Lecture-Notes (Thermodynamics) - niser

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30 CHAPTER 4. ENTROPY AND SECOND LAW OF THERMODYNAMICS From this follows that the efficiencies of the engines compare as ηC ≥ ηX . Consequence: all Carnot engines have the same efficiency since X can be, as a special case, a Carnot engine. ⇒ The Carnot engine is universal, i.e., it depends only on the temperatures involved and not on the working substance. Definition: absolute temperature θ of a heat reservoir (bath) is defined such that the ratio of the absolute temperatures of two reservoirs is given by θ2 θ1 ≡ |Q2| |Q1| −Q2 = 1 − η = , where η is the efficiency of a Carnot engine operating between the two reservoirs. Since |Q2| is strictly greater than zero, |Q2| > 0, according to the second law of thermodynamics, the absolute temperature is bounded from below: θ > 0 . ⋆ This means that the absolute zero θ = 0 is a limiting value that can never be reached since this would violate the second law of thermodynamics. If an ideal gas is considered as a working substance in a Carnot engine, the absolute temperature coincides with the temperature of the ideal gas defined as T = PV NkB = θ. 4.4 Temperature as integrating factor The absolute temperature T can be considered as an integrating factor that converts the inexact differential δQ into an exact differential δQ/T. Let us consider the following theorem stated by Clausius: ⇒ ”In an arbitrary cyclic process P, the following inequality holds: P δQ T ≤ 0; where the equality holds for P reversible.“ Q1
4.4. TEMPERATURE AS INTEGRATING FACTOR 31 Proof. Let us divide the cycle P into n segments so that on each segment its temperature Ti (i = 1, . . .,n) is constant. We consider now a reservoir at temperature T0 > Ti(∀i) (Fig. 4.7) and introduce Carnot engines between the reservoir at T0 and Ti. Figure 4.7: Cycle P connected to the reservoir at T0 via Carnot engines. For each Carnot engine, Wi + Q (0) i + QC i For cycle P, with Q (0) i T0 + QC i Ti = 0 (first law of thermodynamics) and = 0 (definition of absolute temperature). W + n Qi = 0 (first law of thermodynamics), i=1 Qi = −Q C i . Then, the total heat absorbed from the reservoir at T0 is n n Q = −T0 C i = T0 Q (0) T = i=1 Q (0) i and the work performed by the system is n WT = − W + = n Qi + i=1 i=1 n i=1 Wi i=1 Ti Q (0) i − Qi = n i=1 n i=1 Q (0) i Qi Ti = Q(0) T .
Page 1 and 2: Chapter 4 Entropy and second law of
Page 3 and 4: 4.2. SECOND LAW OF THERMODYNAMICS 2
Page 5: 4.3. ABSOLUTE TEMPERATURE 29 4.3 Ab
Page 9 and 10: 4.5. ENTROPY 33 The entropy is defi
Page 11 and 12: 4.7. USEFUL COEFFICIENTS 35 we canc
Page 13 and 14: 4.8. ENTROPY AND DISORDER 37 By int
Page 15 and 16: 4.9. THIRD LAW OF THERMODYNAMICS (N

Lecture-Notes (Thermodynamics) - niser

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