the entropy : Now let's write the things as follows: Suppose that we have a function S of vsriables P, V , n and T. S = S(P, V, n, T). S is an intensive quantity; then for two parts "1" and "2" of a system, we can write the total entropy as follows: S = S1 + S2 But if their numbers of microstates are Ω(1)and Ω(2), the total microstate for the system is Ω(1) x Ω(2). A macrostate depends also on the same variables of states: P, V, n, and T. To recap: S = S(P, V, n, T), S = Σ Si Ω = Ω(P, V, n, T)= Π Ω(i) The function that responds to the above criteria is the natural logarithm (ln) function. So, S = κln(Ω) κ is a constant. ```

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