Lecture-Notes (Thermodynamics) - niser

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36 CHAPTER 4. ENTROPY AND SECOND LAW OF THERMODYNAMICS Using (4.6), we get ∂P ∂T V where = − α = 1 ∂V V ∂T P kT = − 1 ∂V V ∂P T kS = − 1 ∂V V ∂P ∂z = − ∂x y ∂f/∂x ∂f/∂z . 1 (∂T/∂V ) P (∂V/∂P) T S = − (∂V/∂T) P (∂V/∂P) T = α kT (coefficient of thermal expansion), (isothermal compressibility), (adiabatic compressibility). We substitute in ∂U δQ = TdS = CV dT + + P dV ∂V T expressions (4.7) and (4.5) and get the following useful relations. , (4.7) 1) The absorbed heat is expressed in terms of directly measurable coefficients as where T and V are independent variables. 2) If T and P are used as independent variables, then δQ = TdS = CV dT + α TdV , (4.8) kT TdS = CPdT − αTV dP (4.9) 3) If V and P are used as independent variables, dT can be rewritten in terms of dV and dP as ∂T ∂T dT = dV + dP = ∂V P ∂P V 1 kT + dP . αV α The corresponding TdS equation is then TdS = CP dV + αV CPkT α − αTV dP . (4.10) There is also an important connection between CP and CV (which follows from eq. (4.8) and (4.9)): CP − CV = α2 TV = −T kT ∂V 2 ∂T P ∂V ∂P T > 0 . (4.11)
4.8. ENTROPY AND DISORDER 37 By introducing the isochore pressure coefficient it is possible to rewrite CP − CV as Indeed, β = 1 P α kT ⇒ CP − CV = α2 β = 1 P 4.8 Entropy and disorder kT ∂P ∂T V β 2 TV kTP 2 . ⇒ α = βPkT ⇒ TV = β2 P 2 k 2 T kT , TV = β 2 TV kTP 2 . We want now to establish a connection between thermodynamics and statistical mechanics, which we will be learning in the second half of the course. 1. We give a definition of the multiplicity of a macrostate as ⎛ number of microstates multiplicity of = ⎝ that correspond to the a macrostate macrostate 2. The entropy can be defined in terms of the multiplicity as S ≡ k ln (multiplicity) . We shall see in statistical mechanics that this definition is equivalent to the phenomenological concept we have learnt previously in this section. 3. With the above definition of entropy at hand, we can formulate the second law of thermodynamics in this way: 4. ”If an isolated system of many particles is allowed to change, then, with large probability, it will evolve to the macrostate of largest entropy and will remain in that macrostate.” (energy input by heating) ∆Sany system ≥ , T where T is the temperature of the reservoir. 5. If two macroscopic systems are in thermal equilibrium and in thermal contact, the entropy of the composite system equals the sum of the two individual entropies. ⎞ ⎠.
Page 1 and 2: Chapter 4 Entropy and second law of
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Page 5 and 6: 4.3. ABSOLUTE TEMPERATURE 29 4.3 Ab
Page 7 and 8: 4.4. TEMPERATURE AS INTEGRATING FAC
Page 9 and 10: 4.5. ENTROPY 33 The entropy is defi
Page 11: 4.7. USEFUL COEFFICIENTS 35 we canc
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Lecture-Notes (Thermodynamics) - niser

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